Advances on the coefficient bounds for m-fold symmetric bi-close-to-convex functions

Abstract

In 1955, Waadeland considered the class of m-fold symmetric starlike functions of the form $f_{m}(z)=z+\sum_{n=1}^{\infty }a_{mn+1}z^{mn+1}$; $m\geq 1$; $|z|\lt1$ and obtained the sharp coefficient bounds $|a_{mn+1}|\leq\left[ (2/m+n-1)!\right] /\left[ (n!)(2/m-1)!\right] $. Pommerenke in 1962, proved the same coefficient bounds for m-fold symmetric close-to-convex functions. Nine years later, Keogh and Miller confirmed the same bounds for the class of m-fold symmetric Bazilevic functions. Here we will show that these bounds can be improved even further for the m-fold symmetric bi-close-to-convex functions. Moreover, our results improve those corresponding coefficient bounds given by Srivastava et al that appeared in 7(2) (2014) issue of this journal. A function is said to be bi-close-to-convex in a simply connected domain if both the function and its inverse map are close-to-convex there.