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 \(\sqrt{1+z}\).