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Abstract In the article, we prove that λ1=1/2+[(2+log(1+2))/2]1/ν−1/2 $\lambda _{1}=1/2+\sqrt{ [ (\sqrt{2}+ \log (1+\sqrt{2}) )/2 ]^{1/\nu }-1}/2$, μ1=1/2+6ν/(12ν) $\mu _{1}=1/2+\sqrt{6 \nu }/(12\nu )$, λ2=1/2+[(π+2)/4]1/ν−1/2 $\lambda _{2}=1/2+\sqrt{ [(\pi +2)/4 ] ^{1/\nu }-1}/2$ and μ2=1/2+3ν/(6ν) $\mu _{2}=1/2+\sqrt{3\nu }/(6\nu )$ are the best possible parameters on the interval [1/2,1] $[1/2, 1]$ such that the double inequalities Cν[λ1x+(1−λ1)y,λ1y+(1−λ1)x]A1−ν(x,y)0 $x, y>0$ with x≠y $x\neq y$ and ν∈[1/2,∞) $\nu \in [1/2, \infty )$, where A(x,y) $A(x, y)$ is the arithmetic mean, C(x,y) $C(x, y)$ is the contraharmonic mean, and RQA(x,y) $\mathcal{R}_{QA}(x, y)$ and RAQ(x,y) $\mathcal{R}_{AQ}(x, y)$ are two Neuman means.

Abstract In the article, we provide several sharp upper and lower bounds for two Sándor–Yang means in terms of combinations of arithmetic and contra-harmonic means.

Abstract The so-called Schwab-Borchardt mean plays an important role in the theory of (bivariate) means. It includes a lot of standard means, such as the logarithmic mean, the first and second Seiffert means and the Neuman-Sándor mean. In this paper, we investigate an approach which allows us to construct a class of new means. Such class includes the (generalized) Schwab-Borchardt mean and other old/new means derived as well.

Abstract In the article, we prove that the double inequalities M α ( a , b ) < S Q A ( a , b ) < M β ( a , b ) $M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)$ and M λ ( a , b ) < S A Q ( a , b ) < M μ ( a , b ) $M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)$ hold for all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 … $\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$ , β ≥ 5 / 3 $\beta\geq5/3$ , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … $\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$ and μ ≥ 4 / 3 $\mu\geq4/3$ , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b ) / M ( a , b ) − 1 $S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$ and S A Q ( a , b ) = Q ( a , b ) e A ( a , b ) / T ( a , b ) − 1 $S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ are the Sándor-type means, A ( a , b ) = ( a + b ) / 2 $A(a,b)=(a+b)/2$ , Q ( a , b ) = ( a 2 + b 2 ) / 2 $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ , T ( a , b ) = ( a − b ) / [ 2 arctan ( ( a − b ) / ( a + b ) ) ] $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$ , and M ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / ( a + b ) ) ] $M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]$ are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means.

Five Wilker Huygens-type inequalities involving Gudermannian and the inverse Gudermannian functions are obtained. The Schwab-Borchardt mean plays a crucial role in the proofs. Also, an analytical inequality for the sums of powers, established earlier by this author, is an indispensable tool in the the proofs of the main results of this paper.

The so-called Schwab-Borchardt mean plays an important role in the theory of (bivariate) means. It includes a lot of standard means, such as the logarithmic mean, the first and second Seiffert means and the Neuman-Sándor mean. In this paper, we investigate an approach which allows us to construct a class of new means. Such class includes the (generalized) Schwab-Borchardt mean and other old/new means derived as well. [ABSTRACT FROM AUTHOR]

In the article, we present the sharp bounds for the Neuman mean NAG(a; b), NGA(a; b), NQA(a; b) and NAQ(a; b) in terms of the convex combinations of the arithmetic and one-parameter harmonic and contraharmonic means. As applications, we find several sharp inequalities for the first Seiffert, second Seiffert, Neuman-Sándor and logarithmic means. [ABSTRACT FROM AUTHOR]

In this article, we present the sharp bounds for the Neuman means derived from the Schwab-Borchardt, geometric, arithmetic and quadratic means in terms of the arithmetic and geometric combinations of harmonic, arithmetic and contra-harmonic means. [ABSTRACT FROM AUTHOR]

In the article, we prove that the double inequalities $M_{\alpha }(a,b)< S_{QA}(a,b)< M_{\beta}(a,b)$ and $M_{\lambda }(a,b)< S_{AQ}(a,b)< M_{\mu}(a,b)$ hold for all $a, b>0$ with $a\neq b$ if and only if $\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$ , $\beta\geq5/3$, $\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$ and $\mu\geq4/3$, where $S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$ and $S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ are the Sándor-type means, $A(a,b)=(a+b)/2$, $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$, $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$, and $M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]$ are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means. [ABSTRACT FROM AUTHOR]

In the article, we establish several Lazarević and Cusa type inequalities involving the hyperbolic sine and cosine functions with two parameters. As applications, we find some new bounds for certain bivariate means. [ABSTRACT FROM AUTHOR]

A new bivariate mean is introduced and studied. The mean under discussion is defined as the degenerate case of the completely symmetric elliptic integral of the second kind. Its relationship with the Schwab-Borchardt mean is established. Four particular cases of the mean in question are discussed. Sharp bounds and the Ky Fan inequalities for the quadruple of new means are established. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit formulas for the Neuman means NAH, NHA, NAC, and NCA, and present the best possible upper and lower bounds for these means in terms of the combinations of harmonic mean H, arithmetic mean A, and contraharmonic mean C. [ABSTRACT FROM AUTHOR]

Generalizations of two Seiffert means, usually denoted by P and T, are defined and investigated. The means under discussion are symmetric and homogeneous of degree one in each variable. Computable lower and upper bounds for the new means are also established. Several inequalities involving means discussed in this paper are obtained. In particular, two Wilker's type inequalities involving those means are derived. [ABSTRACT FROM AUTHOR]

Inequalities for the convex combinations of the arithmetic mean and a particular Schwab-Borchardt mean are obtained. A particular Schwab-Borchardt mean used in this paper is either the logarithmic mean or one of the Seiffert means or the Neuman-Sándor mean. The bounding means belong to a one-parameter family of bivariate means introduced in [17]. Another problem discussed in this paper deals with finding bounds for each of the four means mentioned above, where now the bounding expressions are convex combinations of the arithmetic mean and a one-parameter bivariate mean. Several particular inequalities involving four bivariate means mentioned above are also derived. [ABSTRACT FROM AUTHOR]

Abstract: Let us consider the second trigonometric mean defined by Seiffert and the hyperbolic mean defined by Neuman and Sándor. There are some known inequalities between these means and some power means . We prove that the evaluationsare optimal. In some details of the proofs we have used the computer algebra Mathematica. [Copyright &y& Elsevier]

A one-parameter family of bivariate means is introduced. They are defined in terms of the inverse functions of Jacobian elliptic functions cn and nc. It is shown that the new means are symmetric and homogeneous of degree one in their variables. Members of this family of means interpolate an inequality which connects two Schwab-Borchardt means. Computable lower and upper bounds for the new mean are also established. [ABSTRACT FROM AUTHOR]