JIPAM

On the Maximum Modulus of Polynomials. II  
 
  Authors: M. A. Qazi,  
  Keywords: Polynomials, Inequality, Zeros.  
  Date Received: 15/02/07  
  Date Accepted: 23/08/07  
  Subject Codes:

30D15, 41A10, 41A17.

  
  Editors: Narendra K. Govil,  
 
  Abstract:

Let $ f (z) := sum_{nu=0}^n a_nu z^nu$ be a polynomial of degree $ n$ having no zeros in the open unit disc, and suppose that $ max_{vert zvert=1} vert f(z)vert = 1$. How small can $ max_{vert zvert=rho} vert f(z)vert$ be for any $ rho in [0 , 1)$? This problem was considered and solved by Rivlin [4]. There are reasons to consider the same problem under the additional assumption that $ f^prime (0) = 0$. This was initiated by Govil [2] and followed up by the present author [3]. The exact answer is known when the degree $ n$ is even. Here, we make some observations about the case where $ n$ is odd. ;


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