Monotonic Refinements of a Ky Fan Inequality

Volume 2, Issue 2, 2001

Article 19

MONOTONIC REFINEMENTS OF A KY FAN INEQUALITY

DEPARTMENT OF APPLIED MATHEMATICS
THE HONG KONG POLYTECHNIC UNIVERSITY
HUNG HOM, KOWLOON
HONG KONG, CHINA
E-Mail: makkchon@inet.polyu.edu.hk

Received 3 October, 2000; accepted 2 February, 2001.
Communicated by: F. 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

on $\left[ 0,1\right]$ such that for all $%% t\in \lbrack 0,1],$

$\displaystyle H_{n}\leq H(t)\leq G_{n}\leq K(t)\leq A_{n}$

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 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, ReiddelDordrecht, 1988.
[7] A.M. FINK, J.E. PECARIC and D.S. MITRINOVIC, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, 1993.

Key words:
Ky Fan inequality, Monotonic refinements of inequalities, Arithmetic, geometric and harmonic means.

2000 Mathematics Subject Classification:
26D15, 26A48

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