Abstract
In the paper, we study dynamic h-convexity for interval valued functions. Some generalizations of Jensen’s inequality in interval valued analysis for h-convex functions on time scales are proved in the paper. In seek of applications of generalized Jensen’s inequality, Hermite-Hadamard type inequalities for h-convex functions on time scales are established. Further discrete analogues of newly proved results are also presented in the paper. Some numerical examples are also provided to check the validity of the results.
1. Introduction
The theory of convex function has a long history and has been the subject of considerable research in mathematics for over a century. Convexity plays a significant part in social sciences, administration technology and theoretical development in science and current analysis. The revival of curiosity in convex functions and optimization in applied sciences and engineering can be backed down to the early 1980s. Throughout the centuries that followed, this investigation led in the emergence of convex function concept as a distinct topic of mathematical analysis.
The classical definition of convexity for a function is where and
By using different methodologies, the research on convexity is being expanded day after day; see Awan, Noor, Noor, and Safdar (Citation2017). In optimization, economics, and nonlinear programming, substantially extended convexity is commonly employed. Convex functions have seen rapid generalizations and extensions in recent years; some extensions are included in Niculescu and Persson (Citation2006), Pečarić and Tong (Citation1992), and Zalinescu (Citation2002). Almost all fields of mathematics and other concerns of applied sciences and engineering employ inequalities as a mechanism. Jensen inequality is widely recognized in the domain of analysis and mathematics. It is named after Johan Jensen, a Danish mathematician. The Jensen type inequalities are not only used to get the majority of classical inequalities but also to predict the effects of environmental changes (Cabrerizo & Marañón, Citation2022; Ruel & Ayres, Citation1999). Developments in the Jensen type inequalities can be seen in Andrić (Citation2021) and Mughal, Almusawa, Haq, and Baloch (Citation2021). Some generalizations of the Jensen type inequalities with applications can be seen in Rodić (Citation2022). Inequalities of Jensen type for class of h-convex functions can be acquired in Lara, Merentes, and Nikodem (Citation2016).
In Sarikaya, Saglam, and Yildirim (Citation2008), Sarikaya et al. proved a variety of the Hermite-Hadamard inequality for functions which are h-convex. More extensions and refinements of the Hermite-Hadamard type inequalities for functions which are h-convex have been thoroughly analyzed in Dragomir (Citation2015) and Noor, Noor, and Awan (Citation2015). An application of Hermite Hadamard inequality for elastic torsion problem can be found in Dragomir and Keady (Citation1998). Moreover, applications of Hermite Hadamard type inequalities to f-divergence measures as well as to some special means of real numbers and estimates for the error term of trapezoidal formula are given in Khan, Ali, and Khan (Citation2017). Stefan Hilger established the time scales calculus in his PhD thesis 1988 (Hilger, Citation1990), by demonstrating ways to bring together discrete-time and continuous-time dynamical networks. Several scholars are currently interested in the calculus of time scales, which is helpful in developing theory and methodologies in biology (see, e.g. Bohner & Warth, Citation2007). Delta and nabla calculi are fundamental approaches to study time scales theory. Sheng, Fadag, Henderson, and Davis (Citation2006) introduced the diamond alpha calculus by combining delta and nabla calculi. Agarwal et al. drove Jensen type inequality in Agarwal, Bohner, and Peterson (Citation2001) with the help of delta integrals on time scales. In case of diamond-α integrals Jensen type inequality is studied in Ammi, Ferreira, and Torres (Citation2008). In particular, for it has the following form:
Theorem 1.1.
(Jensen’s inequality) Assume that and . Imagine is right dense continuous and is convex. Then
Middle point Hermite-Hadamard inequality given in Dinu (Citation2008, 5.1) is the following:
Suppose is a time scale and Assume that is a continuous convex function. Then, (1) (1)
Hermite Hadamard type inequality on time scales can be seen in Mohammed, Ryoo, Kashuri, Hamed, and Abualnaja (Citation2021).
The background of interval analysis might be linked all the way return to Archimedes’ calculation of the circle’s circumference. It was mostly neglected for quite some time due to deficiency of applications in various fields. Significant work in this field did not appear until 1950s. R. E. Moore’s acclaimed work (Moore, Citation1996) was the first textbook on interval analysis.
Many research publications in interval analysis are based on the presentation of an uncertain variable as an interval (Gallego-Posada & Puerta-Yepes, Citation2018; Markov, Citation1979; Yadav, Bhurjee, Karmakar, & Dikshit, Citation2017). Further, a thorough examination of different interval valued inequalities can be found in An, Ye, Zhao, and Liu (Citation2019), Guo, Ye, Zhao, and Liu (Citation2019), Liu, Ye, Zhao, and Liu (Citation2019), and Younus, Asif, and Farhad (Citation2015).
Recently, Jensen’s and Hermite Hadamard type inequalities for interval valued functions can be seen in Khan, Srivastava, Mohammed, Nonlaopon, and Hamed (Citation2022).
The plan of paper is as follows: Sec. 2, consists of some basic information about interval valued arithmetics, Riemann integrable function, Riemann delta integral for interval valued functions, diamond alpha derivatives integrals and dynamic h-convexity for interval valued functions. In Sec. 3, we proved interval valued Jensen’s inequalities for h-convex functions on time scales. We apply obtained inequality for h-convex functions to produce interval valued Hermite-Hadamard inequality for h-convex functions on time scales. Further many examples are provided to check the validity of results. Finally concluding remark of the paper is given in Sec. 4.
2. Preliminaries
2.1. Interval valued arithmetics
The following arithmetics are chosen from Markov (Citation1979) and Stefanini (Citation2010).
Assume that
For and respectively. By definition, we get Moreover, where is called gH-difference.
For width of is referred as By using we can write
More explicitly, for C we have
Since is not totally order set (e.g. see Chalco-Cano, Flores-Franulič, & Román-Flores, Citation2012; Markov, Citation1979; Moore, Citation1979; Younus et al., Citation2015). For such that we say that:
and if and where
and
and if if and where
and
and if and
Assume that be the collection of these partial orders on The following results include some of the properties of these partial orders.
Lemma 2.1.
Assume that If then
Lemma 2.2.
Assume that If , then
Lemma 2.3.
Assume that C . If and then
Lemma 2.4.
If then and
Lemma 2.5.
If then and
Lemma 2.6.
If then
Notations: denotes the collection of all intervals of denotes the collection of all positive intervals of and denotes the collection of all negative intervals of
Riemann integrable function (Zhao, An, Ye, & Liu, Citation2018)
Let be any finite ordered subset of obtained by division of having pattern
A division’s mesh represents subintervals of maximum length that make up i.e.
Suppose that is the collection of all for which In every interval select any point ξj, makes the sum where (or is known as Riemann sum of related to
Definition 2.7
(Dinghas, Citation1956). A function is known as Riemann integrable (R-integrable) on if B for every there is for which for each Riemann sum S of is in and independent of the option of for Here B is known as R-integral of on and is defined as The bunch of all R-integrable functions on are indicated by
Interval valued Riemann integrable function
Definition 2.8
(Zhao et al., Citation2018). A function is known as interval Riemann integrable (IR-integrable) on if there is B for all there exist for which for every Riemann sum S of related to each and independent of the option of for Where, B is known as the IR-integral of on and is defined as The bunch of all IR-integrable functions on is indicated by
Riemann delta integral for interval valued functions
Definition 2.9
(Zhao et al., Citation2018). A function is known as IR on if an for which
In this situation, B is known as the IR of η on and is indicated as The collection of all IR functions on is indicated as
Theorem 2.10
(Zhao, Ye, Liu, & Torres, Citation2019). If then and (2) (2)
Theorem 2.11
(Zhao et al., Citation2019). Suppose that , and α is any real number, then
and
and
For and
If on then
2.2. Arithmetics of diamond alpha derivative and integral
Let be a time scale. If has a right-scattered minimum m, then define otherwise If T has a left-scattered maximum M, then define otherwise Finally, put and for s, t denote and
One defines the diamond-α dynamic derivation (Sheng et al., Citation2006) of a function at t to be the number denoted by (when it exist), with the property that, for any there is a neighbourhood U of t such that for all (3) (3)
A function is called diamond-α differentiable on if exists for all
Moreover if is differentiable on in the sense of Δ and then is diamond-α differentiable at and the diamond-α derivative for is given by (4) (4)
We present here some properties of diamond-α derivative (Sheng et al., Citation2006). For that be diamond-α differentiable at Then, for
is diamond-α differentiable at
If c and is diamond-α differentiable at and
Let a, b and The diamond-α integral of ζ1 from a to b is defined by (5) (5)
For Equation(4)(4) (4) and Equation(5)(5) (5) are symmetric derivative and symmetric integral (respectively) involving delta and nabla derivativesintegrals. The present study is restricted to use of diamond- derivative or integral.
Similar to Theorem 2.10, one can define interval valued nabla integral in the form:
Theorem 2.12.
If then and (6) (6)
Theorem 2.10 together with Theorem 2.12 gives us the following result:
Theorem 2.13.
If then and (7) (7)
Proof.
Divide Equation(2)(2) (2) and Equation(6)(6) (6) by 2 and add the resultants to get Equation(7)(7) (7) . □
Remark 2.14.
Theorem 2.11 remains valid if we replace delta with nabla or diamond- integrals.
Example 2.15.
Let be a time scale and is the collection of positive integers including zero. Suppose that is characterized by
If then
Hence
Remark 2.16.
Example 2.15 is given in Zhao et al. (Citation2019) for delta integral.
We use the following notations in the next section. Let I be any interval on and is interval on time scale.
Convex function on time scales
Definition 2.17.
A function is convex function on (Dinu, Citation2008), if (8) (8) and all such that If the inequality is strict for distinct and then function is strictly convex on If is convex, then function is said to be concave on is affine on if it is both concave and convex on
Note:
Next we provide some definitions of p-function, h-convex function, h-convex function for interval valued functions on time scales which are helpful to establish our main results in Sec. 3.
P-function on time scales
Definition 2.18.
A function is a P-function on time scale if is positive and and we get
h-Convexity on time scales
Definition 2.19.
Let be a positive function, and A function is in collection or a function is known as h-convex function, if is positive and and we get (9) (9)
A function is called h-concave if inequality is reversed i.e.
3. Main results
In order to establish the main results, it needs to define h-convexity for interval valued functions on time scales.
3.1. h-Convexity for interval valued functions on time scales
Definition 3.1.
Suppose that is a positive function such that and A function Λ is in collection or a function is known as h-convex function, if Λ is positive and and we get (10) (10)
A function Λ is called h-concave if set inclusion is reversed.
Example 3.2.
Let Define and then is not convex interval valued function but h-convex interval valued function for For more simplicity, choose then when we get equality and when one of v or u is we get
Notations. The set of all h-concave interval valued functions on time scales is indicated by The collection of all h-convex interval valued functions is indicated by The collection of all h-affine interval valued functions is indicated by
3.2. Interval valued Jensen’s inequalities for h-convex functions on time scales
Theorem 3.3.
Suppose is interval valued function which is h-convex on time scales and . Then iff and
Proof.
Let and Then and (11) (11)
It comes that and
This reveals that and Conversely, if and then from definition of P-function and set inclusion one obtains □
Example 3.4.
Let and then Theorem 3.3 is satisfied.
Corollary 3.5.
If we choose time scale to be set of real numbers in Theorem , it becomes (Zhao et al., Citation2018, Theorem 3.7).
Theorem 3.6.
Suppose is interval valued function which is h-concave on time scales and . Then iff and
Proof.
Suppose that and Then and (12) (12)
It comes that and
This reveals that and Conversely, if and then from definition of P-function and set inclusion one obtains □
Example 3.7.
Let Define and then is not concave interval valued function but h-concave interval valued function.
Corollary 3.8.
If we choose time scale to be a set of real numbers in Theorem , it becomes (Zhao et al., Citation2018, Theorem 3.8).
3.3. Applications to interval valued Hermite Hadamard inequalities for h-convex functions
Theorem 3.9.
Suppose that is a interval valued function which is h-convex and and and is a positive function. If , then (13) (13) where and
If , then (14) (14)
Proof.
We split interval into and Then for we have (15) (15)
Integration of the set inclusion with respect to η on gives (16) (16)
Now for we have (17) (17)
Integration of the set inclusion with respect to η on gives (18) (18)
By adding inclusions Equation(16)(16) (16) and Equation(18)(18) (18) , one gets
By using and
Now
This completes the proof. □
Corollary 3.10.
If we choose time scale to be a set of real numbers in Theorem , it becomes (Zhao et al., Citation2018, Theorem 4.3).
Example 3.11.
Consider Assume that and be referred by for all We have and
Then one gets
Consequently, Theorem 3.9 is verified.
Corollary 3.12.
When and in Theorem , then set inclusions Equation(13)(13) (13) and Equation(14)(14) (14) turn into the following inclusions respectively, (19) (19) (20) (20) where and
Example 3.13.
If we choose h(t) = t and in Corollary 3.12, Equation(19)(19) (19) becomes (21) (21)
Theorem 3.14.
Let be two interval valued functions which are h-convex with and and are positive right dense continuous functions. If , then (22) (22) where and
Proof.
Since and
Therefore one has that and
Which can be written as and
Since and are non-negative, therefore Integrating over we get
By using where and □
Corollary 3.15.
If we choose time scale to be set of real numbers in Theorem 3.14, it becomes (Zhao et al., Citation2018, Theorem 4.5).
Corollary 3.16.
When and in Theorem 3.14, set inclusion Equation(22)(22) (22) becomes where and
Example 3.17.
Let In this case Further, choose in Corollary 3.16 to get
Corollary 3.18.
When and ζ = 1, in Theorem 3.14, then set inclusions Equation(22)(22) (22) take the form where and
Example 3.19.
Choose In this case Also assume in Corollary 3.18 to get
Theorem 3.20.
Let be two interval valued functions which are h-convex with and are positive right dense continuous functions and . If , then (23) (23)
Proof.
By hypothesis, one has that
Then
Integration over [0,1], gives
It concludes the proof. □
Corollary 3.21.
If we choose time scale to be set of real numbers in Theorem , it becomes (Zhao et al., Citation2018, Theorem 4.6).
Corollary 3.22.
When and in Theorem 3.20, then set inclusion Equation(23)(23) (23) becomes
Example 3.23.
Let In this case Further, choose in Corollary 3.22 to get
4. Conclusion
We have studied the h-convex (affine, concave) functions for interval valued functions. We proved Jensen’s type inequalityin for interval valued h-convex functions on time scales. In seek of applications, we proved interval valued Hermite-Hadamard type inequalities on time scales. Present results generalize the existing inequalities given in Zhao et al. (Citation2018). We have discussed all the obtained inequalities by choosing for Which are also new up to knowledge of authors. Some numerical examples are also provided to illustrate the results.
Disclosure statement
No potential conflict of interest was reported by the authors.
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