The following inequality appears in the proof of certain isoperimetric-type inequalities for analytic functions in two dimensions:

$$\sum_{m=0}^{\infty}\frac{|c_m|^2}{m+1} \leq \pi \left(\sum_{m=0}^{\infty}|a_m|^2 \right)^2,$$ where $$c_m=a_0a_m+a_1 a_{m-1}+ \dots +a_ma_0.$$

It sounds like a basic inequality, but I wasn't able to prove it. Does this inequality have a name? Where I can find a proof or how can one prove it?