JIPAM

A Refinement of Jensen's Inequality  
 
  Authors: Jamal Rooin,  
  Keywords: Product measure, Fubini's Theorem, Jensen's inequality.  
  Date Received: 27/08/04  
  Date Accepted: 16/03/05  
  Subject Codes:

Primary: 26D15, 28A35.

  
  Editors: Charles E. M. Pearce,  
 
  Abstract:

We refine Jensen's inequality as

$displaystyle varphileft(int_X f d muright) leq int_Yvarphileft(int_X f(x)omega(x,y)dmu(x)right)dlambda(y)leq int_X(varphicirc f)dmu, $

where $ (X,mathcal{A},mu)$ and $ (Y,mathcal{B},lambda)$ are two probability measure spaces, $ omega:Xtimes Yrightarrow[0,infty)$ is a weight function on $ Xtimes Y$, $ I$ is an interval of the real line, $ fin L^1(mu), f(x)in I$ for all $ xin X$ and $ varphi$ is a real-valued convex function on $ I$. ;


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