Certain Bounds for the Differences of Means

Volume 4, Issue 4, 2003

Article 76

CERTAIN BOUNDS FOR THE DIFFERENCES OF MEANS

PENG GAO

DEPARTMENT OF MATHEMATICS,
UNIVERSITY OF MICHIGAN,
ANN ARBOR, MI 48109, USA.
E-Mail: penggao@umich.edu URL: http://www.math.uvic.ca/faculty/harimsri/

Received 17 August, 2002; Accepted 3 June, 2003.
Communicated by: K.B. Stolarsky

ABSTRACT. Let $P_{n,r}(\mathbf{x})$ be the generalized weighted power means. We consider bounds for the differences of means in the following form:

$\displaystyle \max \left\{\frac {C_{u, v,\beta}}{x^{2\beta-\alpha}_1},\frac {C_{u, v,\beta}}{ x^{2\beta-\alpha}_n} \right\}\sigma_{n, w^{\prime},\beta}$	$\displaystyle \geq \frac { P^{\alpha}_{n,u} - P^{\alpha}_{n,v}}{\alpha}$
	$\displaystyle \geq \min \left\{\frac {C_{u, v,\beta} }{x^{2\beta-\alpha}_1},\frac {C_{u, v,\beta}}{x^{2\beta-\alpha}_n} \right\}\sigma_{n,w,\beta}.$

Here $\beta \neq 0,\sigma_{n,t,\beta}(\mathbf{ x})=\sum_{i=1}^{n} \omega_i[x^{\beta}_i-P^{\beta}_{n,t}(\mathbf{x})]^2$ and $C_{u, v,\beta}= \frac {u-v}{2\beta^2}$ . Some similar inequalities are also considered. The results are applied to inequalities of Ky Fan's type.

Key words:
Ky Fan's inequality, Levinson's inequality, Generalized weighted power means, Mean value theorem.

2000 Mathematics Subject Classification:
Primary 26D15, 26D20.

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