Notes on Convex Functions of Order α

Marx and Strohhacker showed in 1933 that f(z) / z is subordinate to for a normalized convex function f on the unit disk In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to if f is convex of order for and conjectured that this is true also for Here, is the standard extremal function in the class of normalized convex functions of order and We prove the conjecture and study geometric properties of convex functions of order In particular, we prove that is starlike whenever both f and g are convex of order 3 / 5.