Optimal evaluations for the S\'{a}ndor-Yang mean by power mean

In this paper, we prove that the double inequality $M_{p}(a,b)0$ with $a\neq b$ if and only if $p\leq 4\log 2/(4+2\log 2-\pi)=1.2351\cdots$ and $q\geq 4/3$, where $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ $(r\neq 0)$ and $M_{0}(a,b)=\sqrt{ab}$ is the $r$th power mean, $B(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ is the S\'{a}% ndor-Yang mean, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ and $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$.Comment: 9 page

Refinements of the inequalities between Neuman-Sandor, arithmetic, contra-harmonic and quadratic means

In this paper, we prove that the inequalities $\alpha [1/3 Q(a,b)+2/3 A(a,b)]+(1-\alpha)Q^{1/3}(a,b)A^{2/3}(a,b)<M(a,b) <\beta [1/3 Q(a,b)+2/3 A(a,b)]+(1-\beta)Q^{1/3}(a,b)A^{2/3}(a,b)$ and $\lambda [1/6 C(a,b)+5/6 A(a,b)]+(1-\lambda)C^{1/6}(a,b)A^{5/6}(a,b)<M(a,b)<\mu [1/6 C(a,b)+5/6 A(a,b)]++(1-\mu)C^{1/6}(a,b)A^{5/6}(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $\alpha\leq (3-3\sqrt[6]{2}\log(1+\sqrt{2}))/[(2+\sqrt{2}-3\sqrt[6]{2})\log(1+\sqrt{2})]=0.777...$, $\beta\geq 4/5$, $\lambda\leq (6-6\sqrt[6]{2}\log(1+\sqrt{2}))/(7-6\sqrt[6]{2}\log(1+\sqrt{2}))=0.274...$, and $\mu\geq 8/25$. Here, $M(a,b)$, $A(a,b)$, $C(a,b)$, and $Q(a,b)$ denote the Neuman-S\'{a}ndor, arithmetic, contra-harmonic, and quadratic means of $a$ and $b$, respectively.Comment: 9 page

Asymptotical Bounds for Complete Elliptic Integrals of the Second Kind

In this paper, we establish several asymptotical bounds for the complete elliptic integrals of the second kind $\mathcal{E}(r)$, and improve the well-known conjecture $\mathcal{E}(r)>\pi[(1+(1-r^2)^{3/4})/2]^{2/3}/2$ for all $r\in(0,1)$ proposed by M. Vuorinen.Comment: 15 page

A note on the Neuman-S\'andor Mean

In this article, we present the best possible upper and lower bounds for the Neuman-S\'andor mean in terms of the geometric combinations of harmonic and quadratic means, geometric and quadratic means, harmonic and contra-harmonic means, and geometric and contra-harmonic means.Comment: 7 page

Sharp inequalities for the Neuman-Sandor mean in terms of arithmetic and contra-harmonic means

In this paper, we find the greatest values $\alpha$ and $\lambda$, and the least values $\beta$ and $\mu$ such that the double inequalities $$C^{\alpha}(a,b)A^{1-\alpha}(a,b)<M(a,b)<C^{\beta}(a,b)A^{1-\beta}(a,b)$$ and &[C(a,b)/6+5 A(a,b)/6]^{\lambda}[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-\lambda}<M(a,b) &\qquad<[C(a,b)/6+5 A(a,b)/6]^{\mu}[C^{1/6}(a,b)A^{5/6}(a,b)]^{1-\mu} hold for all $a,b>0$ with $a\neq b$, where $M(a,b)$, $A(a,b)$ and $C(a,b)$ denote the Neuman-S\'andor, arithmetic, and contra-harmonic means of $a$ and $b$, respectively.Comment: 9 page

Optimal bounds for the Neuman-Sandor means in terms of geometric and contra-harmonic means

In this article, we prove that the double inequality $$\alpha G(a,b)+(1-\alpha)C(a,b)<M(a,b)<\beta G(a,b)+(1-\beta)C(a,b)$$ holds true for all $a,b>0$ with $a\neq b$ if and only if $\alpha\geq 5/9$ and $\beta\leq 1-1/[2\log(1+\sqrt{2})]=0.4327...$, where $G(a,b),C(a,b)$ and $M(a,b)$ are respectively the geometric, contra-harmonic and Neuman-S\'andor means of $a$ and $b$.Comment: 6 page

Some Best Possible Inequalities Concerning Certain Bivariate Means

In this paper, some inequalities of bounds for the Neuman-S\'{a}ndor mean in terms of weighted arithmetic means of two bivariate means are established. Bounds involving weighted arithmetic means are sharp.Comment: 8 page

Sharp Inequalities between Harmonic, Seiffert, Quadratic and Contraharmonic Means

In this paper, we present the greatest values $\alpha$, $\lambda$ and $p$, and the least values $\beta$, $\mu$ and $q$ such that the double inequalities $\alpha D(a,b)+(1-\alpha)H(a,b)<T(a,b)<\beta D(a,b)+(1-\beta) H(a,b)$, $\lambda D(a,b)+(1-\lambda)H(a,b)<C(a,b)<\mu D(a,b)+(1-\mu) H(a,b)$ and $p D(a,b)+(1-p)H(a,b)0$ with $a\neq b$, where $H(a,b)=2ab/(a+b)$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, $Q(a,b)=\sqrt{(a^2+b^2)/2}$, $C(a,b)=(a^2+b^2)/(a+b)$ and $D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respectively.Comment: 11 page

Optimal two parameter bounds for the Seiffert mean

In this note we obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).Comment: 6 page

Ramanujan's cubic transformation inequalities for zero-balanced hypergeometric functions

In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.Comment: 9 page