Year: 2014
Analysis in Theory and Applications, Vol. 30 (2014), Iss. 3 : pp. 260–274
Abstract
For a polynomial $p(z)$ of degree $n$ which has no zeros in $|z|< 1$, Dewan et al., (K. K. Dewan and Sunil Hans, Generalization of certain well known polynomial inequalities, J. Math. Anal. Appl., 363 (2010), 38-41) established$$\Big|zp'(z)+\frac{n\beta}{2}p(z)\Big|\leq\frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$for any complex number $\beta$ with $|\beta|\leq 1$ and $|z|=1$. In this paper we consider the operator $B$, which carries a polynomial $p(z)$ into$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p'(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p''(z)}{2!}, $$ where $\lambda_0,$ $\lambda_1$, and $\lambda_2$ are such that all the zeros of $u(z)=\lambda_0+c(n,1)\lambda_1z+c(n,2)\lambda_2z^2$ lie in the half plane $|z|\leq |z-{n}/{2}|.$ By using the operator $B$, we present a generalization of result of Dewan. Our result generalizes 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.2014.v30.n3.2
Analysis in Theory and Applications, Vol. 30 (2014), Iss. 3 : pp. 260–274
Published online: 2014-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 15
Keywords: $B$-operator inequality polynomial maximum modulus restricted zeros.
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Integral mean estimates for polar derivative of polynomials
Zireh, Ahmad
Journal of Interdisciplinary Mathematics, Vol. 21 (2018), Iss. 1 P.29
https://doi.org/10.1080/09720502.2014.962841 [Citations: 0]