A best-possible double inequality between Seiffert and harmonic means

In this paper, we establish a new double inequality between the Seiffert and harmonic means.

The achieved results is inspired by the papers of Sándor (Arch. Math., 76, 34-40, 2001) and Hästö (Math. Inequal. Appl., 7, 47-53, 2004), and the methods from Wang et al. (J. Math. Inequal., 4, 581-586, 2010). The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.

2010 Mathematics Subject Classification: 26E60.

For a, b > 0 with a ≠ b, the Seiffert mean P(a, b) was introduced by Seiffert [1] as follows:

Recently, the bivariate mean values have been the subject of intensive research. In particular, many remarkable inequalities for the Seiffert mean can be found in the literature [1-9].

Let H(a, b) = 2ab/(a+b), , L(a, b) = (a - b)/(log a - log b), I(a, b) = 1/e(b^b/a^a)^1/(b-a), A(a, b) = (a+b)/2, C(a, b) = (a²+b²)/(a+b), and M_p(a, b) = ((a^p + b^p)/2)^1/p(p ≠ 0) and be the harmonic, geometric, logarithmic, identric, arithmetic, contraharmonic, and p-th power means of two different positive numbers a and b, respectively. Then, it is well known that

For all a, b > 0 with a ≠ b, Seiffert [1] established that L(a, b) < P(a, b) < I(a, b); Jagers [4] proved that M_1/2(a, b) < P (a, b) < M_2/3(a, b) and M_2/3(a, b) is the best-possible upper power mean bound for the Seiffert mean P(a, b); Seiffert [7] established that P(a, b) > A(a, b)G(a, b)/L(a, b) and P(a, b) > 2 A(a, b)/π; Sándor [6] presented that and ; Hästö [3] proved that P(a, b) > M_{log 2/ log π}(a, b) and M_{log 2/ log π}(a, b) is the best-possible lower power mean bound for the Seiffert mean P(a, b).

Very recently, Wang and Chu [8] found the greatest value α and the least value β such that the double inequality A^α(a, b)H^1-α(a, b) < P(a, b) < A^β(a, b)H^1-β(a, b) holds for a, b > 0 with a ≠ b; For any α ∈ (0, 1), Chu et al. [10] presented the best-possible bounds for P^α(a, b)G^1-α(a, b) in terms of the power mean; In [2], the authors proved that the double inequality αA(a, b) + (1 - α)H(a, b) < P(a, b) < βA(a, b) + (1 - β)H(a, b) holds for all a, b > 0 with a ≠ b if and only if α ≤ 2/π and β ≥ 5/6; Liu and Meng [5] proved that the inequalities

and

hold for all a, b > 0 with a ≠ b if and only if α₁ ≤ 2/9, β₁ ≥ 1/π, α₂ ≤ 1/π and β₂ ≥ 5/12.

For fixed a, b > 0 with a ≠ b and x ∈ [0, 1/2], let

Then, it is not difficult to verify that h(x) is continuous and strictly increasing in [0, 1/2]. Note that h(0) = H(a, b) < P(a, b) and h(1/2) = A(a, b) > P(a, b). Therefore, it is natural to ask what are the greatest value α and least value β in (0, 1/2) such that the double inequality H(αa + (1 - α)b, αb + (1 - α)a) < P(a, b) < H(βa + (1 - β)b, βb + (1 - β)a) holds for all a, b > 0 with a ≠ b. The main purpose of this paper is to answer these questions. Our main result is the following Theorem 1.1.

Theorem 1.1. If α, β ∈ (0, 1/2), then the double inequality

holds for all a, b > 0 with a ≠ b if and only if and .

Proof of Theorem 1.1. Let and . We first prove that inequalities

and

hold for all a, b > 0 with a ≠ b.

Without loss of generality, we assume that a > b. Let and p ∈ (0, 1/2); then, from (1.1), one has

Let

then, simple computations lead to

and

where

Note that

Let , , , , and . Then, simple computations lead to

and

We divide the proof into two cases.

Case 1. . Then equations (2.6), (2.13), (2.15), (2.17), (2.19), (2.22) and (2.25) become

and

From (2.24), we clearly see that f₇(t) is strictly increasing in [1, +∞), and then (2.32) leads to the conclusion that f₇(t) > 0 for t ∈ [1, +∞). Thus, f₆(t) is strictly increasing in [1, +∞).

It follows from (2.23) and (2.31) together with the monotonicity of f₆(t) that there exists t₁ > 1 such that f₆(t) < 0 for t ∈ (1, t₁) and f₆(t) > 0 for t ∈ (t₁, +∞). Thus, f₅(t) is strictly decreasing in [1, t₁] and strictly increasing in [t₁, +∞).

From (2.20) and (2.30), together with the piecewise monotonicity of f₅(t), we clearly see that there exists t₂ > t₁ > 1 such that f₄(t) is strictly decreasing in [1, t₂] and strictly increasing in [t₂, +∞). Then, equation (2.18) and inequality (2.29) lead to the conclusion that there exists t₃ > t₂ > 1 such that f₃(t) is strictly decreasing in [1, t₃] and strictly increasing in [t₃, +∞).

It follows from (2.16) and (2.28) together with the piecewise monotonicity of f₃(t) we conclude that there exists t₄ > t₃ > 1 such that f₂(t) is strictly decreasing in [1, t₄] and strictly increasing in [t₄, +∞). Then, equation (2.14) and inequality (2.27) lead to the conclusion that there exists t₅ > t₄ > 1 such that is strictly decreasing in [1, t₅] and strictly increasing in [t₅, +∞).

From equations (2.11) and (2.12), together with the piecewise monotonicity of , we know that there exists t₆ > t₅ > 1 such that f₁(t) is strictly decreasing in [1, t₆] and strictly increasing in [t₆, +∞). Then, equations (2.7)-(2.10) lead to the conclusion that there exists t₇ > t₆ > 1 such that f(t) is strictly decreasing in [1, t₇] and strictly increasing in [t₇, +∞).

Therefore, inequality (2.1) follows from equations (2.3)-(2.5) and (2.26) together with the piecewise monotonicity of f(t).

Case 2. . Then, equations (2.13), (2.15), (2.17), (2.19) and (2.21) become

and

for t > 1.

From inequality (2.37), we know that f₅(t) is strictly increasing in [1, +∞), and then inequality (2.36) leads to the conclusion that f₅(t) > 0 for t ∈ [1, +∞). Thus, f₄(t) is strictly increasing in [1, +∞).

It follows from inequality (2.35) and the monotonicity of f₄(t) that f₃(t) is strictly increasing in [1, +∞).

Therefore, inequality (2.2) follows easily from equations (2.3)-(2.5), (2.7), (2.9), (2.11), (2.33), and (2.34) together with the monotonicity of f₃(t).

Next, we prove that is the best-possible parameter such that inequality (2.1) holds for all a, b > 0 with a ≠ b. In fact, if , then equation (2.6) leads to

Inequality (2.38) implies that there exists T = T(p) > 1 such that

for t ∈ (T, +∞).

From equations (2.3) and (2.4), together with inequality (2.39), we clearly see that P(a, b) < H(pa + (1 - p)b, pb + (1 - p)a) for a/b ∈ (T², +∞).

Finally, we prove that is the best-possible parameter such that inequality (2.2) holds for all a, b > 0 with a ≠ b. In fact, if , then equation (2.13) leads to

Inequality (2.40) implies that there exists δ = δ (p) > 0 such that

for t ∈ (1, 1 + δ).

Therefore, P(a, b) > H(pa + (1 - p)b, pb + (1 - p)a) for a/b ∈ (1, (1 + δ)²) follows from equations (2.3)-(2.5), (2.7), (2.9), and (2.11) together with inequality (2.41).

The authors declare that they have no competing interests.

Y-MC provided the main idea in this paper. M-KW carried out the proof of inequality (2.1) in this paper. Z-KW carried out the proof of inequality (2.2) in this paper. All authors read and approved the final manuscript.

This study is partly supported by the Natural Science Foundation of China (Grant no. 11071069), the Natural Science Foundation of Hunan Province (Grant no. 09JJ6003), and the Innovation Team Foundation of the Department of Education of Zhejiang Province(Grant no. T200924).

1. Seiffert, H-J: Problem 887. Nieuw Arch Wisk (4). 11(2), 176–176 (1993)

2. Chu, Y-M, Qiu, Y-F, Wang, M-K, Wang, G-D: The optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean. In: J Inequal Appl. 2010, 7 Article ID 436457

3. Hästö, PA: Optimal inequalities between Seiffert's mean and power means. Math Inequal Appl. 7(1), 47–53 (2004)

4. Jagers, AA: Solution of Problem 887. Nieuw Arch Wisk (4). 12, 230–231 (1994)

5. Liu, H, Meng, X-J: The optimal convex combination bounds for Seiffert's mean. In: J Inequal Appl. 2011, 9 Article ID 686834

6. Sándor, J: On certain inequalities for means III. Arch Math. 76(1), 34–40 (2001). Publisher Full Text

7. Seiffert, H-J: Ungleichungen für einen bestimmten Mittelwert. Nieuw Arch Wisk (4). 13(2), 195–198 (1995)

8. Wang, S-S, Chu, Y-M: The best bounds of the combination of arithmetic and harmonic means for the Seiffert's mean. Int J Math Anal. 4(21-24), 1079–1084 (2010)

9. Wang, M-K, Qiu, Y-F, Chu, Y-M: Sharp bounds for Seiffert means in terms of Lehmer means. J Math Inequal. 4(4), 581–586 (2010)

10. Chu, Y-M, Qiu, Y-F, Wang, M-K: Sharp power mean bounds for the combination of Seiffert and geometric means. In: Abstr Appl Anal. 2010, 12 Article ID 108920