Notes on convex functions of order $\alpha$

Abstract

Marx and Strohhäcker 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 $3/5.$

12 pages