Some invariant differential operator properties for hyperbolic univalent convex functions

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 .