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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.

In the article, we prove that the double inequalities α1Q(a, b) + (1 - α1)NGA(a, b) < T(a, b) < β1Q(a, b) + (1 - β1)NGA(a, b); α2Q(a, b) + (1 - α2)NQA(a, b) < T(a, b) < β2Q(a, b) + (1 - β2)NQA(a, b); α3C(a, b) + (1 - α3)NGA(a, b) < T(a, b) < β3C(a, b) + (1 - β3)NGA(a, b); α4C(a, b) + (1 - α4)NQA(a, b) < T(a, b) < β4C(a, b) + (1 - β4)NQA(a, b) hold for all a,b> 0 with a = b if and only if a1 ... where Q(a; b), C(a; b) and T(a; b) are respectively the quadratic, contra-harmonic and Toader means, and NGA(a; b) and NQA(a; b) are the Neuman means. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove that the double inequalities α N Q A ( a , b ) + ( 1 − α ) G ( a , b ) < T D [ A ( a , b ) , G ( a , b ) ] < β N Q A ( a , b ) + ( 1 − β ) G ( a , b ) , λ N A Q ( a , b ) + ( 1 − λ ) G ( a , b ) < T D [ A ( a , b ) , G ( a , b ) ] < μ N A Q ( a , b ) + ( 1 − μ ) G ( a , b ) $$\begin{aligned}& {\alpha }N_{QA}(a,b)+({1-\alpha })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\beta }N_{QA}(a,b)+({1-\beta })G(a,b), \\& {\lambda }N_{AQ}(a,b)+({1-\lambda })G(a,b)< TD \bigl[A(a,b),G(a,b) \bigr]< {\mu }N_{AQ}(a,b)+({1-\mu })G(a,b) \end{aligned}$$ hold for all a , b > 0 $a,b>0$ with a ≠ b $a\neq b$ if and only if α ≤ 3 / 8 $\alpha \leq 3/8$ , β ≥ 4 / [ π ( log ( 1 + 2 ) + 2 ) ] = 0.5546 ⋯ $\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots $ , λ ≤ 3 / 10 $\lambda \leq 3/10$ and μ ≥ 8 / [ π ( π + 2 ) ] = 0.4952 ⋯ $\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots $ , where T D ( a , b ) $TD(a,b)$ , G ( a , b ) $G(a,b)$ , A ( a , b ) $A(a,b)$ and N Q A ( a , b ) $N_{QA}(a,b)$ , N A Q ( a , b ) $N_{AQ}(a,b)$ are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

In the article, we prove that the double inequalities αNAG(a, b) + (1 - α) Q(a, b) < T D (a, b) < βNAG(a, b) + (1 - β) Q(a, b), λNAG(a, b) + (1 - λ) C(a, b) < T D(a, b) < µNAG(a, b) + (1 - µ) C(a, b) hold for all a, b > 0 with a 6= b if and only if α ≥ 3/10, β ≤ 2 (√ 2π - 4)/h(2√2 - 1)πi=0.1542 ···, λ ≥ 9/16 and µ ≤ 4 (π - 2)/(3π) = 0.4845 ···, where Q(a, b),C(a, b), NAG (a, b) and T D(a, b) are the quadratic, contra-harmonic, secondNeuman and Toader means of a and b, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the double inequalities hold for all $a,b>0$ with $a\neq b$ if and only if $\alpha \leq 3/8$ , $\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots $ , $\lambda \leq 3/10$ and $\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots $ , where $TD(a,b)$ , $G(a,b)$ , $A(a,b)$ and $N_{QA}(a,b)$ , $N_{AQ}(a,b)$ are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively. [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 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 $\alpha N_{AQ}(a,b)+(1-\alpha)A(a,b)< T^{\ast}(a,b)<\beta N_{AQ}(a,b)+(1-\beta)A(a,b)$, $Q[\lambda a+(1-\lambda)b, \lambda b+(1-\lambda)a]< T^{\ast}(a,b)< Q[\mu a+(1-\mu)b, \mu b+(1-\mu)a] $ hold for all $a, b>0$ 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. [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]