Sharp bounds in terms of the power of the contra-harmonic mean for Neuman-Sandor mean

In the paper, the authors obtain sharp bounds in terms of the power of the contra-harmonic mean for Neuman-S\'andor mean.


Introduction
For positive numbers a, b > 0 with a = b, the second Seiffert mean T (a, b), the root-mean-square S(a, b), Neuman-Sándor mean M (a, b), and the contra-harmonic mean C(a, b) are respectively defined in [9,13] by It is well known [7,8,10] that the inequalities M (a, b) < T (a, b) < S(a, b) < C(a, b) hold for all a, b > 0 with a = b .
In [2,3], the inequalities and were proved to be valid for 1 2 < α, β, λ, µ < 1 and for all a, b > 0 with a = b if and only if respectively. In [12], the double inequality was proved to be valid for 1 2 < α, β < 1 and for all a, b > 0 with a = b if and only if .
For more information on this topic, please refer to recently published papers [4,5,6,11] and references cited therein. For t ∈ 1 2 , 1 and p ≥ 1 2 , let where A(a, b) = a+b 2 is the classical arithmetic mean of a and b. Then, by definitions in (1.1) and (1.2), it is easy to see that and Q t,p (a, b) is strictly increasing with respect to t ∈ 1 2 , 1 . Motivating by results mentioned above, we naturally ask a question: What are the greatest value t 1 = t 1 (p) and the least value t 2 = t 2 (p) in 1 2 , 1 such that the double inequality holds for all a, b > 0 with a = b and for all p ≥ 1 2 ? The aim of this paper is to answer this question. The solution to this question may be stated as the following Theorem 1.1.
holds for all a, b > 0 with a = b if and only if

Lemmas
In order to prove Theorem 1.1, we need the following lemmas. on (a, b). If g ′ (x) = 0 and f ′ (x) g ′ (x) is strictly increasing (or strictly decreasing, respectively) on (a, b), so are the functions Lemma 2.2. The function x is strictly increasing and convex on (0, ∞).
Proof. This follows from the following arguments: Proof. It is ready that . An easy computation yields where Furthermore, we have where h(x) is defined by (2.2). From Lemma 2.2, it follows that the quotient is strictly decreasing on (0, 1). Accordingly, from Lemma 2.1 and (2.7), it is deduced that the ratio g1(x) g2(x) is strictly decreasing on (0, 1). Moreover, making use of L'Hôpital's rule leads to When u ≥ 1 6p , combining (2.6) and (2.9) with the monotonicity of g1(x) g2(x) shows that the function f u,p (x) is strictly increasing on (0, 1). Therefore, the positivity of f u,p (x) on (0, 1) follows from (2.4) and the increasingly monotonicity of f u,p (x).