Optimal bounds for Neuman-Sándor mean in terms of the convex combination of the logarithmic and the second Seiffert means

In the article, we prove 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}$$ \alpha L(a,b)+(1-\alpha)T(a,b)< \mathit{NS}(a,b)< \beta L(a,b)+(1-\beta)T(a,b) $$\end{document}αL(a,b)+(1−α)T(a,b)<NS(a,b)<βL(a,b)+(1−β)T(a,b) holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a,b>0$\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\ne 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\ge1/4$\end{document}α≥1/4 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta\le1-\pi/[4\log(1+\sqrt{2})]$\end{document}β≤1−π/[4log(1+2)], where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{NS}(a,b)$\end{document}NS(a,b), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L(a,b)$\end{document}L(a,b) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T(a,b)$\end{document}T(a,b) denote the Neuman-Sándor, logarithmic and second Seiffert means of two positive numbers a and b, respectively.

Recently, the means NS, T, L and other means have been the subject of extensive research. In particular, many remarkable inequalities for the Neuman-Sándor, second Seiffert and logarithmic means can be found in the literature [ √ ab, and H(a, b) = ab/(a + b) denote the first Seiffert, root-square, arithmetic, identric, geometric, and the harmonic means of two positive numbers a and b with a = b, respectively. Then it is well known that the inequality In [] and [], the authors proved that the double inequalities , it was showed that the inequality holds for all a, b >  with a = b if and only if α  > / and Let L p (a, b) = (a p+ + b p+ )/(a p + b p ) be the Lehmer mean of two positive numbers a and b with a = b. In [], the authors proved the double inequality  holds.

Gao [] proved the optimal double inequality
holds for all a, b >  with a = b.

Yang [] proved the inequality
holds for all a, b >  with a = b if and only if p ≥ / √  and  < q ≤ /. And the inequality was proved by Lin in [].
In [], the authors present bounds for L in terms of G and A The purpose of this paper is to answer the following questions: What are the least value α and the greatest value β such that

Lemmas
It is well known that, for x ∈ (, ), To establish our main result, we need several lemmas as follows.
Then H(x) is strictly increasing on (, ). Moreover, the inequality holds for any x ∈ (, .) and the inequality holds for any x ∈ (, ).

Lemma . We have
Then it is easy to verify that η(x) is decreasing on (, μ), where

Main results
Theorem . The double inequality Then it follows that where H(x), S(x) and T(x) are defined as in Lemmas .-., respectively.
On one hand, from inequalities (.), (.) and (.), we clearly see that for any x ∈ (, ). It leads to for any x ∈ (, ). Thus, from (.) it follows that On the other hand, from inequalities (.), (.) and (.), we have for x ∈ (, .). According to Lemma ., we have Finally, we prove that L(a, b)/ + T(a, b)/ and λL(a, b) + (λ)T(a, b) are the best possible lower and upper mean bound for the Neuman-Sándor mean M(a, b).
For any  ,  > , let t  = / - , t  = λ +  . Then one can get Let x  →  + and x  →  -, then the Taylor expansion leads to

Conclusion
In the article, we give the sharp upper and lower bounds for Neuman-Sándor mean in terms of the linear convex combination of the logarithmic and second Seiffert means.