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? 



Proof of a discrete isoperimetric inequality
Suppose you have a power series with coefficients $a\_n$
$$ f(z):= \sum\_{k=1}^\infty a\_k z^k .$$
Then the coefficients of $f^2$ are exactly $c\_n$. Also if we denote by $\odot$ the Hadamard multiplication of powerseries (coefficient-wise or equivalently convolution of the boundary values), $c\_k^2$ are the coefficients of $f^2\odot f^2$. We apply first Hardy's inequality (Theory of Hp spaces, Duren, Corollary of Theorem 3.15) and then Young's inequality for convolution
\begin{align\*} \sum\_{k=1}^\infty \frac{|c\_k|^2}{k+1} & \leq \pi \Vert f^2 \odot f^2 \Vert\_{H^1} \\ 
& \leq \pi \Vert f^2 \Vert\_{H^1} \Vert f^2 \Vert\_{H^1} \\
&= \pi \Vert f \Vert\_{H^2} ^4 \\ & = \pi \Big( \sum\_{k=0}^\infty |a\_k|^2 \Big)^2 \end{align\*}.
If your inequality is true I'm missing a constant of $\pi$.


