IJPAM: Volume 92, No. 4 (2014)

OPTIMAL CONVEX COMBINATION BOUNDS OF
THE CONTRAHARMONIC AND HARMONIC MEANS
FOR THE WEIGHTED GEOMETRIC MEAN

Shaoqin Gao$^1$, Lingling Song$^2$, Mengna You$^3$
$^{1,2,3}$College of Mathematics and Computer Sciences
Hebei University
Baoding, 071002, P.R. CHINA


Abstract. We find the greatest value $\alpha$ and the least value $\beta$ such that the double inequality

\begin{displaymath}\alpha C(a,b) + (1 - \alpha )H(a,b) < S(a,b) < \beta C(a,b) + (1 - \beta )H(a,b)\end{displaymath}

holds for all $a,b>0$ with $a\neq b$. Here $C(a,b)$, $H(a,b)$ and $S(a,b)$ denote the contraharmonic, harmonic, and the weighted geometric means of two positive numbers $a$ and $b$ respectively.

Received: January 26, 2014

AMS Subject Classification: 65N15

Key Words and Phrases: optimal convex combination bound, contraharmonic mean, harmonic mean, the weighted geometric mean

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DOI: 10.12732/ijpam.v92i4.15 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 92
Issue: 4
Pages: 599 - 607

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).