Abstract
The purpose of this paper is to show some interesting properties of hyperbolic univalent convex differential functions in the case of estimating the real and imaginary parts of a hyperbolic univalent convex function using the definition of invariant differential operator of the first and second-order as stated in theorems (2.1) and (2.2). In addition, the same differential operator has been analyzed through an estimation of the differential operator when to be be a holomorphic function for each , except some poles z, and also in case of being a locally univalent function defined in the unit disk .