IJPAM: Volume 105, No. 2 (2015)

BOUNDS FOR A TOADER-TYPE MEAN BY
ARITHMETIC AND CONTRAHARMONIC MEANS

Nali Li$^1$, Tiehong Zhao$^2$, Yile Zhao$^3$
$^{1,2}$Department of Mathematics
Hangzhou Normal University
Hangzhou, 311121, P.R. CHINA


Abstract. In this paper, we present the best possible parameters $\alpha_{i}$ and $\beta_{i}$ with $i=1,2,3,4$ such that the double inequalities
\begin{align*} \alpha_{1}A(a,b)+\left(1-\alpha_{1}\right)C(a,b)&<T[A(a,b), C(a,... ...\ &\hspace{1.2cm}<C[\beta_4a+(1-\beta_4)b,\beta_4b+(1-\beta_4)a] \end{align*}
hold for all $a, b>0$ with $a\neq b$, as consequences, we provide several new bounds for the complete elliptic integral $\mathcal{E}(r)=\int_{0}^{\pi/2}(1-r^{2}\sin^{2}\theta)^{1/2}d\theta$ $(r\in (0, \sqrt{3}/2)$ of the second kind, where $T(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta}d\theta$, $A(a,b)=(a+b)/2$ and $C(a,b)=(a^{2}+b^{2})/(a+b)$ are the Toader, arithmetic and contraharmonic means of $a$ and $b$, respectively.

Received: September 20, 2015

AMS Subject Classification: 26E60

Key Words and Phrases: arithmetic mean, Toader mean, contraharmonic mean

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DOI: 10.12732/ijpam.v105i2.11 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: 2015
Volume: 105
Issue: 2
Pages: 257 - 268


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