A few results concerning the Schur stability of the Hadamard powers and the Hadamard products of complex polynomials

For a complex polynomial $$f\left(s\right)=s^n+a_{n-1}{s}^{n-1}+\dots +a_{1}s+a_0$$ and for a rational number $p$, we consider the Schur stability problem of the $p$-th Hadamard power of $f$ $$f^{\left[p\right]}\left(s\right)=s^n+a_{n-1}^p{s}^{n-1}+\dots +a_1^{p}s+a_0^p\text{.}$$ We show that there exist two numbers $p^{\ast }\ge 0\ge p_{\ast }$ such that $f^{\left[p\right]}$ is Schur stable for every $p>p^{\ast }$ and is not Schur stable for $p<p_{\ast }$ (or vice versa, depending on $f$). Also, we give simple sufficient conditions for the Schur stability of the Hadamard product of two complex polynomials. Numerical examples complete and illustrate the results.