Inequalities Concerning the Maximum Modulus of Polynomials

Abstract

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤k$, $k≤1$, then for every real or complex number $β$, with $|β|≤1$ and $R≥1$, it was shown by A. Zireh et al. [7] that for $|z|=1$,
$$\min\limits_{|z|=1}\left|P(Rz)+\beta(\frac{R+k}{1+k})^nP(z)\right|\geq k^{-n}\left|R^n+\beta(\frac{R+k}{1+k})^n\right|\min\limits_{|z|=k}|P(z)|.$$
In this paper, we shall present a refinement of the above inequality. Besides, we shall also generalize some well-known results.