Inequalities Concerning the Maximum Modulus of Polynomials (Analysis in Theory and Applications, 2018)
Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤k$, $k≤1$, thenfor every real or complex number $β$, with $|β|≤1$ and $R≥1$, it was shown by A. Zireh etal. [7] that for $|z|=1$,$$\min\limits_{|z|=1}\left|P(Rz)+β(\frac{R+k}{1+k})^nP(z)\right|≥ k^{-n}\left|R^n+β(\frac{R+k}{1+k})^n\right|\min\limits_{|z|=k}|P(z)|.$$In this paper, we shall present a refinement of the above inequality. Besides, we shallalso generalize some well-known results.
Keywords: Growth of polynomials; minimum modulus of polynomials; inequalities.