We solve the problem of finding the largest domain D for which, under given ψ and q, the differential subordination ψ(p(z),zp′(z)) ∈ D ⇒ p(z) ≺ q(z), where D and q( ) are regions bounded by conic sections, is satisfied. The shape of the domain D is described by the shape of q( ). Also, we find the best dominant of the differential subordination p(z)+ (zp′(z)/(βp(z)+γ)) ≺ pk(z), when the functio...

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, w...