Sharp bounds for the Neuman-Sándor mean in terms of the power and contraharmonic means

In the paper, the authors obtain sharp bounds for the Neuman–Sándor mean in terms of the power and contraharmonic means.


Introduction
For positive numbers a, b < 0 with a ≠ b, the second Seiffert mean T(a, b), quadratic mean S(a, b), Neuman-Sándor mean M(a, b), and contraharmonic mean C(a, b) are respectively defined in Neuman and Sándor (2003), and Seiffert (1995) by , C(a, b) = a 2 + b 2 a + b

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In the paper, the authors obtain a sharp lower bound and a sharp upper bound for the Neuman-Sándor mean in terms of the power and contraharmonic means.
It is well known Neuman (2012Neuman ( , 2011, and Neuman and Sándor (2006) that the inequalities hold for all a, b < 0 with a ≠ b.
In , Chu, Hou, and Shen (2012), the inequalities and were proved to be valid for 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 The aim of this paper is to answer this question. The solution to this question may be stated as the following Theorem 1.

Lemmas
In order to prove Theorem 1.1, we need the following lemmas.

Lemma 2.2 The function
is strictly increasing and convex on (0, ∞).
Proof This follows from the following arguments: (1.10) is strictly decreasing on (0, 1). Accordingly, from Lemma 2.1 and (2.7), it is deduced that the ratio is strictly decreasing on (0, 1).
Moreover, making use of L'Hôpital's rule leads to and When u ≥ 1 6p , combining (2.6) and (2.9) with the monotonicity of 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).
, combining (2.6) and (2.10) with the monotonicity of reveals that the function f u, p (x) is strictly decreasing on (0, 1). Hence, the negativity of f u,p (x) on (0,1) follows from (2.4) and the decreasingly monotonicity of f u, p (x).
, from (2.6), (2.9), (2.10) and the monotonicity of the ratio , we conclude that there exists a number x 0 ∈ (0, 1) such that f u, p (x) is strictly decreasing in (0, x 0 ) and strictly increasing in (x 0 , 1). Denote the limit in (2.5) by h p (u). Then, from the above arguments, it follows that and Since h p (u) is strictly increasing for u > −1, so it is also in � √ 2 t * −1 √ 2 (2p−1)t * +1 , 1 6p � . Thus, the inequalities in (2.11) and (2.12) imply that the function h p (u) has a unique zero , u 0 � and h p (u) > 0 for u ∈ u 0 , 1 6p . As a result, combining (2.4) and (2.5) with the piecewise monotonicity of f u, p (x) reveals that f u, p (x) < 0 for all x ∈ (0, 1) if and only if < u < u 0 . The proof of Lemma 2.3 is complete.

Proof of Theorem 1.1
Now we are in a position to prove our Theorem 1.1.
Since both Q t, p (a, b) and M(a, b) are symmetric and homogeneous of degree 1, without loss of generality, we assume that a > b. Let x = a−b a+b ∈ (0, 1). From (1.2) and (1.8), we obtain Thus, Theorem 1.1 follows from Lemma 2.3.
Remark 3.1 This is a slightly modified version of the preprint Jiang and Qi (2013b). (2.10)