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 Kq(1−q1+q2)(0<q<1), the class of normalized univalent q-convex functions of order (1−q)/(1+q2). We provide examples that exhibit a Marx-Strohhäcker type inclusion relation, i.e. Kq(1−q1+q2)⊂Sq⁎(11+q), where Sq⁎(11+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⁎(12), of Marx and Strohhäcker.