Notes on convex functions of order $\alpha$

TL;DR

This paper proves a conjecture regarding subordination properties of convex functions of order b5 and explores their geometric characteristics, including conditions for averages of such functions to be starlike.

Contribution

It confirms the conjecture that subordination holds for all b5 in (0,1/2) and analyzes geometric properties of convex functions of order b5.

Findings

Proved the conjecture for all b5 in (0,1/2).

Established that the average of two convex functions of order 3/5 is starlike.

Abstract

Marx and Strohh\"acker showed around in 1933 that $f(z)/z$ is subordinate to $1/(1-z)$ for a normalized convex function $f$ on the unit disk $|z|<1.$ Brickman, Hallenbeck, MacGregor and Wilken proved in 1973 further that $f(z)/z$ is subordinate to $k_\alpha(z)/z$ if $f$ is convex of order $\alpha$ for $1/2\le\alpha<1$ and conjectured that this is true also for $0<\alpha<1/2.$ Here, $k_\alpha$ is the standard extremal function in the class of normalized convex functions of order $\alpha$ and $k_0(z)=z/(1-z).$ We prove the conjecture and study geometric properties of convex functions of order $\alpha.$ In particular, we prove that $(f+g)/2$ is starlike whenever $f$ and $g$ both are convex of order $3/5.$