Notes on convex functions of order α

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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 kα(z)/z if f is convex of order α for 1/2≤α<1 and conjectured that this is true also for 0<α<1/2. Here, kα is the standard extremal function in the class of normalized convex functions of order α and k₀(z)=z/(1-z). We prove the conjecture and study geometric properties of convex functions of order α. In particular, we prove that (f+g)/2 is starlike whenever f and g both are convex of order 3/5.