TY - JOUR
T1 - Some Generalization for an Operator Which Preserving Inequalities Between Polynomials
AU - A. Zireh, S. Kumar & K. K. Dewan
JO - Analysis in Theory and Applications
VL - 3
SP - 260
EP - 274
PY - 2014
DA - 2014/10
SN - 30
DO - http://doi.org/10.4208/ata.2014.v30.n3.2
UR - https://global-sci.org/intro/article_detail/ata/4490.html
KW - $B$-operator, inequality, polynomial, maximum modulus, restricted zeros.
AB - <p style="text-align: justify;">For a polynomial $p(z)$ of degree $n$ which has no zeros in $|z|&lt; 1$, 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$$\Big|zp&#39;(z)+\frac{n\beta}{2}p(z)\Big|\leq\frac{n}{2}\Big\{\Big(\Big|\frac{\beta}{2}\Big|+\Big|1+\frac{\beta}{2}\Big|\Big)\max_{|z|=1}|p(z)|-\Big(\Big|1+\frac{\beta}{2}\Big|-\Big|\frac{\beta}{2}\Big|\Big)\min_{|z|=1}|p(z)|\Big\},$$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$$B[p(z)]:=\lambda_0p(z)+\lambda_1\Big(\frac{nz}{2}\Big)\frac{p&#39;(z)}{1!}+\lambda_2\Big(\frac{nz}{2}\Big)^2\frac{p&#39;&#39;(z)}{2!}, $$ 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.</p>