Some new estimations on the spectral radius of the Schur product of matrices

The present study investigates the Schur product of multiple nonnegative matrices ${\mathit{G}}_1$, ${\mathit{G}}_2$, $\cdots$, ${\mathit{G}}_n$. By utilizing the Perron root estimation for nonnegative matrices and applying the Hölder inequality, we establish some upper bounds on $\rho \left({\mathit{G}}_1\circ {\mathit{G}}_2\circ \cdots \circ {\mathit{G}}_n\right)$. These novel findings encompass and extend certain earlier results. Some comparative analysis between our new results and existing results is conducted through numerical experiments. Theoretical analysis and data calculations demonstrate that our results outperform those reported in other studies.