A conjecture on Marx-Strohhäcker type inclusion relation between q-convex and q-starlike functions

Abstract We prove that the class K of normalized univalent convex functions defined in the unit disk E is contained in K q ( 1 − q 1 + q 2 ) ( 0 q 1 ) , the class of normalized univalent q-convex functions of order ( 1 − q ) / ( 1 + q 2 ) . We provide examples that exhibit a Marx-Strohhacker type inclusion relation, i.e K q ( 1 − q 1 + q 2 ) ⊂ S q ⁎ ( 1 1 + q ) , where S q ⁎ ( 1 1 + q ) is the class of q-starlike functions of order 1 / ( 1 + q ) . Note that for q → 1 − this relation coincides with the well-known result, K ⊂ S ⁎ ( 1 2 ) , of Marx and Strohhacker.