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  Volume 9, Issue 3, Article 76
 
Ratio Vectors of Polynomial-Like Functions
    Authors: Alan Horwitz,  
    Keywords: Polynomial, Real roots, Ratio vector, Critical points.  
    Date Received: 26/08/06  
    Date Accepted: 29/07/08  
    Subject Codes:

26C10.

  
    Editors: Sever S. Dragomir,  
 
    Abstract:

Let be a hyperbolic polynomial-like function of the form $ p(x)=(x-r_{1})^{m_{1}}\cdots (x-r_{N})^{m_{N}},$ where $ m_{1},\dots ,m_{N}$ are given positive real numbers and . Let $ x_{1}<x_{2}<\cdots <x_{N-1}$ be the $ N-1$ critical points of lying in $ I_{k}=(r_{k},r_{k+1}),k=1,2,\dots ,N-1$. Define the ratios We prove that . These bounds generalize the bounds given by earlier authors for strictly hyperbolic polynomials of degree . For , we find necessary and sufficient conditions for to be a ratio vector. We also find necessary and sufficient conditions on $ m_{1},m_{2},m_{3}$ which imply that . For , we also give necessary and sufficient conditions for to be a ratio vector and we simplify some of the proofs given in an earlier paper of the author on ratio vectors of fourth degree polynomials. Finally we discuss the monotonicity of the ratios when .

         
       
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