Sharpening Littlewood subordination principle with univalent symbol

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