Volume 2,  Issue 2, 2001

Article 19

MONOTONIC REFINEMENTS OF A KY FAN INEQUALITY

KWOK KEI CHONG

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 $Q(t)$ 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 $(0,1/2],$ 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), 362366.
[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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Other papers in this issue

Power-Monotone Sequences and Fourier Series with Positive Coefficients
J. Nemeth

On Some Fundamental Integral Inequalities and their Discrete Analogues
B.G.  Pachpatte

Subharmonic Functions and their Riesz Measure
Raphaele Supper

A Priori Estimate for a System of Differential Operators
Chikh Bouzar

Improved Inclusion-Exclusion Inequalities for Simplex and Orthant Arrangements
Daniel Q. Naiman and Henry P. Wynn

Monotonic Refinements of a Ky Fan Inequality
Kwok K. Chong

Sub-super Solutions and the Existence of Extremal Solutions in Noncoercive Variational Inequalities
V.K. Le

Refinements of Carleman's Inequality
Bao-Quan Yuan 

Inequalities related to the Chebychev Functional Involving Integrals Over Different Intervals
I. Budimir, P. Cerone, and  J. Pecaric

Some Distortion Inequalities Associated with the Fractional Derivatives of Analytic and Univalent Functions
H.M. Srivastava, Yi Ling and Gejun Bao

Necessary and Sufficient Condition for Existence and Uniqueness of the Solution of Cauchy Problem for Holomorphic Fuchsian Operators
Mekki Terbeche

Bounds for Entropy and Divergence for Distributions over a Two-Element Set
Flemming Topsoe

A Pick Function Related to an Inequality for the Entropy Function
Christian Berg


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