IJPAM: Volume 81, No. 2 (2012)

SOME CONVEX COMBINATION BOUNDS FOR
ARITHMETIC AND THE SECOND SEIFFERT MEANS

Gao Hongya$^1$, Qin Yanli$^2$, Fang Jian$^3$
$^{1,2}$ College of Mathematics and Computer Science
Hebei University
Baoding, 071002, P.R. CHINA
$^3$Department of Mathematics and Computer Science
Hebei Normal College for Nationalities
Chengde, 067000, P.R. CHINA


Abstract. We find the greatest value $\alpha_1$ and the least value $\beta_1$ such that the double inequality

\begin{displaymath}
\alpha_1 C(a,b)+(1-\alpha_1)N(a,b)<A(a,b)<\beta_1
C(a,b)+(1-\beta_1)N(a,b)
\end{displaymath}

holds for all $a,b>0$ with $a\neq b$. We also find the estimate for the least value $\alpha_2$ and the greatest value $\beta_2$ such that the double inequality

\begin{displaymath}
\alpha_2 C(a,b)+(1-\alpha_2)N(a,b)<T(a,b)<\beta_2
C(a,b)+(1-\beta_2)N(a,b)
\end{displaymath}

holds for all $a,b>0$ with $a\neq b$. Here $C(a,b)$, $N(a,b)$, $A(a,b)$ and $T(a,b)$ denote the contraharmonic, square-root, arithmetic, and the second Seiffert means of two positive numbers $a$ and $b$, respectively.

Received: June 28, 2011

AMS Subject Classification: 26D15

Key Words and Phrases: convex combination bound, arithmetic mean, contraharmonic mean, square-root mean, the second Seiffert mean

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 81
Issue: 2