Non-Carathéodory analytic functions with respect to symmetric points

ABSTRACT

The authors introduce new classes of analytic function with respect $(\mathit{\eta },\mathit{\tau })$-symmetric points subordinate to a domain that is not Carathéodory. To use the existing infrastructure or framework, usually, the study of analytic function have been limited to a differential characterization subordinate to functions which are Carathéodory. Here, we try to obtain various interesting properties of functions which are not Carathéodory. Integral representation, interesting conditions for starlikeness and inclusion relations for functions in these classes are obtained.

KEYWORDS:

• Univalent function
• symmetrical functions
• differential subordination

1. Introduction, definitions and preliminaries

Let $\mathcal{A}$ be the class of function of the form

(1.1) $\mathit{\phi }(\mathit{\omega })=\mathit{\omega }+\sum _{\mathit{n}=2}^{\mathrm{\infty }}{a}_n{\omega }^n,$(1.1)

which are analytic in the unit disc $\mathbb{U}=\left\{\mathit{\omega }:|\mathit{\omega }|<1\right\}$. Let $\mathcal{S}$ denote the class of functions $\mathit{\phi }\in \mathcal{A}$ which are univalent in $\mathbb{U}$. We call $\mathcal{P}$ to denote the class of functions with normalization $\mathit{p}(0)=1$ which satisfies $\mathrm{Re}\left(\mathit{p}(\mathit{\omega })\right)>0$, $\mathit{\omega }\in \mathbb{U}$. Starlike and convex functions, the well-known geometrically defined subclasses of $\mathcal{S}$ have the following analytic characterizations respectively

$\frac{\mathit{\omega }{\mathit{\phi }}^{\prime }(\mathit{\omega })}{\mathit{\phi }(\mathit{\omega })}\in \mathcal{P}\mathrm{and}1+\frac{\omega {\phi }^{{ }^{\prime \prime }}(\mathit{\omega })}{{\phi }^{\prime }(\mathit{\omega })}\in \mathcal{P}.$

We denote the class of starlike and convex functions by ${\mathcal{S}}^{\ast }$ and $\mathcal{C}$ respectively. Ma-Minda (Ma and Minda 1992) studied an analytic function $\mathit{\psi }$ which satisfies the conditions

1. $\mathrm{Re}\mathit{\psi }>0,\mathbb{U};$

2. $\mathit{\psi }(0)=1,{\psi }^{\prime }(0)>0;$

3. (iii) $\mathit{\psi }$ maps $\mathbb{U}$ onto a starlike region with respect to $1$ and symmetric with respect to the real axis.

Also, they assumed that $\mathit{\psi }(\mathit{z})$ has a series expansion of the form

(1.2) $\mathit{\psi }(\mathit{\omega })=1+L_1\mathit{\omega }+L_2{\omega }^2+L_3{\omega }^3+\cdots ,(L_1>0;\mathit{\omega }\in \mathbb{U}).$(1.2)

and introduced and studied the following subclasses of using subordination of analytic functions:

${\mathcal{S}}^{\ast }(\mathit{\psi }):=\left\{\mathit{\phi }\in \mathcal{A}:\frac{\mathit{\omega }{\mathit{\phi }}^{\prime }(\mathit{\omega })}{\mathit{\phi }(\mathit{\omega })}\prec \mathit{\psi }(\mathit{\omega })\right\}$

and

$\mathcal{C}(\mathit{\psi }):=\left\{\mathit{\phi }\in \mathcal{A}:\left(1+\frac{\omega {\phi }^{{ }^{\prime \prime }}(\mathit{\omega })}{{\phi }^{\prime }(\mathit{\omega })}\right)\prec \mathit{\psi }(\mathit{\omega })\right\}.$

By choosing $\mathit{\psi }$ to map unit disc on to some specific regions like parabolas, cardioid, lemniscate of Bernoulli, booth lemniscate in the right-half of the complex plane, various interesting subclasses of starlike and convex functions can be obtained. For detailed study, refer to (Gandhi 2020; Dziok et al. 2011b, 2011a, 2013; Mendiratta et al. 2014; Raina and Sokół 2015, 2016, 2019; Srivastava et al. 2019, 2019, 2019, 2019; Mustafa and Murugusundaramoorthy 2021; Khan et al. 2022; Mustafa and Korkmaz 2022; Araci et al. 2023).

From the above discussion, it can be seen that the entire architecture supports the study of analytic functions that are subordinate to Carathéodory function. Here in this study, we will deviate by introducing certain differential characterization subordinate to a non-Carathéodory functions.

To begin with, we let $\mathcal{N}\mathcal{P}$ to denote the class of functions that are analytic in the unit disc and equals $1$ at $\mathit{\omega }=0$. Both Carathéodory and non-Carathéodory functions satisfy the same normalization $\mathit{p}(0)=1$, the only difference is that the requirement of the function to map the unit disc onto a right-half plane is relaxed in case of non-Carathéodory functions. Recently, Karthikeyan et al. (2023) introduced a class belonging to a class of non-Carathéodory functions $\mathcal{N}\mathcal{P}$ defined by

(1.3) $\mathrm{\Lambda }[\mathit{\alpha },\mathit{\beta };\mathit{p}(\mathit{\omega })]=\frac{{\mathit{e}}^{-\mathit{i\alpha }}\left[cos\mathit{\alpha }+\mathit{i}sin\mathit{\alpha }\mathit{p}(\mathit{\omega })\right]}{1-2cos\mathit{\beta \omega }{e}^{-\mathit{i\alpha }}+e^{-2\mathit{i\alpha }}{\omega }^2},$(1.3)

with $\mathit{\alpha },\mathit{\beta }\in \left[0,\mathit{\pi }\right]$ and $\mathit{p}(\mathit{\omega })\in \mathcal{P}$. Here, in this paper, we slightly modify the equation (1.3)(1.3) $\mathrm{\Lambda }[\mathit{\alpha },\mathit{\beta };\mathit{p}(\mathit{\omega })]=\frac{{\mathit{e}}^{-\mathit{i\alpha }}\left[cos\mathit{\alpha }+\mathit{i}sin\mathit{\alpha }\mathit{p}(\mathit{\omega })\right]}{1-2cos\mathit{\beta \omega }{e}^{-\mathit{i\alpha }}+e^{-2\mathit{i\alpha }}{\omega }^2},$(1.3) to accommodate or unify the studies of well-known classes of analytic functions, which we define as follows.

(1.4) ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{p}(\mathit{\omega })]=\frac{{\mathit{e}}^{-\mathit{i\alpha }}\left[cos\mathit{\alpha }+\mathit{i}sin\mathit{\alpha }(1-{\omega }^2)\mathit{p}(\mathit{\omega })\right]}{1-2cos\mathit{\beta \omega }{e}^{-\mathit{i\alpha }}+e^{-2\mathit{i\alpha }}{\omega }^2}.$(1.4)

It can be easily seen that for a choice of $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$, ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{p}(\mathit{\omega })]=\mathit{p}(\mathit{\omega })\in \mathcal{P}$. Figure 1a,b illustrates the impact of ${\mathrm{\Lambda }}^{\ast }[\frac{\pi }{3},0;\mathit{p}(\mathit{\omega })]$ on the regions $\mathit{p}(\mathit{\omega })=\frac{1+\mathit{\omega }}{1-\mathit{\omega }}$ and $\mathit{p}(\mathit{\omega })=\sqrt{1+\mathit{\omega }}$ respectively. Further, the images shows that function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{p}(\mathit{\omega })]$ belongs to the class which is not Carathéodory.

Figure 1. Mapping of $|\mathit{\omega }|<1$ under ${\mathrm{\Lambda }}^{\ast }[\mathit{\pi }/3,0;\mathit{p}(\mathit{\omega })]$.

Mittag-Leffler function, a special transcendental function has been on the spotlight due to its role in treating problems related to integral and differential equations of fractional order. Refer to Srivastava (2021c, 2021a, 2021b), Srivastava (1968), Srivastava and Tomovski (2009) and Srivastava et al. (2018, 2019, 2022) for detailed studies which involved Mittag-Leffler function. The function $E_{\theta ,\vartheta }^{\rho }(\mathit{\omega })$ is popularly known as Prabhakar function or generalized Mittag-Leffler three parameter function. Explicitly, the generalized Mittag-Leffler three parameter function is defined by

(1.5) $E_{\theta ,\vartheta }^{\rho }(\mathit{\omega })=\sum _{\mathit{n}=0}^{\mathrm{\infty }}\frac{{(\mathit{\rho })}_n{\omega }^n}{\mathrm{\Gamma }\left(\mathit{\theta n}+\mathit{\vartheta }\right)\mathit{n}!},(\mathit{\omega },\mathit{\theta },\mathit{\vartheta },\mathit{\rho }\in \mathbb{C},\mathit{Re}(\mathit{\theta })>0).$(1.5)

where $\mathbb{C}$ denotes the sets of complex numbers and $(\mathit{x}{)}_n$ will be used to denote the usual Pochhammer symbol.

Using the Mittag-leffler function, Murat et al. (2023) defined the following operator $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }:\mathbb{U}\to \mathbb{U}$ by

(1.6) $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }(\mathit{\omega })=\mathit{\omega }+\sum _{\mathit{n}=2}^{\mathrm{\infty }}{\left[n+\frac{r}{2}(1+{(-1)}^{\mathit{n}+1})\right]}^{\mathit{m}}\frac{\mathrm{\Gamma }(\vartheta )(\rho {)}_{\mathit{n}-1}}{\mathrm{\Gamma }\left(\mathit{\vartheta }+\mathit{\theta }(\mathit{n}-1)\right)(\mathit{n}-1)!}{a}_n{\omega }^n,$(1.6)

$\left(\mathit{m},\mathit{r}\in {\mathbb{N}}_0=\{0,1,2,\dots \},\mathit{\omega },\mathit{\theta },\mathit{\vartheta },\mathit{\rho }\in \mathbb{C},\mathit{Re}(\mathit{\theta })>0\right).$

The operator $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }$ was motivated by the operator $D_r^m\mathit{\phi }(\mathit{\omega })$ defined by Ibrahim and Darus (2019) and Ibrahim (2020).

It is well known that if $\mathit{\phi }(\mathit{\omega })$ given by (1.1) is in $\mathcal{S}$, then ${\left[\phi ({\omega }^{\tau })\right]}^{1/\mathit{\tau }}$, ($\mathit{\tau }$ is a positive integer) is also in $\mathcal{S}$. For every integer $\mathit{\eta }$, a function $\mathit{\phi }\in \mathit{A}$ is said to be $(\mathit{\eta },\mathit{\tau })$-symmetrical if for each $\mathit{\omega }\in \mathbb{U}$

(1.7) $\mathit{\phi }(\mathit{\epsilon \omega })={\epsilon }^{\eta }\mathit{\phi }(\mathit{\omega }),$(1.7)

where $\mathit{\tau }\ge 2$ is a fixed integer, $\mathit{\eta }=0,1,2,\dots ,\mathit{\tau }-1$ and $\mathit{\epsilon }=exp(2\mathit{\pi i}/\mathit{\tau })$. The family of $(\mathit{\eta },\mathit{\tau })$-symmetrical functions denoted by ${\mathcal{F}}_{\tau }^{\eta }$ was defined and studied by Liczberski and Po$\mathit{\hspace{0.17em}}$ ubi$\mathrm{n}{ }^{\mathrm{\prime }}$ski in Liczberski and Połubiński (1995). We observe that ${\mathcal{F}}_2^1$, ${\mathcal{F}}_2^0$ and ${\mathcal{F}}_{\tau }^1$ are well-known families of odd functions, even functions and $\mathit{\tau }$-symmetrical functions respectively.

Also, for every integer $\mathit{\eta }$, let ${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })$ be defined by the following equality

(1.8) ${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}\frac{\phi ({\epsilon }^{\nu }\mathit{\omega })}{{\epsilon }^{\mathit{\nu \eta }}},(\mathit{\phi }\in \mathcal{A}).$(1.8)

Obviously, ${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })$ inherits the all linearity properties $\mathit{\phi }(\mathit{\omega })$. The characterization (1.8) was first presented by Liczberski and Po$\mathit{\hspace{0.17em}}$ ubi$\mathrm{n}{ }^{\mathrm{\prime }}$ski in (Liczberski and Połubiński 1995).

The following identities can be derived from (1.8), provided $\mathit{\nu }$ is an integer,:

(1.9) $\begin{array}{c}{\phi }_{\mathit{\eta },\mathit{\tau }}({\epsilon }^{\nu }\mathit{\omega })={\epsilon }^{\mathit{\nu \eta }}{\phi }_{\mathit{\eta },\mathit{\tau }}(\mathit{\omega }),\\ {\phi }_{\eta ,\tau }^{\prime }({\epsilon }^{\nu }\mathit{\omega })={\epsilon }^{\mathit{\nu \eta }-\mathit{\nu }}{\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}\frac{{\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega })}{{\epsilon }^{\mathit{\nu \eta }-\mathit{\nu }}},\\ {\phi }_{\mathit{\eta },\mathit{\tau }}^{{ }^{\prime \prime }}({\epsilon }^{\nu }\mathit{\omega })={\epsilon }^{\mathit{\nu \eta }-2\mathit{\nu }}{\phi }_{\mathit{\eta },\mathit{\tau }}^{{ }^{\prime \prime }}(\mathit{\omega })=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}\frac{{\phi }^{\prime \prime }({\epsilon }^{\nu }\mathit{\omega })}{{\epsilon }^{\mathit{\nu \eta }-2\mathit{\nu }}}.\end{array}$(1.9)

We assume that $\mathit{\tau }\in \mathbb{N}$, $\mathit{\epsilon }=exp(2\mathit{\pi i}/\mathit{\tau })$ and

(1.10) $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\epsilon }^{-\mathit{\nu \eta }}\left[D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }(\mathit{\omega })\right]=\mathit{w}+\cdots ,$(1.10)

From (1.10), we can get

(1.11) $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\sum _{\mathrm{\ell }=1}^{\mathrm{\infty }}{a}_{\mathrm{\ell }}{\mathrm{\Phi }}_{\mathrm{\ell }}{\mathrm{\Gamma }}_{\mathrm{\ell },\mathit{\eta }}{w}^{\mathrm{\ell }},(a_1={\mathrm{\Phi }}_1=1),{\mathrm{\Gamma }}_{\mathrm{\ell },\mathit{\eta }}=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\epsilon }^{(\mathrm{\ell }-\mathit{\eta })\mathit{\nu }},$(1.11)

where

${\mathrm{\Phi }}_n={\left[n+\frac{r}{2}\left(1+{\left(-1\right)}^{\mathit{n}+1}\right)\right]}^{\mathit{m}}\frac{\mathrm{\Gamma }\left(\vartheta \right){(\mathit{\rho })}_{\mathit{n}-1}}{\mathrm{\Gamma }\left(\mathit{\vartheta }+\mathit{\theta }\left(\mathit{n}-1\right)\right)\left(\mathit{n}-1\right)!},\mathit{n}\underset{\_}{>}2$

We will define a comprehensive subclass of analytic functions involving the well-known Mittag-Leffler function. The major deviation of this paper is that, we have obtained coefficient inequalities, inclusion relationships, integral representation and closure property using differential subordination for a subclass of non-Carathéodory functions.

Motivated by (Srivastava et al. 2018; Karthikeyan et al. 2021), we now define the following:

Definition 1.1.

For $0\le \mathit{\delta }\le 1$, $\mathit{\theta },\mathit{\vartheta }\in \mathbb{C},\mathit{Re}(\mathit{\theta })>0$ and $\mathit{m}\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\}$, $\mathit{\phi }\in \mathcal{A}$ is said to be in the class ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$ if it satisfies

(1.12) $\begin{array}{c}\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })],\end{array}$(1.12)

where $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }$ is defined as in (1.6) and $\mathit{\psi }\in \mathcal{P}$ is defined as in (1.2).

Remark 1.1. The defined class of functions involves lots of parameters, so we can obtain several classes as its special case. Here, we will present a few of them:

1. If we let $\mathit{\theta }=\mathit{\tau }=\mathit{m}=0$ and $\mathit{\rho }=1$, the class ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$ will reduce to the classes ${\mathcal{S}}_s^{(\eta ,\tau )}(\mathit{\psi })$ and ${\mathcal{C}}_s^{(\eta ,\tau )}(\mathit{\psi })$ by choosing $\mathit{\delta }=0$ and $\mathit{\delta }=1$ respectively. The class ${\mathcal{S}}_s^{(\eta ,\tau )}(\mathit{\psi })$ and ${\mathcal{C}}_s^{(\eta ,\tau )}(\mathit{\psi })$ were recently introduced and studied by Karthikeyan in. (Karthikeyan 2013)

2. If we let $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$, $\mathit{\theta }=\mathit{\tau }=\mathit{m}=0$, $\mathit{\rho }=1$ and $\mathit{\psi }(\mathit{\omega })=1+\mathit{F\omega }/1+\mathit{G\omega }$ in Definition 1.1, then the class ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$ reduces to the class ${\mathcal{S}}^{(\mathit{\eta },\tau )}[\mathit{F},\mathit{G}]$ defined and studied by Al Sarari et al. [2016, Definition 5]

3. If we let $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$, $\mathit{\theta }=\mathit{\tau }=\mathit{m}=0$, $\mathit{\eta }=\mathit{\rho }=1$ and $\mathit{\psi }(\mathit{\omega })=1+\mathit{F\omega }/1+\mathit{G\omega }$ in Definition 1.1, then the class ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$ reduces to the class ${\mathcal{S}}_s^{(\tau )}[\mathit{F},\mathit{G}]$ defined and studied by Kwon, and Sim. (Kwon and Sim 2013)

The analytic characterization of ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$ is very similar to the one that was defined by Karthikeyan et al. in (Karthikeyan et al. 2021). The major deviation here is that we have defined the class with respect to $(\mathit{\eta },\mathit{\tau })$-symmetric points whereas the class studied by Karthikeyan et al. (2021) involved the odd function. The class ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$ neither unifies nor generalizes the study of Karthikeyan et al. (2021) Further, we have deviated in obtaining the results like subordination condition of starlikeness.

2. Coefficient inequalities

In this section, we impose another condition that the first coefficient of the series namely $L_1$ in (1.2) to be real. To obtain our main results, we need the following Lemma.

Lemma 2.1.

Let the function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ be convex in $\mathbb{U}$ where the function $\mathit{\psi }$ is defined as in (1.2). If $\mathit{p}(\mathit{\omega })=1+\sum _{\mathit{n}=1}^{\mathrm{\infty }}{p}_n{\omega }^n$ is analytic in $\mathbb{U}$ and satisfies the subordination condition

(2.1) $\mathit{p}(\mathit{\omega })\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })],$(2.1)

then

(2.2) $\left|p_n\right|\le \sqrt{4{cos}^2\mathit{\beta }+L_1^2{sin}^2\mathit{\alpha }},\mathit{n}\ge 1.$(2.2)

Proof.

If the function $\mathit{\psi }$ has the power series expansion (1.2), then

${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]=1+e^{-\mathit{i\alpha }}\left(2cos\mathit{\beta }+\mathit{i}{L}_{1}sin\mathit{\alpha }\right)\mathit{\omega }+\dots ,\mathit{\omega }\in \mathbb{U}.$

The equation (2.1)(2.1) $\mathit{p}(\mathit{\omega })\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })],$(2.1) is equivalent to

$\mathit{p}(\mathit{\omega })-1\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]-1.$

Since the convexity of ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ remains unaffected by translation, from a well-known Lemma of Rogosinski ([1943, Theorem VII]) it follows the conclusion (2.2).□

Here we present the coefficient inequality of ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$.

Theorem 2.2.

If $\mathit{\phi }\in {\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$, then for $\mathit{n}\ge 2$,

(2.3) $|a_n|\le \frac{1}{\left[1+\mathit{\delta }(\mathit{n}-1)\right]\left|{\mathrm{\Phi }}_n\right|}\prod _{\mathit{t}=1}^{\mathit{n}-1}\frac{\left|{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}+\left|\mathit{t}-{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|}{\left|(\mathit{t}+1)-{\mathrm{\Gamma }}_{\mathit{t}+1,\mathit{\eta }}\right|},$(2.3)

where

${\mathrm{\Phi }}_n={\left[n+\frac{r}{2}\left(1+{\left(-1\right)}^{\mathit{n}+1}\right)\right]}^{\mathit{m}}\frac{\mathrm{\Gamma }\left(\vartheta \right){(\mathit{\rho })}_{\mathit{n}-1}}{\mathrm{\Gamma }\left(\mathit{\vartheta }+\mathit{\theta }\left(\mathit{n}-1\right)\right)\left(\mathit{n}-1\right)!},\mathit{n}\ge 2$

Proof.

By the definition of ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$, we have

(2.4) $\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}=\mathit{p}(\mathit{\omega }),$(2.4)

where $\mathit{p}(\mathit{\omega })\in \mathcal{P}$ is subordinate to $\mathit{p}(\mathit{\omega })\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$.

With $a_1={\mathrm{\Phi }}_1={\mathrm{\Gamma }}_{1,\mathit{\eta }}=1$, (2.4) can be written as

$(1-{\mathrm{\Gamma }}_{1,\mathit{\eta }})\mathit{\omega }+\sum _{\mathit{n}=2}^{\mathrm{\infty }}\left(\mathit{n}-{\mathrm{\Gamma }}_{\mathit{n},\mathit{\eta }}\right)\left[1+\mathit{\delta }(\mathit{n}-1)\right]{\mathrm{\Phi }}_n{a}_n{\omega }^n=\left(\sum _{\mathit{n}=1}^{\mathrm{\infty }}{p}_n{\omega }^n\right)\left(\sum _{\mathit{n}=1}^{\mathrm{\infty }}\left[1+\mathit{\delta }(\mathit{n}-1)\right]{\mathrm{\Gamma }}_{\mathit{n},\mathit{\eta }}{\mathrm{\Phi }}_n{a}_n{\omega }^n\right).$

From the above equality, we have

$\left(\mathit{n}-{\mathrm{\Gamma }}_{\mathit{n},\mathit{\eta }}\right)\left[1+\mathit{\delta }(\mathit{n}-1)\right]{\mathrm{\Phi }}_n{a}_n=[{\mathrm{\Gamma }}_{\mathit{n}-1,\mathit{\eta }}\left[1+\mathit{\delta }(\mathit{n}-2)\right]{\mathrm{\Phi }}_{\mathit{n}-1}{a}_{\mathit{n}-1}{p}_1+\cdots +p_{\mathit{n}-1}{\mathrm{\Gamma }}_{1,\mathit{\eta }}{\mathrm{\Phi }}_1{a}_1]$

$=\sum _{\mathit{t}=1}^{\mathit{n}-1}\left|p_{\mathit{n}-\mathit{t}}\left[1+(\mathit{t}-1)\mathit{\delta }\right]{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}{\mathrm{\Phi }}_t{a}_t\right|\le \sum _{\mathit{t}=1}^{\mathit{n}-1}\left(1+\mathit{\delta }(\mathit{t}-1)\right)|p_{\mathit{n}-\mathit{t}}{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}|{\mathrm{\Phi }}_t|a_t|.$

The assertion of 2.1 implies $|p_n|\le \sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }},\mathit{n}\ge 1$. On computation, we have

(2.5) $|a_n|\le \frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}\sum _{\mathit{t}=1}^{\mathit{n}-1}\left(1+\mathit{\delta }(\mathit{t}-1)\right)|{\mathrm{\Phi }}_t{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}||a_t|}{\left|\mathit{n}-{\mathrm{\Gamma }}_{\mathit{n},\mathit{\eta }}\right|\left[1+\mathit{\delta }(\mathit{n}-1)\right]|{\mathrm{\Phi }}_n|}.$(2.5)

Let $\mathit{n}=2$ in (2.5), then

(2.6) $|a_2|\le \frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|\left[1+\mathit{\delta }\right]|{\mathrm{\Phi }}_2|}.$(2.6)

Letting $\mathit{n}=2$ in (2.3), we get

(2.7) $|a_2|\le \frac{1}{\left[1+\mathit{\delta }\right]\left|{\mathrm{\Phi }}_2\right|}\prod _{\mathit{t}=1}^{2-1}\frac{\left|{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}+\left|\mathit{t}-{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|}{\left|(\mathit{t}+1)-{\mathrm{\Gamma }}_{\mathit{t}+1,\mathit{\eta }}\right|}=\frac{1}{\left[1+\mathit{\delta }\right]\left|{\mathrm{\Phi }}_2\right|}\frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}.$(2.7)

From (4.2) and (2.7), we conclude that (2.3) is correct for $\mathit{n}=2$. Now let $\mathit{n}=3$ in (2.5), we have

$|a_3|\le \frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|3-{\mathrm{\Gamma }}_{3,\mathit{\eta }}\right|\left[1+2\mathit{\delta }\right]|{\mathrm{\Phi }}_3|}\left[\left|{\mathrm{\Gamma }}_{1,\mathit{\eta }}\right|+(1+2\mathit{\delta })\left|{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|{\mathrm{\Phi }}_2|a_2|\right]$

$\le \frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|3-{\mathrm{\Gamma }}_{3,\mathit{\eta }}\right|\left[1+2\mathit{\delta }\right]|{\mathrm{\Phi }}_3|}\left[1+\frac{\left|{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}\right].$

If we let $\mathit{n}=3,$ in (2.3), we have

$\begin{array}{rl}& |a_3|\le \frac{1}{\left[1+2\mathit{\delta }\right]\left|{\mathrm{\Phi }}_3\right|}\left[\frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}\times \frac{\left|{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}+\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}{\left|3-{\mathrm{\Gamma }}_{3,\mathit{\eta }}\right|}\right]\\ & \le \frac{1}{\left[1+2\mathit{\delta }\right]\left|{\mathrm{\Phi }}_3\right|}\left[\frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|3-{\mathrm{\Gamma }}_{3,\mathit{\eta }}\right|}\times \frac{\left|{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}+\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}{\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}\right]\\ & \le \frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left[1+2\mathit{\delta }\right]\left|{\mathrm{\Phi }}_3\right|\left|3-{\mathrm{\Gamma }}_{3,\mathit{\eta }}\right|}\left[1+\frac{\left|{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|2-{\mathrm{\Gamma }}_{2,\mathit{\eta }}\right|}\right].\end{array}$

Hence the hypothesis is correct for $\mathit{n}=3$. Assume that (2.3) is valid for $\mathit{n}=2,3,\dots \mathit{r}$. So from (2.3), we have

$\begin{array}{c}|a_r|\le \frac{1}{\left[1+\mathit{\delta }(\mathit{r}-1)\right]\left|{\mathrm{\Phi }}_r\right|}\prod _{\mathit{t}=1}^{\mathit{r}-1}\frac{\left|{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}+\left|\mathit{t}-{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|}{\left|(\mathit{t}+1)-{\mathrm{\Gamma }}_{\mathit{t}+1,\mathit{\eta }}\right|}.\end{array}$

By induction hypothesis, we have

(2.8) $\begin{array}{rl}& |a_r|\le \frac{1}{\left[1+\mathit{\delta }(\mathit{r}-1)\right]\left|{\mathrm{\Phi }}_r\right|}\prod _{\mathit{t}=1}^{\mathit{r}-1}\frac{\left|{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}+\left|\mathit{t}-{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|}{\left|(\mathit{t}+1)-{\mathrm{\Gamma }}_{\mathit{t}+1,\mathit{\eta }}\right|}\\ & =\frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|\mathit{r}-{\mathrm{\Gamma }}_{\mathit{r},\mathit{\eta }}\right|\left[1+\mathit{\delta }(\mathit{r}-1)\right]|{\mathrm{\Phi }}_r|}\sum _{\mathit{t}=1}^{\mathit{r}-1}\left(1+\mathit{\delta }(\mathit{t}-1)\right)|{\mathrm{\Phi }}_t{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}||a_t|.\end{array}$(2.8)

Now letting $\mathit{n}=\mathit{r}+1$ in (2.5), we have

(2.9) $\begin{array}{rl}& |a_{\mathit{r}+1}|\le \frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|(\mathit{r}+1)-{\mathrm{\Gamma }}_{\mathit{r}+1,\mathit{\eta }}\right|\left[1+\mathit{\delta }(\mathit{r})\right]|{\mathrm{\Phi }}_{\mathit{r}+1}|}\sum _{\mathit{t}=1}^r\left(1+\mathit{\delta }(\mathit{t}-1)\right)|{\mathrm{\Phi }}_t{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}||a_t|\\ & =\frac{\sqrt{4\stackrel{2}{cos}\mathit{\beta }+L_1^2\stackrel{2}{sin}\mathit{\alpha }}}{\left|(\mathit{r}+1)-{\mathrm{\Gamma }}_{\mathit{r}+1,\mathit{\eta }}\right|\left[1+\mathit{\delta }(\mathit{r})\right]|{\mathrm{\Phi }}_{\mathit{r}+1}|}\text{break}\left[\sum _{\mathit{t}=1}^{\mathit{r}-1}\left(1+\mathit{\delta }(\mathit{t}-1)\right)|{\mathrm{\Phi }}_t{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}||a_t|+\left(1+\mathit{\delta }(\mathit{r}-1)\right)|{\mathrm{\Phi }}_r{\mathrm{\Gamma }}_{\mathit{r},\mathit{\eta }}||a_r|\right]\end{array}$(2.9)

Using (2.8) in (2.9), we can obtain

$\left|a_{\mathit{r}+1}\right|\le \frac{\sqrt{4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }}}{\left|(\mathit{r}+1)-{\Gamma }_{\mathit{r}+1,\mathit{\eta }}\right|\left[1+\mathit{\delta }(\mathit{r})\right]\left|{\Phi }_{\mathit{r}+1}\right|}\sum \mathit{t}=1\mathit{r}-11+\mathit{\delta }(\mathit{t}-1)\left|{\Phi }_t{\Gamma }_{\mathit{t},\mathit{\eta }}\right|\left|a_t\right|+\frac{\left|{\mathit{\Gamma }}_{\mathit{r},\mathit{\eta }}\right|\sqrt{4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }}}{\left|\mathit{r}-{\Gamma }_{\mathit{r},\mathit{\eta }}\right|}{\displaystyle \sum _{\mathit{t}=1}^{\mathit{r}-1}\left(1+\mathit{\delta }(\mathit{t}-1)\right)}\left|{\Phi }_t{\Gamma }_{\mathit{t},\mathit{\eta }}\right|\left|a_t\right|=\sqrt{4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }}\sum \mathit{t}=1\mathit{r}-1\left(1+\mathit{\delta }(\mathit{t}-1)\right)\left|{\Phi }_t{\Gamma }_{\mathit{t},\mathit{\eta }}\right|\left|a_t\right|(\mathit{r}+1)-\mathit{\Gamma r}+1,\mathit{\eta }\left[1+\mathit{\delta }(\mathit{r})\right]\left|{\Phi }_{\mathit{r}+1}\right|\left[\frac{\left|{\mathit{\Gamma }}_{\mathit{r},\mathit{\eta }}\right|\sqrt{4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }}+\left|\mathit{r}-{\Gamma }_{\mathit{r},\mathit{\eta }}\right|}{\left|\mathit{r}-{\Gamma }_{\mathit{r},\mathit{\eta }}\right|}\right]=4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }\sum \mathit{t}=1\mathit{r}-1\left(1+\mathit{\delta }(\mathit{t}-1)\right)\left|{\Phi }_t{\Gamma }_{\mathit{t},\mathit{\eta }}\right|\left|a_t\right|\left|\mathit{r}-{\Gamma }_{\mathit{r},\mathit{\eta }}\right|\left[1+\mathit{\delta }(\mathit{r})\right]\left|{\Phi }_{\mathit{r}+1}\right|\left|{\Gamma }_{\mathit{r},\mathit{\eta }}\right|4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }+\left|\mathit{r}-{\Gamma }_{\mathit{r},\mathit{\eta }}\right|\left|(\mathit{r}+1)-{\Gamma }_{\mathit{r}+1,\mathit{\eta }}\right|=\frac{1}{\left[1+\mathit{\delta }(\mathit{r})\right]\left|{\mathit{\Phi }}_{\mathit{r}+1}\right|}\prod \text{\_}\mathit{t}=1\text{^}\mathit{r}-1\left|{\Gamma }_{\mathit{t},\mathit{\eta }}\right|4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }+\left|\mathit{t}-{\Gamma }_{\mathit{t},\mathit{\eta }}\right|\left|(\mathit{t}+1)-{\Gamma }_{\mathit{t}+1,\mathit{\eta }}\right|\times \left|{\Gamma }_{\mathit{r},\mathit{\eta }}\right|\sqrt{4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }}+\left|\mathit{r}-{\Gamma }_{\mathit{r},\mathit{\eta }}\right|(\mathit{r}+1)-\mathit{\Gamma r}+1,\mathit{\eta }=\frac{1}{\left[1+\mathit{\delta }(\mathit{r})\right]\left|{\mathit{\Phi }}_{\mathit{r}+1}\right|}{\displaystyle \prod _{\mathit{t}=1}^r\frac{\left|{\mathit{\Gamma }}_{\mathit{t},\mathit{\eta }}\right|\sqrt{4{\mathit{cos}}^2\mathit{\beta }+L_1^2{\mathit{sin}}^2\mathit{\alpha }}+\left|\mathit{t}-{\Gamma }_{\mathit{t},\mathit{\eta }}\right|}{\left|(\mathit{t}+1)-{\Gamma }_{\mathit{t}+1,\mathit{\eta }}\right|}},$

implies that inequality (2.3) is true for $\mathit{n}=\mathit{r}+1$. Hence, the proof of the Theorem.□

If we let $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$, $\mathit{\theta }=\mathit{\tau }=\mathit{m}=0$, $\mathit{\lambda }=\mathit{\rho }=1$ and $\mathit{\psi }(\mathit{\omega })=(1+\mathit{F\omega })/(1+\mathit{G\omega })$ in Theorem 2.2, then we get the following result.

Corollary 2.3.

[Al Sarari et al. 2016, Theorem 2] If $\mathit{\phi }\in {\mathcal{S}}^{(\mathit{\eta },\tau )}(\mathit{F},\mathit{G})$ (see Remark 1.1), then for $\mathit{n}\ge 2,$ $-1\le \mathit{F}<\mathit{G}\le 1$,

$|a_n|\le \prod _{\mathit{t}=1}^{\mathit{n}-1}\frac{|{\mathrm{\Gamma }}_{\mathit{t},\mathit{\eta }}|\left[(\mathit{F}-\mathit{G})-1\right]+\mathit{t}}{\left|\mathit{t}+1-{\mathrm{\Gamma }}_{\mathit{t}+1,\mathit{\eta }}\right|}.$

For $\mathit{F}=1-2\mathit{\xi }cos\mathit{\varrho }$, $0\le \mathit{\xi }<1$, $\mathit{G}=-1$ and $\mathit{\eta }=1$, the Corollary 2.3 reduces to the next special case:

Corollary 2.4.

[Libera. 1967, Theorem 1] If $\mathit{\phi }\in \mathcal{A}$ satisfy the inequality

$\mathrm{Re}\frac{\mathit{\omega }{\mathit{\phi }}^{\prime }(\mathit{\omega })}{\mathit{\phi }(\mathit{\omega })}>\mathit{\xi }cos\mathit{\varrho },\mathit{\omega }\in \mathbb{U},$

then

(2.10) $\left|a_n\right|\le \prod _{\mathit{t}=0}^{\mathit{n}-2}\frac{\left|2(1-\mathit{\xi }){\mathit{e}}^{-\mathit{i\rho }}cos\mathit{\varrho }+\mathit{t}\right|}{\mathit{t}+1}.$(2.10)

The coefficient estimates of (2.10) are sharp

If we let $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$, $\mathit{\theta }=\mathit{\tau }=\mathit{m}=0$, $\mathit{\lambda }=\mathit{\rho }=1$ and $\mathit{\psi }(\mathit{\omega })=\frac{(F+1)p_{\alpha }(\mathit{\omega })-(\mathit{F}-1)}{(\mathit{G}+1)p_{\alpha }(\mathit{\omega })-(\mathit{G}-1)}$ in Theorem 2.2, then we get the following result.

Corollary 2.5.

[Karthikeyan et al. 2020, Corollary 6] If $\mathit{\phi }\in \mathcal{A}$ satisfy the condition

$\frac{\mathit{\omega }{\mathit{\phi }}^{\prime }(\mathit{\omega })}{\mathit{\phi }(\mathit{\omega })}\prec \frac{(F+1)p_{\alpha }(\mathit{\omega })-(\mathit{F}-1)}{(\mathit{G}+1)p_{\alpha }(\mathit{\omega })-(\mathit{G}-1)},$

with $p_{\alpha }(\mathit{\omega })=(1+2\mathit{\alpha })\sqrt{\frac{1+\mathit{b\omega }}{1-\mathit{bz}}}-2\mathit{\alpha }$, $\mathit{b}=\mathit{b}(\mathit{\alpha })=\frac{1+4\alpha -4{\alpha }^2}{{(1+2\mathit{\alpha })}^2}$, $\mathit{\alpha }>0$. Then for $\mathit{n}\ge 2,$ $-1\le \mathit{H}<\mathit{G}\le 1$,

$|a_n|\le \prod _{\mathit{t}=0}^{\mathit{n}-2}\frac{\left|(\mathit{F}-\mathit{G})(1+4\mathit{\alpha })-2\mathit{tG}\right|}{2(1+2\mathit{\alpha })(\mathit{t}+1)}.$

Remark 2.1. Several well-known results can be obtained as special case of Theorem 2.2.

3. Inclusion relationships and integral representations of the classes ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$

If ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$, then by Definition 1.1 there exist a Schwatrz function $\mathit{\sigma }(\mathit{\omega })$ such that

(3.1) $\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}={\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{\omega })\right)].$(3.1)

If we replace $\mathit{\omega }$ by ${\epsilon }^{\nu }\mathit{\omega }(\mathit{\nu }=0,1,2,\dots ,\mathit{\tau }-1)$ in (3.1), then (3.1) will be of the form

(3.2) $\frac{(1-\delta ){\epsilon }^{\nu }\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega })+\mathit{\delta }{\epsilon }^{\nu }\mathit{\omega }{({\epsilon }^{\nu }\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }({\epsilon }^{\nu }\mathit{\omega })+\mathit{\delta }{\epsilon }^{\nu }\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }({\epsilon }^{\nu }\mathit{\omega })}={\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)].$(3.2)

Using (1.9) in (3.2), we get

(3.3) $\frac{(1-\delta ){\epsilon }^{\nu }\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega })+\mathit{\delta }{\epsilon }^{\nu }\mathit{\omega }{({\epsilon }^{\nu }\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega }))}^{\prime }}{{\epsilon }^{\mathit{\nu \eta }}\left[(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })\right]}={\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)].$(3.3)

Let $\mathit{\nu }=0,1,2,\dots ,\mathit{\tau }-1$ in (3.3) respectively and summing them, we get

$\begin{array}{rl}& \frac{(1-\delta )\frac{1}{\tau }{\sum }_{\nu =0}^{\tau -1}{\epsilon }^{\mathit{\nu }-\mathit{\nu \eta }}\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega })+\mathit{\delta }\left[\frac{1}{\tau }{\sum }_{\nu =0}^{\tau -1}{\epsilon }^{\mathit{\nu }-\mathit{\nu \eta }}\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }({\epsilon }^{\nu }\mathit{\omega })+\frac{1}{\tau }{\sum }_{\nu =0}^{\tau -1}{\epsilon }^{2\mathit{\nu }-\mathit{\nu \eta }}{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }({\epsilon }^{\nu }\mathit{\omega })\right]}{\left[(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })\right]}\\ & =\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)].\end{array}$

Or equivalently,

$\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)].$

On summarizing the above discussion, we have the following.

Theorem 3.1.

Let the function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\in \mathcal{N}\mathcal{P}$ satisfy the subordination condition $\frac{1}{\tau }{\sum }_{\nu =0}^{\tau -1}{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)]\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$. If $\mathit{\phi }\in {\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$, then

$\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta }{\epsilon }^{\nu }\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })].$

Let $\mathit{G}(\mathit{\omega })=(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })$ and $\mathit{H}(\mathit{\omega })=(1-\mathit{\delta })D_r^m\text{break}(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })$. If $\mathit{\phi }\in {\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$, then following the steps as in Theorem 3.1, we have

$\frac{\mathit{\omega }{H}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })}=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)].$

Alternatively, the above equality can be rewritten as

$\frac{{\mathit{H}}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })}-\frac{1}{\omega }=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\omega })\right)]-1}{\mathit{\omega }}.$

Integrating this equality, we get

$log\left\{\frac{\mathit{H}(\mathit{\omega })}{\omega }\right\}=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{\omega }\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }({\epsilon }^{\nu }\mathit{\zeta })\right)]-1}{\mathit{\zeta }}\mathit{d\zeta }=\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)]-1}{\mathit{t}}\mathit{dt},$

or equivalently,

$(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })=\mathit{\omega }exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)]-1}{\mathit{t}}\mathit{dt}\right\}.$

We have two cases namely

(1) For $\mathit{\delta }=0$, trivially we have

$D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\mathit{\omega }exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)]-1}{\mathit{t}}\mathit{dt}\right\}.$

$(2)\mathrm{For}0<\mathit{\delta }\le 1,$

$D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\frac{1}{\delta }{\omega }^{\left(1-\frac{1}{\delta }\right)}{\int }_0^{\omega }{u}^{\frac{1}{\delta }-1}exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)]-1}{\mathit{t}}\mathit{dt}\right\}\mathit{du}.$

Summarising the above discussion, we have

Theorem 3.2.

If $\mathit{\phi }\in {\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$, then

(i) for $0<\mathit{\delta }\le 1$,

(3.4) $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\frac{1}{\delta }{\omega }^{\left(1-\frac{1}{\delta }\right)}{\int }_0^{\omega }{u}^{\frac{1}{\delta }-1}exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)]-1}{\mathit{t}}\mathit{dt}\right\}\mathit{du}.$(3.4)

(ii) for $\mathit{\delta }=0$,

(3.5) $D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\mathit{\omega }exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)]-1}{\mathit{t}}\mathit{dt}\right\}.$(3.5)

If we let $\mathit{\theta }=\mathit{\tau }=\mathit{m}=\mathit{\delta }=0$, $\mathit{\lambda }=\mathit{\rho }=1$ and $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$ in Theorem 3.2, we get

Corollary 3.3.

[13, Theorem 2.3] Let $\mathit{\phi }\in {\mathcal{S}}_s^{(\eta ,\tau )}(\mathit{\psi })$, then we have

${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })=\mathit{\omega }exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\omega }}\frac{\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)-1}{t}\mathit{dt}\right\}$

where ${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })$ defined by equality (1.8), $\mathit{\sigma }(\mathit{\omega })$ is analytic in $\mathbb{U}$ and $\mathit{\sigma }(0)=0$, $|\mathit{\sigma }(\mathit{\omega })|<1$.

If we let $\mathit{\theta }=\mathit{\tau }=\mathit{m}=0$, $\mathit{\delta }=\mathit{\lambda }=\mathit{\rho }=1$ and $\mathit{\alpha }=\mathit{\beta }=\frac{\pi }{2}$ in Theorem 3.2, we have the following Corollary.

Corollary 3.4.

[13, Theorem 2.4] Let $\mathit{\phi }\in {\mathcal{C}}_s^{(\eta ,\tau )}(\mathit{\psi })$, then we have

${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })={\int }_0^{\omega }exp\left\{\frac{1}{\tau }\sum _{\mathit{\nu }=0}^{\mathit{\tau }-1}{\int }_0^{{\epsilon }^{\nu }\mathit{\zeta }}\frac{\mathit{\psi }\left(\mathit{\sigma }(\mathit{t})\right)-1}{t}\mathit{dt}\right\}\mathit{d\zeta }$

where ${\phi }_{\mathit{\eta },\tau }(\mathit{\omega })$ defined by equality (1.8), $\mathit{\sigma }(\mathit{\omega })$ is analytic in $\mathbb{U}$ and $\mathit{\sigma }(0)=0$, $|\mathit{\sigma }(\mathit{\omega })|<1$.

4. Subordination conditions for the classes ${\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$

Note that the function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ in general does not belong to the class $\mathcal{P}$ and is not convex. However, if we restrict the radius of the domain or by choosing appropriate values of the parameter, we can see that ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ will belong to class $\mathcal{P}$.

Motivated by the results presented in Chapter 4 of (Bulboacã 2005), here we obtain some conditions for starlikeness. We now state the following result which will be used in the sequel.

Lemma 4.1

([8]). Let $\mathit{g}$ be convex in $\mathbb{U}$, with $\mathit{g}(0)=\mathit{a}$, $\mathit{\gamma }\ne 0$ and $\mathrm{Re}(\mathit{\gamma })>0$. Suppose that $\mathit{h}(\mathit{\omega })$ is analytic in $\mathbb{U}$, which is given by

(4.1) $\mathit{h}(\mathit{\omega })=\mathit{a}+{\vartheta }_n{\omega }^n+{\vartheta }_{\mathit{n}+1}{\omega }^{\mathit{n}+1}+\cdots ,\mathit{\omega }\in \mathbb{U}.$(4.1)

If

$\mathit{h}(\mathit{\omega })+\frac{\mathit{\omega }{\mathit{h}}^{\prime }(\mathit{\omega })}{\mathit{\gamma }}\prec \mathit{g}(\mathit{\omega }),$

then

$\mathit{h}(\mathit{\omega })\prec \mathit{q}(\mathit{\omega })\prec \mathit{g}(\mathit{\omega }),$

where

$\mathit{q}(\mathit{\omega })=\frac{\gamma }{\mathit{n}{\mathit{\omega }}^{\mathit{\gamma }/n}}{\int }_0^{\omega }\mathit{g}(\mathit{t})t^{(\mathit{\gamma }/n)-1}\mathit{dt}.$

The function $\mathit{q}$ is convex and is the best $(\mathit{a},\mathit{n})$-dominant.

For convenience, we denote $\mathit{G}(\mathit{\omega })=(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho })\mathit{\phi }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })$ and $\mathit{H}(\mathit{\omega })=(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })$.

Theorem 4.2.

Let the function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{p}(\mathit{\omega })]$ be defined as in (1.4) be convex univalent in $\mathbb{U}$. Let $\mathit{\phi }\in \mathcal{A}$ satisfy

(4.2) $\begin{array}{rl}& \frac{\mathit{\omega }{\mathit{G}}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })}\left[1+\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+(2\mathit{\delta }+1){\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })+\mathit{\delta }{\omega }^3{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime \prime }}(\mathit{\omega })}{\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })}\right.\\ & \left.-\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }^{{ }^{\prime \prime }}(\mathit{\omega })}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}\right]\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })],\end{array}$(4.2)

then

(4.3) $\begin{array}{rl}& \frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_{\tau }^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}\prec \mathit{q}(\mathit{\omega })\\ & =\frac{1}{\omega }{\int }_0^{\omega }\left(\frac{{\mathit{e}}^{-\mathit{i\alpha }}\left[cos\mathit{\alpha }+\mathit{i}sin\mathit{\alpha }(1-t^2)\mathit{\psi }(\mathit{t})\right]}{1-2cos\mathit{\beta t}{e}^{-\mathit{i\alpha }}+e^{-2\mathit{i\alpha }}{t}^2}\right)\mathit{dt}\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })].\end{array}$(4.3)

and $\mathit{q}(\mathit{\omega })$ is the best dominant.

Proof.

Let $\mathit{p}(\mathit{\omega })$ be defined by

(4.4) $\mathit{p}(\mathit{\omega })=\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_{\tau }^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}=\frac{\mathit{\omega }{\mathit{G}}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })},\mathit{\omega }\in \mathbb{U}.$(4.4)

Then the function $\mathit{p}(\mathit{\omega })$ is of the form $\mathit{p}(\mathit{\omega })=1+p_1\mathit{\omega }+p_2{\omega }^2+\cdots$ and is analytic in $\mathbb{U}$. Differentiating both sides of (4.4) and by simplifying, we have

(4.5) $\begin{array}{rl}& \frac{\mathit{\omega }{\mathit{G}}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })}\left[1+\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+(2\mathit{\delta }+1){\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })+\mathit{\delta }{\omega }^3{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime \prime }}(\mathit{\omega })}{\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })}\right.\\ & \left.-\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }^{{ }^{\prime \prime }}(\mathit{\omega })}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}\right]=\mathit{p}(\mathit{\omega })+\mathit{\omega }{p}^{\prime }(\mathit{\omega }).\end{array}$(4.5)

By hypothesis (4.2), we have

$\mathit{p}(\mathit{\omega })+\mathit{\omega }{p}^{\prime }(\mathit{\omega })\prec \frac{{\mathit{e}}^{-\mathit{i\alpha }}\left[cos\mathit{\alpha }+\mathit{i}sin\mathit{\alpha }(1-{\omega }^2)\mathit{\psi }(\mathit{\omega })\right]}{1-2cos\mathit{\beta \omega }{e}^{-\mathit{i\alpha }}+e^{-2\mathit{i\alpha }}{\omega }^2}.$

Applying Lemma 4.1 to the above equation with $\mathit{\gamma }=1$ and $\mathit{a}=\mathit{n}=1$, we get the assertion (4.2) Hence, the proof of the Theorem 4.2□

Remark 4.1. In Lemma 4.1, there is no need for the superordinate function to be in class $\mathcal{P}$. Hence, the choice of ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\in \mathcal{N}\mathcal{P}$ is admissible.

Theorem 4.3.

Let the function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\in \mathcal{P}$ be convex univalent in $\mathbb{U}$ and let $\mathit{\kappa }(\mathit{\omega }):={\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^2+\mathit{\omega }{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime }$. If the function $\mathit{\phi }\in \mathit{A}$ satisfies the conditions

(4.6) $\frac{\mathit{\omega }{\mathit{G}}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })}\times \left[\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+(2\mathit{\delta }+1){\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })+\mathit{\delta }{\omega }^3{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime \prime }}(\mathit{\omega })}{\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })}\right.$(4.6)

(4.7) $\left.-\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }^{{ }^{\prime \prime }}(\mathit{\omega })}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}+\frac{\mathit{\omega }{\mathit{G}}^{\prime }(\mathit{\omega })}{\mathit{H}(\mathit{\omega })}\right]\prec \mathit{\kappa }(\mathit{\omega }),$(4.7)

then $\mathit{\phi }\in {\mathcal{C}}_{(\eta ,\tau )}^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho };\mathit{\delta };\mathit{\psi })$. Moreover, the function ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ is the best dominant of the left-hand side of (1.12).

Proof.

If we define the function $\mathit{p}(\mathit{\omega })$ by

$\mathit{p}(\mathit{\omega }):=\frac{(1-\delta )\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta \omega }{(\omega D_{\tau }^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega }))}^{\prime }}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })},\mathit{\omega }\in \mathbb{U},$

then from the hypothesis, it follows that $\mathit{p}$ is analytic in $\mathbb{U}$. By a straight forward computation, we have

$\begin{array}{rl}& \mathit{\omega }{p}^{\prime }(\mathit{\omega })=\mathit{p}(\mathit{\omega })\left[\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+(2\mathit{\delta }+1){\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })+\mathit{\delta }{\omega }^3{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime \prime }}(\mathit{\omega })}{\mathit{\omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }^{{ }^{\prime \prime }}(\mathit{\omega })}\right.\\ & \left.-\frac{\omega D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })+\mathit{\delta }{\omega }^2{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }^{{ }^{\prime \prime }}(\mathit{\omega })}{(1-\mathit{\delta })D_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\mathit{\eta },\tau }(\mathit{\omega })+\mathit{\delta \omega }{D}_r^m(\mathit{\theta },\mathit{\vartheta },\mathit{\rho }){\phi }_{\eta ,\tau }^{\prime }(\mathit{\omega })}\right],\end{array}$

and thus, the subordination (4.7) is equivalent to

(4.8) $p^2(\mathit{\omega })+\mathit{\omega }{p}^{\prime }(\mathit{\omega })\prec \mathit{\kappa }(\mathit{\omega }).$(4.8)

Setting $\mathrm{\Omega }(\mathit{\sigma }):={\sigma }^2\mathrm{and}\mathrm{{\rm Y}}(\mathit{\sigma }):=1$, then $\mathrm{\Omega }$ and $\mathrm{{\rm Y}}$ are analytic functions in $\mathbb{C}$, with $\mathrm{{\rm Y}}(0)\ne 0$. Therefore

$\mathit{Q}(\mathit{\omega })=\mathit{\omega }{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime }\mathrm{{\rm Y}}({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })])=\mathit{\omega }{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime }$

and

$\mathit{\kappa }(\mathit{\omega })=\mathrm{\Omega }\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)+\mathit{Q}(\mathit{\omega })={\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^2+\mathit{\omega }{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime },$

and using the assumption that ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ is a convex univalent function in $\mathbb{U}$, it follows that

$\mathrm{\Re }\frac{\mathit{\omega }{\mathit{Q}}^{\prime }(\mathit{\omega })}{\mathit{Q}(\mathit{\omega })}=\mathrm{\Re }\left(1+\frac{\omega {\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{{ }^{\prime \prime }}}{{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime }}\right)>0,\mathit{\omega }\in \mathbb{U},\left({\mathit{Q}}^{\prime }(0)={\left[{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime }\right]}_{\mathit{t}=0}\ne 0\right),$

hence $\mathit{Q}$ is a starlike univalent function in $\mathbb{U}$. Further, the convexity of ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ together with $\mathrm{\Re }[{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]]>0$ (assumed) implies

$\mathrm{\Re }\frac{\mathit{\omega }{\mathit{\kappa }}^{\prime }(\mathit{\omega })}{\mathit{Q}(\mathit{\omega })}=\mathrm{\Re }\left\{2{\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]+\frac{\omega {\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{{ }^{\prime \prime }}}{{\left({\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]\right)}^{\prime }}+1\right\}>0,\mathit{\omega }\in \mathbb{U}.$

Since the conditions of the well-known Miller- Mocanu lemma (see [3, Theorem 3.6.1.]) are satisfied it follows that (4.8) implies $\mathit{p}(\mathit{\omega })\prec {\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$, and ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ is the best dominant of $\mathit{p}$, which prove our conclusions.□

Remark 4.2. Several special cases of Theorem 4.2 and Theorem 4.3 can be obtained by assigning some fixed values to the parameter involved in it.

5. Conclusion

We have obtained the interesting coefficient bounds involving analytic functions with respect to $(\mathit{\eta },\mathit{\tau })$-symmetric points. Indeed, very few researchers have attempted the coefficient problems pertaining to analytic functions with respect to $(\mathit{\eta },\mathit{\tau })$-symmetric points, as it is computationally tedious. Further, most of the studies in this area by various other authors involved the differential characterization subordinate to a Carathéodory function. But in this study, we have obtained interesting subordination conditions, inclusion and integral representation of the functions defined for a class of non-Carathéodory function.

Assertion of the Lemma 2.1 is true only if the superordinate function in (2.1) is convex, so the results that we obtained in Section 2 cannot be applied to functions that are subordinate to non-convex functions. Hence, there is a need to develop some tools or methods to obtain the coefficients for the functions subordinate to non-convex functions. In addition, we note that the impact of ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ is not the same in all conic regions. So, the following question arises: Are there any specific specialized regions in which the impact of ${\mathrm{\Lambda }}^{\ast }[\mathit{\alpha },\mathit{\beta };\mathit{\psi }(\mathit{\omega })]$ will be the same?

The study should be interesting when the ordinary derivative in Definition 1.1 is replaced with a multiplicative derivative. However, the presence of second order derivative in (1.12) will make such a study very complicated. Further, this study can be extended by replacing $\mathit{p}(\mathit{\omega })$ in (1.4) with a Legendre polynomial, $\mathit{q}$-Hermite polynomial, Fibonacci sequence, or Chebyshev polynomial.

Statement on author’s contribution

All three authors contributed equally to this work. All the authors have read and agreed to the published version of the manuscript.

Acknowledgments

Authors would like to thank the referees and the academic editor for their comments and suggestions which helped us remove the mistakes. We also sincerely express our gratitude to the referees for their comments which led to improvements in the readability of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

No data was used to support this study.

Additional information

Funding

This research study received no external.