An inequality associated with $\mathcal{Q}_p$ functions

Abstract: The M\"obius invariant space $\mathcal{Q}p$, $0<p<\infty$, consists of functions $f$ which are analytic in the open unit disk $\mathbb{D}$ with $$ |f|{\mathcal{Q}p}=|f(0)|+\sup{w\in \D} \left(\int_\D |f'(z)|^2(1-|\sigma_w(z)|^2)^p dA(z)\right)^{1/2}<\infty, $$ where $\sigma_w(z)=(w-z)/(1-\overline{w}z)$ and $dA$ is the area measure on $\mathbb{D}$. It is known that the following inequality $$ |f(0)|+\sup_{w\in \D} \left(\int_\D \left|\frac{f(z)-f(w)}{1-\overline{w}z}\right|^2 (1-|\sigma_w(z)|^2)^p dA(z)\right)^{1/2} \lesssim |f|_{\mathcal{Q}_p} $$ played a key role to characterize multipliers and certain Carleson measures for $\mathcal{Q}_p$ spaces. The converse of the inequality above is a conjectured-inequality in [14]. In this paper, we show that this conjectured-inequality is true for $p>1$ and it does not hold for $0<p\leq 1$.