Notes on convex functions of order $\alpha$

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