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 $\mathbb{D}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$. For $0<p<\mathrm{\infty }$ and $f\in \mathcal{H}(\mathbb{D})$, let $M_p^p(r,f)={\int }_0^{2\pi }\mathrm{\mid }f(r{e}^{i\theta }){\mathrm{\mid }}^p\frac{d\theta }{2\pi }$ and $M_{\mathrm{\infty }}(r,f)={\sup }_{\mathrm{\mid }z\mathrm{\mid }=r}\mathrm{\mid }f(z)\mathrm{\mid }$. For $0<p<q\le \mathrm{\infty }$, 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 $0\le r<\rho \le 1$ 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 $T_{g}f(z)={\int }_0^{z}f(\zeta )g^{\mathrm{\prime }}(\zeta )d\zeta$, $S_{g}f(z)={\int }_0^z{f}^{\mathrm{\prime }}(\zeta )g(\zeta )d\zeta$ and $M_{g}f(z)=f(z)g(z)$, are bounded between two different mixed-norm spaces $A_{\omega }^{p,q}=\{g\in \mathcal{H}(\mathbb{D}):{\int }_0^1{M}_p^q(r,g)\omega (r)dr<\mathrm{\infty }\}$ 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.