Sharpening Littlewood subordination principle with univalent symbol

Let u$u$ be a subharmonic function on the open unit disc D$\mathbb {D}$, centered at the origin of the complex plane, and let φ:D→D$\varphi:\mathbb {D}\rightarrow \mathbb {D}$ be a holomorphic function such that φ(0)=0$\varphi (0)=0$. A classical result, known as Littlewood subordination principle, states Iu∘φ(r)⩽Iu(r)$\mathrm{I}_{u\circ \varphi }(r)\leqslant \mathrm{I}_u(r)$, where Iu∘φ(r)$\mathrm{I}_{u\circ \varphi }(r)$ and Iu(r)$\mathrm{I}_u(r)$ are integral means over the circle of radius 0⩽r