Convolution and subordination in the convex hull of convex mappings

In this work we give an extension of the Ruscheweyh and Stankiewicz theorem [S. Ruscheweyh, J. Stankiewicz, Subordination under convex univalent functions, Bull. Polish Acad. Sci. Math. 33 (1985) 499–502] on the subordination under convex functions in the unit disc Δ={z:|z|<1}Δ={z:|z|<1}. We prove that if f≺F∈co¯K and g≺G∈co¯K, then f⋆g≺F⋆Gf⋆g≺F⋆G where ≺≺ denotes the subordination, ⋆⋆ denotes the Hadamard product and co¯K is the closed convex hull of the class of convex functions.