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