Lower bounds for the complex polynomial Hardy--Littlewood inequality

The Hardy--Littlewood inequality for complex homogeneous polynomials asserts that given positive integers $m\geq2$ and $n\geq1$, if $P$ is a complex homogeneous polynomial of degree $m$ on $\ell_{p}^{n}$ with $2m\leq p\leq\infty$ given by $P(x_{1},\ldots,x_{n})=\sum_{|\alpha|=m}a_{\alpha }\mathbf{{x}^{\alpha}}$, then there exists a constant $C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq1$ (which is does not depend on $n$) such that $$\left( {\sum\limits_{\left\vert \alpha\right\vert =m}}\left\vert a_{\alpha }\right\vert ^{\frac{2mp}{mp+p-2m}}\right) ^{\frac{mp+p-2m}{2mp}}\leq C_{\mathbb{C},m,p}^{\mathrm{pol}}\left\Vert P\right\Vert ,$$ with $\Vert P\Vert:=\sup_{z\in B_{\ell_{p}^{n}}}|P(z)|$. In this short note, among other results, we provide nontrivial lower bounds for the constants $C_{\mathbb{C},m,p}^{\mathrm{pol}}$. For instance we prove that, for $m\geq2$ and $2m\leq p<\infty$, $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m}{p}}$$ for $m$ even, and $$C_{\mathbb{C},m,p}^{\mathrm{pol}}\geq2^{\frac{m-1}{p}}$$ for $m$ odd. Estimates for the case $p=\infty$ (this is the particular case of the complex polynomial Bohnenblust--Hille inequality) were recently obtained by D. Nu\~nez-Alarc\'on in 2013.