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Monotonic Refinements of a Ky Fan Inequality  
 
  Authors: Kwok Kei Chong,  
  Keywords: Ky Fan inequality, Monotonic refinements of inequalities, Arithmetic, geometric and harmonic means  
  Date Received: 03/10/00  
  Date Accepted: 02/02/01  
  Subject Codes:

26D15,26A48

  
  Editors: Feng Qi,  
 
  Abstract:

It is well-known that inequalities between means play a very important role in many branches of mathematics. Please refer to [1,3,7], etc. The main aims of the present article are:

(i)
to show that there are monotonic and continuous functions $H(t),;K(t),;P(t)$ and $Q(t)$ on $ left[ 0,1right] $ such that for all $ tin lbrack 0,1],$
$displaystyle H_{n}leq H(t)leq G_{n}leq K(t)leq A_{n} $    
and
$displaystyle H_{n}/(1-H_{n})leq P(t)leq G_{n}/G_{n}^{prime }leq Q(t)leq A_{n}/A_{n}^{prime }, $   

where $A_{n},G_{n}$ and $H_{n}$ are respectively the weighted arithmetic, geometric and harmonic means of the positive numbers $x_{1},x_{2},...,x_{n}$in $(0,1/2],$ with positive weights $alpha _{1},alpha _{2},...,alpha_{n}; $ while $A_{n}^{prime }$ and $G_{n}^{prime }$ are respectively the weighted arithmetic and geometric means of the numbers $ 1-x_{1},;1-x_{2},...,1-x_{n}$ with the same positive weights $alpha _{1},alpha _{2},...,alpha_{n}; $

(ii)
to present more general monotonic refinements for the Ky Fan inequality as well as some inequalities involving means; and

(iii)
to present some generalized and new inequalities in this connection.

[1] H. ALZER, Inequalities for arithmetic, geometric and harmonic means, Bull. London Math. Soc., 22 (1990), 362–366.
[3] P.S. BULLEN, D.S. MITRINOVIC and J.E. PECARIC, Means and Their Inequalities, Reiddel Dordrecht, 1988.
[7] A.M. FINK, J.E. PECARIC and D.S. MITRINOVIC, Classical and New Inequalities in Analysis , Kluwer Academic Publishers, 1993.;


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