Contractive Hardy–Littlewood inequality

Let $H^p(\mathbb{T})$ denote the Hardy space on the unit circle, with $0<p\leq 2$. For $f(z)=a_0+a_1z+a_2z^2+\ldots\in H^p(\mathbb{T})$, define

$$c_\alpha(n)=\binom{n+\alpha-1}{n}.$$

Contractive Hardy–Littlewood inequality. One has

$$\sum_{n=0}^{\infty}\frac{|a_n|^2}{c_{2/p}(n)}\leq \lVert f\rVert_p^2.$$

This is a more precise, contractive form of the classical Hardy–Littlewood inequality. The source presents it as a conjecture previously made in the cited work; the paper's abstract says that the inequality is proved, so the conjecture is resolved.