Integral transforms of functions with restricted derivatives
We show that functions whose derivatives lie in a half-plane are preserved under the Pommerenke, Chandra–Singh, Libera, Alexander and Bernardi integral transforms. We determine precisely how these transforms act on such functions. We prove that if the derivative of a function lies in a convex region then the derivative of its Pommerenke, Chandra–Singh, Libera, Alexander and Bernardi transforms lie in a strictly smaller convex region which can be determined. We also consider iterates of these transforms. Best possible results are obtained.