We answer the question: for any p, q ∈ ℝ with p ≠ q and p ≠ −q, what are the greatest value λ = λ(p, q) and the least value μ = μ(p, q), such that the double inequality holds for all a, b > 0 with a ≠ b? Where M_p(a, b) is the pth power mean of two positive numbers a and b.

1. Introduction

For p ∈ ℝ, the pth power mean M_p(a, b) of two positive numbers a and b is defined by

(1.1)

It is well known that M_p(a, b) is continuous and strictly increasing with respect to p ∈ ℝ for fixed a, b > 0 with a ≠ b. Many classical means are special case of the power mean, for example, M₋₁(a, b) = H(a, b) = 2ab/(a + b), , and M₁(a, b) = A(a, b) = (a + b)/2 are the harmonic, geometric, and arithmetic means of a and b, respectively. Recently, the power mean has been the subject of intensive research. In particular, many remarkable inequalities and properties for the power mean can be found in literature [1–15].

Let L(a, b) = (a − b)/(log a − log b) and I(a, b) = 1/e(a^a/b^b) ^1/(a−b) be the logarithmic and identric means of two positive numbers a and b with a ≠ b, respectively. Then it is well known that

(1.2)

for all a, b > 0 with a ≠ b.

In [16–22], the authors presented the sharp power mean bounds for L, I, (IL) ^1/2, and (L + I)/2 as follows:

(1.3)

for all a, b > 0 with a ≠ b.

Alzer and Qiu [12] proved that the inequality

(1.4)

holds for all a, b > 0 with a ≠ b if and only if p ≤ log 2/(1 + log 2) = 0.40938….

The following sharp bounds for the sum αA(a, b)+(1 − α)L(a, b), and the products A^α(a, b)L^1−α(a, b) and G^α(a, b)L^1−α(a, b) in terms of power means were proved in [5, 8] as follows:

(1.5)

for any α ∈ (0,1) and all a, b > 0 with a ≠ b.

In [2, 7], the authors answered the question: for any α ∈ (0,1), what are the greatest values p₁ = p₁(α), p₂ = p₂(α), p₃ = p₃(α), and p₄ = p₄(α), and the least values q₁ = q₁(α), q₂ = q₂(α), q₃ = q₃(α), and q₄ = q₄(α), such that the inequalities

(1.6)

hold for all a, b > 0 with a ≠ b?

In [4], the authors presented the greatest value p = p(α, β) and the least value q = q(α, β) such that the double inequality

(1.7)

holds for all a, b > 0 with a ≠ b and α, β > 0 with α + β < 1.

It is the aim of this paper to answer the question: for any p, q ∈ ℝ with p ≠ q and p ≠ −q, what are the greatest value λ = λ(p, q) and the least value μ = μ(p, q), such that the double inequality

(1.8)

holds for all a, b > 0 with a ≠ b?

2. Main Result

In order to establish our main result, we need a lemma which we present in this section.

Lemma 2.1. Let p, q ≠ 0, p ≠ q and x > 1. Then

(2.1)

for p + q > 0, and

(2.2)

for p + q < 0.

Proof. From (1.1), we have

(2.3)

Let

(2.4)

then simple computations lead to

(2.5)

(2.6)

Equation (2.6) implies that

(2.7)

for p + q > 0, and

(2.8)

for p + q < 0.

Therefore, inequality (2.1) follows from (2.3)–(2.5) and inequality (2.7), and inequality (2.2) follows from (2.3)–(2.5) and inequality (2.8).

Let

(2.9)

Then we clearly see that , and it is not difficult to verify that the identity holds for all a, b > 0 if . Let

(2.10)

Then we have Theorem 2.2 as follows.

Theorem 2.2. The double inequality

(2.11)

holds for all a, b > 0 with a ≠ b, and M_λ(a, b) and M_μ(a, b) are the best possible lower and upper power mean bounds for the geometric mean of M_p(a, b) and M_q(a, b).

Proof. From (1.1), we clearly see that M_p(a, b) is symmetric and homogenous of degree 1. Without loss of generality, we assume that b = 1, a = x > 1 and p > q. We divide the proof of inequality (2.11) into three cases.

Case 1. (p, q) ∈ E₁⋃ E₂. Then from Lemma 2.1, we clearly see that

(2.12)

for (p, q) ∈ E₁, and

(2.13)

for (p, q) ∈ E₂.

From (1.1), we get

(2.14)

Let

(2.15)

then simple computations lead to

(2.16)

(2.17)

where

(2.18)

(2.19)

(2.20)

where

(2.21)

(2.22)

(2.23)

where

(2.24)

(2.25)

(2.26)

where

(2.27)

(2.28)

(2.29)

If (p, q) ∈ E₁, then (2.15), (2.18), (2.21), (2.22), (2.24), (2.25), (2.27), and (2.28) lead to

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

(2.35)

(2.36)

(2.37)

We divide the discussion into two subcases.

Subcase 1.1. (p, q) ∈ E₁. Then (2.26) and (2.29) together with inequalities (2.36) and (2.37) imply that I(x) is strictly decreasing in [1, +∞). In fact, if (q² − p² + 2pq)/(p + q) ≥ 0, then (2.29) and inequality (2.37) imply that J(x) < 0 for x ∈ [1, +∞). If (q² − p² + 2pq)/(p + q) < 0, then (2.29) and inequality (2.36) lead to the conclusion that J(x) < 0 for x ∈ [1, +∞).

From inequalities (2.34) and (2.35) together with the monotonicity of I(x), we know that there exists λ₁ > 1 such that I(x) > 0 for x ∈ [1, λ₁) and I(x) < 0 for x ∈ (λ₁, +∞). Then (2.23) leads to the conclusion that H(x) is strictly increasing in [1, λ₁] and strictly decreasing in [λ₁, +∞).

It follows from (2.32) and (2.33) together with the piecewise monotonicity of H(x) that there exists λ₂ > λ₁ > 1 such that H(x) > 0 for [1, λ₂] and H(x) < 0 for (λ₂, +∞). Then (2.20) leads to the conclusion that G(x) is strictly increasing in [1, λ₂] and strictly decreasing in [λ₂, +∞).

From (2.17), (2.19) and (2.31) together with the piecewise monotonicity of G(x), we clearly see that there exists λ₃ > λ₂ > 1 such that F(x) is strictly increasing in [1, λ₃] and strictly decreasing in [λ₃, +∞).

Therefore, follows from (2.14)–(2.16) and (2.30) together with the piecewise monotonicity of F(x).

Subcase 1.2. (p, q) ∈ E₂. Then (2.30) and (2.35) again hold, and (2.18), (2.21), (2.22), and (2.28) lead to

(2.38)

(2.39)

(2.40)

(2.41)

It follows from (2.29) and inequalities (q² − p² + 2pq)/(p + q) < 0 and (2.41) that J(x) > 0 for x ∈ [1, +∞). Then (2.26) and inequality (2.35) lead to the conclusion that I(x) > 0 for x ∈ [1, +∞). Therefore, H(x) is strictly increasing in [1, +∞) follows from (2.23).

It follows from (2.20) and (2.39) together with inequality (2.40) and the monotonicity of H(x) that there exists μ₁ > 1 such that G(x) is strictly decreasing in [1, μ₁] and strictly increasing in [μ₁, +∞).

From (2.17), (2.19) and (2.38) together with the piecewise monotonicity of G(x), we clearly see that there exists μ₂ > μ₁ > 1 such that F(x) is strictly decreasing in [1, μ₂] and strictly increasing in [μ₂, +∞).

Therefore, follows from (2.14)–(2.16) and (2.30) together with the piecewise monotonicity of F(x).

Case 2. (p, q) ∈ E₃⋃ E₅. Clearly, we have for (p, q) ∈ E₃ and for (p, q) ∈ E₅. Therefore, we need only to prove that

(2.42)

for r > 0, and

(2.43)

for r < 0.

From (1.1), one has

(2.44)

Let

(2.45)

then simple computations lead to

(2.46)

(2.47)

If r > 0 (or r < 0, resp.), then (2.47) leads to the conclusion that f(x) is strictly decreasing (or increasing, resp.) in [1, +∞). Therefore, inequalities (2.42) and (2.43) follow from (2.44)–(2.46) and the monotonicity of f(x).

Case 3. (p, q) ∈ E₄⋃ E₆. Then from Lemma 2.1, we clearly see that for (p, q) ∈ E₄ and for (p, q) ∈ E₆. Therefore, we need only to prove that

(2.48)

for (p, q) ∈ E₄, and

(2.49)

for (p, q) ∈ E₆.

From (1.1), we get

(2.50)

Let

(2.51)

then simple computations lead to

(2.52)

(2.53)

If (p, q) ∈ E₄ (or E₆, resp.), then (2.53) implies that f(x) is strictly increasing (or decreasing, resp.) in [1, +∞). Therefore, inequalities (2.48) and (2.49) follow from (2.50)–(2.52) and the monotonicity of f(x).

Next, we prove that M_λ(a, b) and M_μ(a, b) are the best possible lower and upper power mean bounds for the geometric mean of M_p(a, b) and M_q(a, b). We divide the proof into six cases.

Case A. (p, q) ∈ E₁. Then for any ϵ ∈ (0, (p + q)/2) and x > 0, from (1.1), one has

(2.54)

(2.55)

Letting x → 0 and making use of Taylor expansion, we get

(2.56)

Equations (2.54) and (2.56) together with inequality (2.55) imply that for any ϵ ∈ (0, p + q/2), there exist δ₁ = δ₁(ϵ) > 0 and X₁ = X₁(p, q, ϵ) > 1 such that for x ∈ (0, δ₁) and for x ∈ (X₁, +∞).

Case B. (p, q) ∈ E₂. Then for ϵ ∈ (0, −(p + q)/2) and x > 0, making use of (1.1) and Taylor expansion, we have

(2.57)

(2.58)

Equation (2.57) and inequality (2.58) imply that for any ϵ ∈ (0, −(p + q)/2), there exist δ₂ = δ₂(ϵ) > 0 and X₂ = X₂(p, q, ϵ) > 1 such that for x ∈ (0, δ₂) and for x ∈ (X₂, +∞).

Case C. (p, q) ∈ E₃. Then for ϵ ∈ (0, p/2) and x > 0, making use of (1.1) and Taylor expansion, we have

(2.59)

Equation (2.59) leads to the conclusion that for any ϵ ∈ (0, p/2), there exist δ₃ = δ₃(ϵ) > 0 and X₃ = X₃(p, ϵ) > 1 such that for x ∈ (0, δ₃) and for x ∈ (X₃, +∞).

Case D. (p, q) ∈ E₄. Then for ϵ ∈ (0, (p + q)/2) and x > 0, making use of (1.1) and Taylor expansion, we have

(2.60)

Equation (2.60) implies that for any ϵ ∈ (0, (p + q)/2), there exist δ₄ = δ₄(ϵ) > 0 and X₄ = X₄(p, q, ϵ) > 1 such that for x ∈ (0, δ₄) and for x ∈ (X₄, +∞).

Case E. (p, q) ∈ E₅. Then for any ϵ ∈ (0, −q/2) and x > 0, making use of (1.1) and Taylor expansion, one has

(2.61)

Equation (2.61) leads to the conclusion that for any ϵ ∈ (0, −q/2), there exist δ₅ = δ₅(ϵ) > 0 and X₅ = X₅(q, ϵ) > 1 such that for x ∈ (0, δ₅) and for x ∈ (X₅, +∞).

Case F. (p, q) ∈ E₆. Then for any ϵ ∈ (0, −(p + q)/2) and x > 0, making use of (1.1) and Taylor expansion, one has

(2.62)

Equation (2.62) shows that for any ϵ ∈ (0, −(p + q)/2), there exist δ₆ = δ₆(ϵ) > 0 and X₆ = X₆(p, q, ϵ) > 1 such that for x ∈ (0, δ₆) and for x ∈ (X₆, +∞).

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

References

1 Chu Y.-M., Wang S.-S., and Zong C., Optimal lower power mean bound for the convex combination of harmonic and logarithmic means, Abstract and Applied Analysis. (2011) 2011, 9, 520648, https://doi.org/10.1155/2011/520648, 2817270, ZBL1217.26040.
10.1155/2011/520648
Google Scholar
2 Chu Y. M. and Long B. Y., Sharp inequalities between means, Mathematical Inequalities & Applications. (2011) 14, no. 3, 647–655, https://doi.org/10.7153/mia%2D14%2D55, 2850179, ZBL1223.26049.
10.7153/mia-14-55
Web of Science® Google Scholar
3 Wang M.-K., Chu Y.-M., Qiu Y.-F., and Qiu S.-L., An optimal power mean inequality for the complete elliptic integrals, Applied Mathematics Letters. (2011) 24, no. 6, 887–890, https://doi.org/10.1016/j.aml.2010.12.044, 2776155, ZBL1216.33044.
10.1016/j.aml.2010.12.044
Web of Science® Google Scholar
4 Long B.-Y. and Chu Y.-M., Optimal power mean bounds for the weighted geometric mean of classical means, Journal of Inequalities and Applications. (2010) 2010, 6, 905679, https://doi.org/10.1155/2010/905679, 2603078, ZBL1187.26016.
10.1155/2010/905679
Google Scholar
5 Xia W.-F., Chu Y.-M., and Wang G.-D., The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means, Abstract and Applied Analysis. (2010) 2010, 9, 604804, https://doi.org/10.1155/2010/604804, 2629623, ZBL1190.26038.
10.1155/2010/604804
Google Scholar
6 Chu Y.-M. and Xia W.-F., Two optimal double inequalities between power mean and logarithmic mean, Computers & Mathematics with Applications. (2010) 60, no. 1, 83–89, 2651886, https://doi.org/10.1016/j.camwa.2010.04.032, ZBL1205.26041.
10.1016/j.camwa.2010.04.032
Web of Science® Google Scholar
7 Chu Y.-M., Qiu Y.-F., and Wang M.-K., Sharp power mean bounds for the combination of Seiffert and geometric means, Abstract and Applied Analysis. (2010) 2010, 12, 108920, https://doi.org/10.1155/2010/108920, 2720028, ZBL1197.26054.
10.1155/2010/108920
Google Scholar
8 Shi M.-Y., Chu Y.-M., and Jiang Y.-P., Optimal inequalities among various means of two arguments, Abstract and Applied Analysis. (2009) 2009, 10, 694394, https://doi.org/10.1155/2009/694394, 2563995, ZBL1187.26017.
10.1155/2009/694394
Google Scholar
9 Chu Y.-M. and Xia W.-F., Two sharp inequalities for power mean, geometric mean, and harmonic mean, Journal of Inequalities and Applications. (2009) 2009, 6, 741923, https://doi.org/10.1155/2009/741923, 2563560, ZBL1187.26013.
10.1155/2009/741923
Google Scholar
10 Wu S. H., Generalization and sharpness of the power means inequality and their applications, Journal of Mathematical Analysis and Applications. (2005) 312, no. 2, 637–652, 2179102, https://doi.org/10.1016/j.jmaa.2005.03.050, ZBL1083.26018.
10.1016/j.jmaa.2005.03.050
Web of Science® Google Scholar
11 Hästö P. A., Optimal inequalities between Seiffert′s mean and power means, Mathematical Inequalities & Applications. (2004) 7, no. 1, 47–53, https://doi.org/10.7153/mia%2D07%2D06, 2035348, ZBL1049.26006.
10.7153/mia-07-06
Web of Science® Google Scholar
12 Alzer H. and Qiu S.-L., Inequalities for means in two variables, Archiv der Mathematik. (2003) 80, no. 2, 201–215, 1979035, https://doi.org/10.1007/s00013%2D003%2D0456%2D2, ZBL1020.26011.
10.1007/s00013-003-0456-2
Web of Science® Google Scholar
13 Alzer H., A power mean inequality for the gamma function, Monatshefte für Mathematik. (2000) 131, no. 3, 179–188, 1801745, https://doi.org/10.1007/s006050070007, ZBL0964.33002.
10.1007/s006050070007
Web of Science® Google Scholar
14 Pečarić J. E., Generalization of the power means and their inequalities, Journal of Mathematical Analysis and Applications. (1991) 161, no. 2, 395–404, 1132116, https://doi.org/10.1016/0022%2D247X(91)90339%2D2, ZBL0753.26009.
10.1016/0022-247X(91)90339-2
Web of Science® Google Scholar
15 Bullen P. S., Mitrinović D. S., and Vasić P. M., Means and Their Inequalities, 1988, 31, D. Reidel, Dordrecht, The Netherlands, 947142.
10.1007/978-94-017-2226-1
Google Scholar
16 Burk F., Notes: the geometric, logarithmic, and arithmetic mean inequality, The American Mathematical Monthly. (1987) 94, no. 6, 527–528, 1541119, https://doi.org/10.2307/2322844.
10.1080/00029890.1987.12000678
Web of Science® Google Scholar
17 Alzer H., Ungleichungen für mittelwerte, Archiv der Mathematik. (1986) 47, no. 5, 422–426, 870279, https://doi.org/10.1007/BF01189983, ZBL0585.26014.
10.1007/BF01189983
Google Scholar
18 Alzer H., Ungleichungen für (e/a)^a(b/e)^b, Elemente der Mathematik. (1985) 40, 120–123.
Google Scholar
19 Stolarsky K. B., The power and generalized logarithmic means, The American Mathematical Monthly. (1980) 87, no. 7, 545–548, 600913, https://doi.org/10.2307/2321420, ZBL0455.26008.
10.1080/00029890.1980.11995086
Web of Science® Google Scholar
20 Pittenger A. O., Inequalities between arithmetic and logarithmic means, Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika. (1980) 678–715, 15–18, 623217, ZBL0469.26009.
Google Scholar
21 Pittenger A. O., The symmetric, logarithmic and power means, Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika. (1980) 678–715, 19–23, 623218, ZBL0469.26010.
Google Scholar
22 Lin T. P., The power mean and the logarithmic mean, The American Mathematical Monthly. (1974) 81, 879–883, 0354972, https://doi.org/10.2307/2319447, ZBL0292.26015.
10.1080/00029890.1974.11993684
Web of Science® Google Scholar

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A Best Possible Double Inequality for Power Mean

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2. Main Result

Acknowledgments

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A Best Possible Double Inequality for Power Mean

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2. Main Result

Acknowledgments

References

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