An analog of H\"older's inequality for the spectral radius of Hadamard products

We prove new inequalities related to the spectral radius $\rho$ of Hadamard products (denoted by $\circ$) of complex matrices. Let $p,q\in [1,\mathrm{\infty }]$ satisfy $\frac{1}{p}+\frac{1}{q}=1$, we show an analog of H"older’s inequality on the space of $n\times n$ complex matrices $$\rho (A\circ B)\le \rho (\mathrm{\mid }A{\mathrm{\mid }}^{\circ p}{)}^{\frac{1}{p}}\rho (\mathrm{\mid }B{\mathrm{\mid }}^{\circ q}{)}^{\frac{1}{q}}\text{for all }A,B\in {\mathbb{C}}^{n\times n}\text{,}$$ where $\mathrm{\mid }\cdot \mathrm{\mid }$ denotes entry-wise absolute values, and $(\cdot {)}^{\circ p}$ represents the entry-wise Hadamard power. We derive a sharper inequality for the special case $p=q=2$. Given $A,B\in {\mathbb{C}}^{n\times n}$, for some $\beta \in (0,1]$ depending on $A$ and $B$, $$\rho (A\circ B)\le \beta \rho (\mathrm{\mid }A\circ A\mathrm{\mid }{)}^{\frac{1}{2}}\rho (\mathrm{\mid }B\circ B\mathrm{\mid }{)}^{\frac{1}{2}}\mathrm{.}$$ Analysis for another special case $p=1$ and $q=\mathrm{\infty }$ is also included.