Growth of the Maximum Modulus of Polynomials With Prescribed Zeros - Dewan, K. K.; Hans, Sunil (Sarajevo Journal of Mathematics, 2024)
If $p(z)$ be a polynomial of degree $n$ which does not vanish in the disk $\left|z\right|<k$, then for $k=1$, it is well known that \begin{align*} \max_{\left|z\right|=r<1}\left|p(z)\right|&\geq \left(\frac{r+1}{2}\right)^n\max_{\left|z\right|=1}\left|p(z)\right|,\\ \intertext{and} \max_{\left|z\right|=R>1}\left|p(z)\right|&\leq \frac{R^n+1}{2}\max_{\left|z\right|=1}\left|p(z)\right|.\end{align*} In this paper, we consider a class of lacunary polynomials and present certain generalizations as well as improvements of the above inequalities for the two cases $k\geq 1$ and $k<1$. 2000 Mathematics Subject Classification. 30A10, 30C10, 30C15