On Visser's inequality concerning coefficient estimates for a polynomial (2411.19831v1)

Abstract: If $P(z)=\sum_{j=0}^{n}a_jz^j$ is a polynomial of degree $n$ having no zero in $|z|<1,$ then it was recently proved that for every $p\in[0,+\infty]$ and $s=0,1,\ldots,n-1,$ \begin{align*} \left|a_nz+\frac{a_s}{\binom{n}{s}}\right|{p}\leq \frac{\left|z+\delta{0s}\right|p}{\left|1+z\right|_p}\left|P\right|{p}, \end{align*} where $\delta_{0s}$ is the Kronecker delta. In this paper, we consider the class of polynomials having no zero in $|z|<\rho,$ $\rho\geq 1$ and obtain some generalizations of above inequality.