Determination of Some Properties of Starlike and Close-to-Convex Functions According to Subordinate Conditions with Convexity of a Certain Analytic Function

Investigation of the theory of complex functions is one of the most fascinating aspects of the theory of complex analytic functions of one variable. It has a huge impact on all areas of mathematics. Numerous mathematical concepts are explained when viewed through the theory of complex functions. Let \(f\left(z\right)\in A, f\left(z\right)=z+{\sum }_{n\ge 2}^{\infty }{a}_{n}{z}^{n},\) be an analytic function in an open unit disc U = {z : |z| < 1, z ∈ ℂ} normalized by f(0) = 0 and f′(0) = 1. For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where r is a positive integer of order \({2}^{-r}\left(0<{2}^{-r}\le \frac{1}{2}\right).\) By using subordination, we propose a criterion for f(z) ∈ S^*[a^r, b^r]. The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study their regional transformations. In addition, the properties of convexity are examined for f(z) ∈ A.