Notes on Convex Functions of Order $$\alpha $$ α

Abstract

Marx and Strohhacker showed in 1933 that f(z) / z is subordinate to $$1/(1-z)$$ for a normalized convex function f on the unit disk $$|z|<1.$$ In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved 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 both f and g are convex of order 3 / 5.