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