Polynomial Growth of Complex Polynomial Hardy--Littlewood Constants

Abstract

We prove polynomial growth bounds for the optimal constants in the complex polynomial Hardy--Littlewood inequality whenever $p\ge c{m}^2/\log m$, for every fixed $c>0$. This extends the recently established polynomial growth of the complex polynomial Bohnenblust--Hille constants at $p=\infty$ to finite values of $p$. Moreover, when $p/m^2\to \infty$, the Hardy--Littlewood constants are bounded by $(1+o(1))$ times the corresponding Bohnenblust--Hille constants. For real scalars, whenever $p_m/m\to \infty$, the optimal constants satisfy $H_{m,p_m}^{\mathrm{pol}}(\mathbb{R})=2^{m+o(m)}$.