  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 $0<p\leq 1$.
