Sharpening Littlewood subordination principle with univalent symbol

Abstract

Let (Formula presented.) be a subharmonic function on the open unit disc (Formula presented.), centered at the origin of the complex plane, and let (Formula presented.) be a holomorphic function such that (Formula presented.). A classical result, known as Littlewood subordination principle, states (Formula presented.), where (Formula presented.) and (Formula presented.) are integral means over the circle of radius (Formula presented.) centered at the origin, of the functions (Formula presented.) and (Formula presented.), respectively. In this note, we obtain an unexpected improvement of Littlewood subordination principle in the case when the function (Formula presented.) is univalent, by proving that (Formula presented.) where (Formula presented.). We also list some applications of this result including an improved variant of Rogosinski theorem with univalent symbol.