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Redheffer Type Inequality for Bessel Functions  
 
  Authors: Árpád Baricz,  
  Keywords: Bessel functions, Modified Bessel functions, Redheffer's inequality.  
  Date Received: 23/08/06  
  Date Accepted: 09/02/07  
  Subject Codes:

33C10, 26D05.

  
  Editors: Feng Qi,  
 
  Abstract:

In this short note, by using mathematical induction and infinite product representations of the functions $ $ mathcal{J}_p:$ mathbb{R} $ rightarrow(-$ infty,1]$ and $ $ mathcal{I}_p:$ mathbb{R}$ rightarrow[1,$ infty),$ defined by

$ displaystyle $ mathcal{J}_p(x)=2^p$ Gamma(p+1)x^{-p}J_p(x) $ and$ displaystyle $ $ mathcal{I} _p(x)=2^p$ Gamma(p+1)x^{-p}I_p(x), $
an extension of Redheffer's inequality for the function $ $ mathcal{J}_p$ and a Redheffer-type inequality for the function $ $ mathcal{I}_p$ are established. Here $ J_p$ and $ I_p,$ denotes the Bessel function and modified Bessel function, while $ $ Gamma$ stands for the Euler gamma function. At the end of this work a lower bound for the $ $ Gamma$ function is deduced, using Euler's infinite product formula. Our main motivation to write this note is the publication of C.P. Chen, J.W. Zhao and F. Qi [2], which we wish to complement.;


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