Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that $$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$$

I wonder if the constant $4\pi$ cound be improved on an annulus $a<z<1$. More precisely, does the following inequality

$$\left (\int_0^{2\pi}|f(e^{i\theta})|d \theta +\int_0^{2\pi}|f(ae^{i\theta}|)d \theta \right)^2 \geq C \iint_{a<|z|<1} |f(r e^{i\theta})|^2r dr d \theta$$

hold for some constant $C(a)> 4\pi$ independent of $f$? Can $C(a)$ be computed in terms of $a$? This seems to be a classical problem but I could not find a reference, and was not able to prove it after trying for a couple of days.