IJPAM: Volume 94, No. 4 (2014)

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

Gao Shaoqin$^1$, Song Lingling$^2$, You Mengna$^3$
$^{1,2,3}$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) < P(a,b) < \beta C(a,b) + (1 - \beta )H(a,b)\end{displaymath}

holds for all $a,b > 0$ with $a \ne b$ . Here ${\mathop{\rm C}\nolimits} (a,b)$ , ${\mathop{\rm H}\nolimits} (a,b)$ and ${\mathop{\rm P}\nolimits} (a,b)$ denote the contraharmonic, harmonic, and the Seiffert means of two positive numbers $a$ and $b$ respectively.

Received: April 7, 2014

AMS Subject Classification: 26D15

Key Words and Phrases: optimal convex combination bound, contraharmonic mean, harmonic mean, the Seiffert means

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DOI: 10.12732/ijpam.v94i4.10 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: 94
Issue: 4
Pages: 541 -

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