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  Volume 9, Issue 1, Article 25
An Operator Preserving Inequalities Between Polynomials

    Authors: W. M. Shah, A. Liman,  
    Keywords: Polynomials, $B$ operator, Inequalities in the complex domain.  
    Date Received: 15/07/2007  
    Date Accepted: 01/02/2008  
    Subject Codes:

30A06, 30A64.

    Editors: Ioan Gavrea,  

Let $ P(z)$ be a polynomial of degree at most $ n.$ We consider an operator $ B, $ which carries a polynomial $ P(z)$ into

$displaystyle B left [P(z) right ] := lambda_0 P(z) + lambda_1 left (frac... ...!} + lambda_2 left (frac{nz}{2} right )^2 frac{P^{prime prime}(z)}{2!} ,$    

where $ lambda_0,$ $ lambda_1$ and $ lambda_2$ are such that all the zeros of
$displaystyle u(z) = lambda_0 + c (n,1) lambda_1 z + c (n, 2) lambda_2 z^2$    

lie in the half plane
$displaystyle vert zvert leq left vert z - frac{n}{2} right vert.$    

In this paper, we estimate the minimum and maximum modulii of $ B [P(z)]$ on $ vert zvert = 1$ with restrictions on the zeros of $ P(z)$ and thereby obtain compact generalizations of some well known polynomial inequalities.

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