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In the article, we provide the sharp bounds for the Sándor–Yang mean in terms of certain families of the two-parameter geometric and arithmetic mean and the one-parameter geometric and harmonic means. As applications, we present new bounds for a certain Yang mean and the inverse tangent function.

In the article, we provide a monotonicity rule for the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[P(x)+A(x)]/[P(x)+B(x)]$\end{document}[P(x)+A(x)]/[P(x)+B(x)], where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$P(x)$\end{document}P(x) is a positive differentiable and decreasing function defined on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-R, R)$\end{document}(−R,R) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R>0$\end{document}R>0), and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$\end{document}A(x)=∑n=n0∞anxn and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$\end{document}B(x)=∑n=n0∞bnxn are two real power series converging on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-R, R)$\end{document}(−R,R) such that the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$\end{document}{an/bn}n=n0∞ is increasing (decreasing) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$\end{document}an0/bn0≥(≤)1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{n}>0$\end{document}bn>0 for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq n_{0}$\end{document}n≥n0. As applications, we present new bounds for the complete elliptic integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$\end{document}E(r)=∫0π/21−r2sin2tdt (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< r

In the article, we present the best possible parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda=\lambda (p)$\end{document}λ=λ(p) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu=\mu(p)$\end{document}μ=μ(p) on the interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0, 1/2]$\end{document}[0,1/2] such that the double inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& G^{p}\bigl[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a \bigr]A^{1-p}(a,b) \\& \quad< E(a,b) < G^{p}\bigl[\mu a+(1-\mu)b, \mu b+(1-\mu)a \bigr]A^{1-p}(a,b) \end{aligned}$$ \end{document}Gp[λa+(1−λ)b,λb+(1−λ)a]A1−p(a,b)0$\end{document}a,b>0 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\neq b$\end{document}a≠b, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(a, b)=(a+b)/2$\end{document}A(a,b)=(a+b)/2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G(a,b)=\sqrt{ab}$\end{document}G(a,b)=ab and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E(a,b)=[2\int_{0}^{\pi /2}\sqrt{a\cos^{2}\theta+b\sin^{2}\theta}\,d\theta/\pi]^{2}$\end{document}E(a,b)=[2∫0π/2acos2θ+bsin2θdθ/π]2 are the arithmetic, geometric and special quasi-arithmetic means of a and b, respectively.

In this paper, we prove that the double inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& {\alpha }N_{QA}(a,b)+({1-\alpha })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\beta }N_{QA}(a,b)+({1-\beta })G(a,b), \\& {\lambda }N_{AQ}(a,b)+({1-\lambda })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\mu }N_{AQ}(a,b)+({1-\mu })G(a,b) \end{aligned}$$ \end{document}αNQA(a,b)+(1−α)G(a,b)0$\end{document}a,b>0 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\neq b$\end{document}a≠b if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \leq 3/8$\end{document}α≤3/8, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots $\end{document}β≥4/[π(log(1+2)+2)]=0.5546⋯ , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda \leq 3/10$\end{document}λ≤3/10 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots $\end{document}μ≥8/[π(π+2)]=0.4952⋯ , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$TD(a,b)$\end{document}TD(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G(a,b)$\end{document}G(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(a,b)$\end{document}A(a,b) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N_{QA}(a,b)$\end{document}NQA(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N_{AQ}(a,b)$\end{document}NAQ(a,b) are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

We prove that the double inequalities $H_{\alpha}(a, b) < X(a, b)0$ with $a\neq b$ if and only if $\alpha\leq1/2$ , $\beta\geq\log3/(1+\log2)=0.6488\cdots$ , $\lambda\leq2\log3/(2\log\pi-\log2) =1.3764\cdots$ , and $\mu\geq2$ , where $H_{p}(a, b)$ , $X(a, b)$ , and $U(a, b)$ are, respectively, the pth power-type Heronian mean, Sandor mean, and Yang mean of a and b.

In this paper, we present sharp bounds for the two Neuman means SHA and SCA derived from the Schwab-Borchardt mean in terms of convex combinations of either the weighted arithmetic and geometric means or the weighted arithmetic and quadratic means, and the mean generated either by the geometric or by the quadratic mean. MSC: 26E60

In this paper, we present the Schur convexity and monotonicity properties for the ratios of the Hamy and generalized Hamy symmetric functions and establish some analytic inequalities. The achieved results is inspired by the paper of Hara et al. [J. Inequal. Appl. 2, 387-395, (1998)], and the methods from Guan [Math. Inequal. Appl. 9, 797-805, (2006)]. The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal. 2010 Mathematics Subject Classification: Primary 05E05; Secondary 26D20.

For , the generalized logarithmic mean , arithmetic mean and geometric mean of two positive numbers and are defined by , ; , , , ; , , ; , , ; and , respectively. In this paper, we give an answer to the open problem: for , what are the greatest value and the least value , such that the double inequality holds for all ?

In the article, we present the best possible parameters $\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}\in(0, 1)$ and $\alpha_{3}, \alpha_{4}, \beta_{3}, \beta_{4}\in(0, 1/2)$ such that the double inequalities $$\begin{aligned}& \alpha_{1}A(a,b)+(1-\alpha_{1})H(a,b)< X(a,b)< \beta _{1}A(a,b)+(1-\beta_{1})H(a,b), \\ & \alpha_{2}A(a,b)+(1-\alpha_{2})G(a,b)< X(a,b)< \beta _{2}A(a,b)+(1-\beta_{2})G(a,b), \\ & H\bigl[\alpha_{3}a+(1-\alpha_{3})b, \alpha_{3}b+(1- \alpha _{3})a\bigr]< X(a,b)< H\bigl[\beta_{3}a+(1- \beta_{3})b, \beta_{3}b+(1-\beta_{3})a\bigr], \\ & G\bigl[\alpha_{4}a+(1-\alpha_{4})b, \alpha_{4}b+(1- \alpha _{4})a\bigr]< X(a,b)< G\bigl[\beta_{4}a+(1- \beta_{4})b, \beta_{4}b+(1-\beta_{4})a\bigr] \end{aligned}$$ hold for all $a, b>0$ with $a\neq b$ . Here, $X(a,b)$ , $A(a,b)$ , $G(a,b)$ and $H(a,b)$ are the Sandor, arithmetic, geometric and harmonic means of a and b, respectively.

In this paper, we present the best possible parameters $\alpha, \beta \in\mathbb{R}$ and $\lambda, \mu\in(1/2, 1)$ such that the double inequalities $\alpha N_{AQ}(a,b)+(1-\alpha)A(a,b)< T^{\ast}(a,b)0$ with $a\neq b$ , where $T^{\ast}(a,b)$ , $A(a,b)$ , $Q(a,b)$ and $N_{QA}(a,b)$ are the Toader, arithmetic, quadratic, and Neuman means of a and b, respectively.

In this paper, we find the greatest values α, λ and the least values β, μ such that the double inequalities $\alpha[G(a,b)/3+2A(a,b)/3]+(1-\alpha)G^{1/3}(a,b)A^{2/3}(a,b)< P(a,b) 0$ with $a\neq b$ . Here $G(a,b)$ , $A(a,b)$ , $C(a,b)$ , $P(a,b)$ and $T(a,b)$ denote the geometric, arithmetic, contraharmonic, first Seiffert and second Seiffert means of two positive numbers a and b, respectively.

We present the best possible parameters p 1 , p 2 , p 3 , p 4 , q 1 , q 2 , q 3 , q 4 ∈ [ 0 , 1 ] such that the double inequalities G p 1 ( a , b ) < S H A ( a , b ) < G q 1 ( a , b ) , Q p 2 ( a , b ) < S C A ( a , b ) < Q q 2 ( a , b ) , H p 3 ( a , b ) < S A H ( a , b ) < H q 3 ( a , b ) , C p 4 ( a , b ) < S A C ( a , b ) < C q 4 ( a , b ) hold for all a , b > 0 with a ≠ b , where S H A , S C A , S A H , S A C are the Neuman means, and G p , Q p , H p , C p are the one-parameter means. MSC:26E60.