Radius of convexity of partial sums of odd functions in the close-to-convex family

TL;DR

This paper investigates the convexity radius of partial sums of a specific class of odd, close-to-convex functions, establishing the exact radius within which all partial sums are convex.

Contribution

It proves the convexity of partial sums within a precise radius for a class of close-to-convex functions and shows this radius is optimal.

Findings

Partial sums are convex in |z|<√2/3.

The radius √2/3 is the best possible.

Provides a sharp bound for convexity of partial sums.

Abstract

We consider the class of all analytic and locally univalent functions $f$ of the form $f(z)=z+\sum_{n=2}^\infty a_{2n-1} z^{2n-1}$, $|z|<1$, satisfying the condition $$ {\rm Re}\,\left(1+\frac{zf^{\prime\prime}(z)}{f^\prime (z)}\right)>-\frac{1}{2}. $$ We show that every section $s_{2n-1}(z)=z+\sum_{k=2}^na_{2k-1}z^{2k-1}$, of $f$, is convex in the disk $|z|<\sqrt{2}/3$. We also prove that the radius $\sqrt{2}/3$ is best possible, i.e. the number $\sqrt{2}/3$ cannot be replaced by a larger one.