Year: 2013
Analysis in Theory and Applications, Vol. 29 (2013), Iss. 1 : pp. 37–46
Abstract
Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|< 1$, then for every real or complex number $\beta$ with $|\beta|\leq 1$, and $|z|=1$, $R\geq 1$, it is proved by Dewan et al. [4] that$$\Big|P(Rz)+\beta\Big(\frac{R+1}{2}\Big)^n P(z)\Big|\leq\frac{1}{2}\Big\{\Big(\Big|R^n+\beta\Big(\frac{R+1}{2}\Big)^n\Big|+\Big|1+\beta\Big(\frac{R+1}{2}\Big)^n\Big|\Big)\max_{|z|=1}|P(z)|$$ $$-\Big(\Big|R^n+\beta\Big(\frac{R+1}{2}\Big)^n\Big|-\Big|1+\beta\Big(\frac{R+1}{2}\Big)^n\Big|\Big)\min_{|z|=1}|P(z)|\Big\}.$$ In this paper we generalize the above inequality for polynomials having no zeros in $|z| < k$, $k\leq 1$. Our results generalize certain well-known polynomial inequalities.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2013.v29.n1.5
Analysis in Theory and Applications, Vol. 29 (2013), Iss. 1 : pp. 37–46
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 10
Keywords: Polynomial inequality maximum modulus growth of polynomial.