Estimates for Initial Coefficients of Certain Starlike Functions with Respect to Symmetric Points

Abstract

It is well known that the class of all analytic functions f defined on the unit disk satisfying $\mathfrak {R}(zf'(z)/(f(z)-f(-z)))>0$ is a subclass of close-to-convex functions and the $n^{th}$ Taylor coefficient of these functions are bounded by one. However, no bounds for the $n^{th}$ coefficients of functions f satisfying $2zf'(z)/(f(z)-f(-z))\prec \varphi (z)$ are known except for $n= 2,3$. The sharp bounds for the fourth coefficient of analytic univalent functions f satisfying the subordination $2zf'(z)/(f(z)-f(-z))\prec \varphi (z)$ is obtained. The bound for the fifth coefficients is also obtained in certain special cases including $\varphi$ is $e^z$ and $\sqrt{1+z}$.