Authors: N. A. Rather, Suhail Gulzar
Let $ P(z) $ be a polynomial of degree $ n $ having all zeros in $|z|\leq k$ where $k\leq 1,$ then it was proved by Dewan \textit{et al} \cite{d} that for every real or complex number $\alpha$ with $|\alpha|\geq k$ and each $r\geq 0$ $$ n(|\alpha|-k)\left\{\int\limits_{0}^{2\pi}\left|P\left(e^{i\theta}\right)\right|^r d\theta\right\}^{\frac{1}{r}}\leq\left\{ \int\limits_{0}^{2\pi}\left|1+ke^{i\theta}\right|^r d\theta\right\}^{\frac{1}{r}}\underset{|z|=1}{Max}|D_\alpha P(z)|. $$ \indent In this paper, we shall present a refinement and generalization of above result and also extend it to the class of polynomials $P(z)=a_nz^n+\sum_{\nu=\mu}^{n}a_{n-\nu}z^{n-\nu},$ $1\leq\mu\leq n,$ having all its zeros in $|z|\leq k$ where $k\leq 1$ and thereby obtain certain generalizations of above and many other known results.
Comments: 8 Pages.
Download: PDF
[v1] 2013-02-20 21:39:58
Unique-IP document downloads: 129 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.