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L^r inequalities for the derivative of a polynomial


Abstract


Let p(z) be a polynomial of degree n having no zero in |z|< k, k\leq 1, then Govil [Proc. Nat. Acad. Sci., \textbf{50}, (1980), 50-52] proved

\max\limits_{|z|=1}|p'(z)|\leq \dfrac{n}{1+k^{n}}\max\limits_{|z|=1}|p(z)|,

provided |p'(z)| and |q'(z)| attain their maxima at the same point on the circle |z|=1, where

\label{A}q(z)=z^{n}\overline{p\left(\frac{1}{\overline{z}}\right)}.

In this paper, we not only obtain an integral mean inequality for the above inequality but also extend an improved version of it into L^{r} norm.


DOI Code: 10.1285/i15900932v41n2p19

Keywords: Inequalities; Polynomials; Zeros; Maximum modulus; Lr norm

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