Polynomial Growth of Complex Polynomial Hardy--Littlewood Constants

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