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 $p\geq c m^2/\log m$, 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 $H^{\rm pol}_{m,p_m}(\mathbb R)=2^{m+o(m)}$.