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Inequalities of Jensen-Pečaric-Svrtan-Fan Type  
 
  Authors: Chaobang Gao, Jiajin Wen,  
  Keywords: Jensen's inequality, Pe{c}ari'c-Svrtan's inequality, Fan's inequality, Theory of majorization, Hermite matrix.  
  Date Received: 22/01/07  
  Date Accepted: 06/07/08  
  Subject Codes:

26D15, 26E60.

  
  Editors: Sever S. Dragomir,  
 
  Abstract:

By using the theory of majorization, the following inequalities of Jensen-Pecaric-Svrtan-Fan type are established: Let $ I$ be an interval, $ f: Iightarrow mathbb{R}$ and $ tin I, x,a,bin I^n$. If $ a_1leq cdotsleq a_nleq b_nleq cdotsleq b_1, a_1+b_1leq cdotsleq a_n+b_n; f(t)>0, f^{prime}(t)>0, f^{primeprime}(t)>0, f^{primeprimeprime}(t) for any <IMG WIDTH= then

$displaystyle frac{f(A(a))}{f(A(b))}=frac{f_{n,n}(a)}{f_{n,n}(b)}leq cdots... ...{k,n}(b)}leq cdotsleq frac{f_{1,n}(a)}{f_{1,n}(b)}=frac{A(f(a))}{A(f(b))},$    

the inequalities are reversed for $ f^{primeprime}(t)0, f^{primeprimeprime}(t)0, forall tin I$, where $ A(cdot)$ is the arithmetic mean and
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