A class of generalized convex functions, the hyperbolic-concave functions, is defined, and used to characterize the collection of Hardy-Littlewood maximal functions. These maximal functions and the ...

Let f be an analytic and univalent function on a simply connected domain D, and let λD be the hyperbolic metric on D. We prove the sharp inequality $\left| {\frac{{{f^n}\left( w \right)}}{{f\left( w ...