I have been trying to prove for any $\delta>0,$ $$ \int_0^{2\pi}\left|1+ e^{i\theta}f(e^{i\theta})\right|^{\delta}d\theta\leq \int_0^{2\pi}\left|1+e^{i\theta}\right|^{\delta}d\theta $$ for any analytic function $f(z)$ in $|z|\leq 1$ with $|f(e^{i\theta})|\leq 1.$ Can anyone help me in this?