Application of the subordination principle to the multivalent harmonic mappings with shear construction method

Abstract

The harmonic function in the open unit disc D = { z ∈ C | | z | < 1 } can be written as a sum of an analytic and an anti-analytic function, f = h ( z ) + g ( z ) ¯ , where h ( z ) and g ( z ) are analytic functions in D , and are called the analytic part and co-analytic part of f , respectively. One of the most important questions in the study of the classes of such functions is related to bounds on the modulus of functions (growth) or modulus of the derivative (distortion), because the growth theorem and distortion theorem give the compactness of the classes of these functions. In this paper we consider both of these questions with the shear construction method.