Improvement of a Hardy-Littlewood inequality and applications to the boundedness of analytic paraproducts on mixed norm spaces

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. For $0<p<\infty$ and $f\in\mathcal{H}(\mathbb{D})$, let $M_p^p(r,f)=\int_0^{2\pi}|f(re^{i\theta})|^p \frac{d\theta}{2\pi}$ and $M_\infty(r,f) = \sup_{|z|=r}|f(z)|$. For $0<p<q\leq \infty$, Hardy and Littlewood proved the prevalent inequality $$M_q(r,f)\le C(p,q)\frac{M_p(\rho,f)}{(\rho-r)^{\frac{1}{p}-\frac{1}{q}}}$$ for $0\leq r<\rho\leq 1$ and $f\in\mathcal{H}(\mathbb{D})$. In this paper, we obtain an improvement of this well-known inequality which is employed to characterize the symbols $g\in\mathcal{H}(\mathbb{D})$ such that the analytic paraproducts $T_gf(z)=\int_0^z f(\zeta)g'(\zeta)\,d\zeta$, $S_gf(z)=\int_0^z f'(\zeta)g(\zeta)\,d\zeta$ and $M_gf(z)=f(z)g(z)$, are bounded between two different mixed-norm spaces $A^{p,q}_\omega=\{ g\in \mathcal{H}(\mathbb{D}): \int_0^1 M_p^q(r,g) \omega(r)\,dr<\infty\}$ induced by a radial doubling weight $\omega$. En route to the proof of these characterizations, we consider an open Carleson measure problem posed by Luecking and we solve it in a meaningful particular case.