IJPAM: Volume 41, No. 6 (2007)

ON INTEGRAL MEAN ESTIMATES FOR POLYNOMIALS

Barchand Chanam$^1$, K.K. Dewan$^2$
$^{1,2}$Department of Mathematics
Faculty of Natural Sciences
Jamia Millia Islamia - Central University
New Delhi, 110025, INDIA


Abstract.Let $p(z)$ be a polynomial of degree $n$ having all its zeros in $\vert z\vert\le k$, where $k\ge 1$, then it is known that for each $r\ge 1$,

\begin{eqnarray*} n\left\{\int_0^{2\pi}\vert p(e^{i\theta})\vert^r d\theta\right... ...\right\}^{\frac{1}{r}} \max_{\vert z\vert=1} \vert p'(z)\vert\,. \end{eqnarray*}

In this paper, we first obtain an improvement as well as an extension for $r>0$, of the above inequality by considering the class of polynomials of degree $n\ge 3$.

Received: June 16, 2007

AMS Subject Classification: 30A10, 30C10, 30C15

Key Words and Phrases: polynomials, $L^r$ inequalities, maximum modulus

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 41
Issue: 6