In this paper we find the best possible lower power mean bounds for the Neuman-Sándor mean and present the sharp bounds for the ratio of the Neuman-Sándor and identric means.

1. Introduction

For p ∈ ℝ the pth power mean M_p(a, b), Neuman-Sándor Mean M(a, b) [1], and identric mean I(a, b) of two positive numbers a and b are defined by

()

respectively, where

is the inverse hyperbolic sine function.

The main properties for M_p(a, b) and I(a, b) are given in [2]. It is well known that M_p(a, b) is continuously and strictly increasing with respect to p ∈ ℝ for fixed a, b > 0 with a ≠ b. Recently, the power, Neuman-Sándor, and identric means have been a subject of intensive research. In particular, many remarkable inequalities for these means can be found in the literature [3–26].

Let H(a, b) = 2ab/(a + b),

, L(a, b) = (b − a)/(log b − log a),

, A(a, b) = (a + b)/2, T(a, b) = (a − b)/[2arctan((a − b)/(a + b))],

, and C(a, b) = (a² + b²)/(a + b) be the harmonic, geometric, logarithmic, first Seiffert, arithmetic, second Seiffert, quadratic, and contraharmonic means of two positive numbers a and b with a ≠ b, respectively. Then, it is well known that the inequalities

()

hold for all a, b > 0 with a ≠ b.

The following sharp bounds for L, I, (IL) ^1/2, and (I + L)/2 in terms of power means are presented in [27–32]:

()

for all a, b > 0 with a ≠ b.

Pittenger [31] found the greatest value r₁ and the least value r₂ such that the double inequality

()

holds for all a, b > 0, where L_r(a, b) is the rth generalized logarithmic means which is defined by

()

The following sharp power mean bounds for the first Seiffert mean P(a, b) are given in [10, 33]:

()

for all a, b > 0 with a ≠ b.

In [17], the authors answered the question: for α ∈ (0,1), what are the greatest value p and the least value q such that the double inequality

()

holds for all a, b > 0 with a ≠ b?

Neuman and Sándor [1] established that

()

for all a, b > 0 with a ≠ b.

Let 0 < a, b ≤ 1/2 with a ≠ b, a^′ = 1 − a and b^′ = 1 − b. Then, the Ky Fan inequalities

()

were presented in [1].

In [24], Li et al. found the best possible bounds for the Neuman-Sándor mean M(a, b) in terms of the generalized logarithmic mean L_r(a, b). Neuman [25] and Zhao et al. [26] proved that the inequalities

()

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

, β ≥ 1/3,

, μ ≥ 1/6, α₁ ≥ 2/9,

, α₂ ≥ 1/3, and

In [7], Sándor and Trif proved that the inequalities

()

hold for all a, b > 0 with a ≠ b.

Neuman and Sándor [15] and Gao [20] proved that α₁ = 1, β₁ = e/2, α₂ = 1, , α₃ = 1, β₃ = 3/e, α₄ = e/π, β₄ = 1, α₅ = 1, and β₅ = 2e/π are the best possible constants such that the double inequalities α₁ < A(a, b)/I(a, b) < β₁, α₂ < I(a, b)/M_2/3(a, b) < β₂, α₃ < I(a, b)/He(a, b) < β₃, α₄ < P(a, b)/I(a, b) < β₄, and α₅ < T(a, b)/I(a, b) < β₅ hold for all a, b > 0 with a ≠ b, where is the Heronian mean of a and b.

In [34], Sándor established that

()

for all a, b > 0 with a ≠ b.

It is not difficult to verify that the inequality

()

holds for all a, b > 0 with a ≠ b.

From inequalities (10), (14), and (15), one has

()

for all a, b > 0 with a ≠ b.

It is the aim of this paper to find the best possible lower power mean bound for the Neuman-Sándor mean M(a, b) and to present the sharp constants α and β such that the double inequality

()

holds for all a, b > 0 with a ≠ b.

2. Main Results

Theorem 1. is the greatest value such that the inequality

()

holds for all a, b > 0 with a ≠ b.

Proof. From (1) and (2), we clearly see that both M(a, b) and M_p(a, b) are symmetric and homogenous of degree one. Without loss of generality, we assume that b = 1 and a = x > 1.

Let , then from (1) and (2) one has

()

Let

()

Then, simple computations lead to

()

where

()

where

()

where

()

for x > 1.

Equation (33) and inequality (34) imply that is strictly decreasing on [1, +∞). Then, the inequality (31) and (32) lead to the conclusion that there exists x₁ > 1, such that is strictly increasing on [1, x₁] and strictly decreasing on [x₁, +∞).

From (29) and (30) together with the piecewise monotonicity of , we clearly see that there exists x₂ > x₁ > 1, such that f₂(x) is strictly increasing on [1, x₂] and strictly decreasing on [x₂, +∞).

It follows from (26)–(28) and the piecewise monotonicity of f₂(x) that there exists x₃ > x₂ > 1, such that f₁(x), is strictly increasing on [1, x₃] and strictly decreasing on [x₃, +∞).

From (23)–(25) and the piecewise monotonicity of f₁(x) we see that there exists x₄ > x₃ > 1, such that f(x) is strictly increasing on (1, x₄] and strictly decreasing on [x₄, +∞).

Therefore, for x > 1 follows easily from (19)–(22) and the piecewise monotonicity of f(x).

Next, we prove that is the greatest value such that for all x > 1.

For any ε > 0 and x > 1, from (1) and (2), one has

()

Inequality (35) implies that for any ε > 0, there exists X = X(ε) > 1, such that for x ∈ (X, +∞).

Remark 2. 4/3 is the least value such that inequality (16) holds for all a, b > 0 with a ≠ b, namely, M_4/3(a, b) is the best possible upper power mean bound for the Neuman-Sándor mean M(a, b).

In fact, for any ε ∈ (0,4/3) and x > 0, one has

()

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

()

Equations (36) and (37) imply that for any ε ∈ (0,4/3) there exists δ = δ(ε) > 0, such that M(1 + x, 1) > M_(4/3)−ε(1 + x, 1) for x ∈ (0, δ).

Theorem 3. For all a, b > 0 with a ≠ b, one has

()

with the best possible constants 1 and

Proof. From (2) and (3), we clearly see that both M(a, b) and I(a, b) are symmetric and homogenous of degree one. Without loss of generality, we assume that b = 1 and a = x > 1. Let

()

Then, simple computations lead to

()

where

()

where

()

for x > 1.

From (46) and (47), we clearly see that is strictly increasing on [1, +∞). Then, (45) leads to the conclusion that is strictly increasing on [1, +∞).

Equations (43) and (44) together with the monotonicity of impliy that f₂(x) > 0 for x > 1. Then, (42) leads to the conclusion that f₁(x) is strictly increasing on [1, +∞).

It follows from equations (40) and (41) together with the monotonicity of f₁(x) that f(x) is strictly increasing on (1, +∞).

Therefore, Theorem 3 follows from (39) and the monotonicity of f(x) together with the facts that

()

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants nos. 11071069 and 11171307, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant no. T200924.

References

1 Neuman E. and Sándor J., On the Schwab-Borchardt mean, Mathematica Pannonica. (2003) 14, no. 2, 253–266, MR2079318, ZBL1053.26015.
Google Scholar
2 Bullen P. S., Mitrinović D. S., and Vasić P. M., Means and Their Inequalities, 1988, 31, D. Reidel Publishing, Dordrecht, The Netherlands, Mathematics and its Applications, MR947142.
10.1007/978-94-017-2226-1
Google Scholar
3 Sándor J., On the identric and logarithmic means, Aequationes Mathematicae. (1990) 40, no. 2-3, 261–270, https://doi.org/10.1007/BF02112299, MR1069798, ZBL0717.26014.
10.1007/BF02112299
Google Scholar
4 Pečarić J. E., Generalization of the power means and their inequalities, Journal of Mathematical Analysis and Applications. (1991) 161, no. 2, 395–404, https://doi.org/10.1016/0022-247X(91)90339-2, MR1132116, ZBL0753.26009.
10.1016/0022-247X(91)90339-2
Web of Science® Google Scholar
5 Sándor J., A note on some inequalities for means, Archiv der Mathematik. (1991) 56, no. 5, 471–473, https://doi.org/10.1007/BF01200091, MR1100572, ZBL0693.26005.
10.1007/BF01200091
Web of Science® Google Scholar
6 Sándor J. and Raşa I., Inequalities for certain means in two arguments, Nieuw Archief voor Wiskunde. (1997) 15, no. 1-2, 51–55, MR1470436, ZBL0938.26011.
Google Scholar
7 Sándor J. and Trif T., Some new inequalities for means of two arguments, International Journal of Mathematics and Mathematical Sciences. (2001) 25, no. 8, 525–532, https://doi.org/10.1155/S0161171201003064, MR1827637, ZBL1002.26018.
10.1155/S0161171201003064
Google Scholar
8 Trif T., On certain inequalities involving the identric mean in n variables, Universitatis Babeş-Bolyai. (2001) 46, no. 4, 105–114, MR1989720, ZBL1027.26024.
Google Scholar
9 Alzer H. and Qiu S.-L., Inequalities for means in two variables, Archiv der Mathematik. (2003) 80, no. 2, 201–215, https://doi.org/10.1007/s00013-003-0456-2, MR1979035, ZBL1020.26011.
10.1007/s00013-003-0456-2
Web of Science® Google Scholar
10 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-07-06, MR2035348, ZBL1049.26006.
10.7153/mia-07-06
Web of Science® Google Scholar
11 Wu S., Generalization and sharpness of the power means inequality and their applications, Journal of Mathematical Analysis and Applications. (2005) 312, no. 2, 637–652, https://doi.org/10.1016/j.jmaa.2005.03.050, MR2179102, ZBL1083.26018.
10.1016/j.jmaa.2005.03.050
Web of Science® Google Scholar
12 Kouba O., New bounds for the identric mean of two arguments, Journal of Inequalities in Pure and Applied Mathematics. (2008) 9, no. 3, article 71, 6, MR2476649, ZBL1172.26314.
Google Scholar
13 Zhu L., New inequalities for means in two variables, Mathematical Inequalities & Applications. (2008) 11, no. 2, 229–235, https://doi.org/10.7153/mia-11-17, MR2410275, ZBL1154.26029.
10.7153/mia-11-17
Web of Science® Google Scholar
14 Zhu L., Some new inequalities for means in two variables, Mathematical Inequalities & Applications. (2008) 11, no. 3, 443–448, https://doi.org/10.7153/mia-11-33, MR2431208, ZBL1154.26029.
10.7153/mia-11-33
Web of Science® Google Scholar
15 Neuman E. and Sándor J., Companion inequalities for certain bivariate means, Applicable Analysis and Discrete Mathematics. (2009) 3, no. 1, 46–51, https://doi.org/10.2298/AADM0901046N, MR2499306, ZBL1199.26087.
10.2298/AADM0901046N
Web of Science® Google Scholar
16 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, MR2563560, ZBL1187.26013.
10.1155/2009/741923
Google Scholar
17 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, MR2720028, ZBL1197.26054.
10.1155/2010/108920
Google Scholar
18 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, MR2603078, ZBL1187.26016.
10.1155/2010/905679
Google Scholar
19 Wang M.-K., Chu Y.-M., and Qiu Y.-F., Some comparison inequalities for generalized Muirhead and identric means, Journal of Inequalities and Applications. (2010) 2010, 10, 295620, https://doi.org/10.1155/2010/295620, MR2592865, ZBL1187.26018.
10.1155/2010/295620
Google Scholar
20 Gao S., Inequalities for the Seiffert′s means in terms of the identric mean, Journal of Mathematical Sciences. (2011) 10, no. 1-2, 23–31, MR2906381.
Google Scholar
21 Chu Y.-M., Wang M.-K., and Wang Z.-K., A sharp double inequalities between harmonic and identric means, Abstract and Applied Analysis. (2011) 2011, 7, 657935, https://doi.org/10.1155/2011/657935.
10.1155/2011/657935
Google Scholar
22 Qiu Y.-F., Wang M.-K., Chu Y.-M., and Wang G.-D., Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean, Journal of Mathematical Inequalities. (2011) 5, no. 3, 301–306, https://doi.org/10.7153/jmi-05-27, MR2858507, ZBL1226.26019.
10.7153/jmi-05-27
Web of Science® Google Scholar
23 Wang M.-K., Wang Z.-K., and Chu Y.-M., An optimal double inequality between geometric and identric means, Applied Mathematics Letters. (2012) 25, no. 3, 471–475, https://doi.org/10.1016/j.aml.2011.09.038, MR2856016, ZBL1247.26040.
10.1016/j.aml.2011.09.038
Web of Science® Google Scholar
24 Li Y. M., Long B. Y., and Chu Y. M., Sharp bounds for the Neuman-Sándor mean in terms of generalized logarithmic mean, Journal of Mathematical Inequalities. (2012) 6, no. 4, 567–577.
10.7153/jmi-06-54
Web of Science® Google Scholar
25 Neuman E., A note on a certain bivariate mean, Journal of Mathematical Inequalities. (2012) 6, no. 4, 637–643.
10.7153/jmi-06-62
Web of Science® Google Scholar
26 Zhao T. H., Chu Y. M., and Liu B. Y., Optimal bounds for Neuman-Sándor mean in terms of the convex combinations of harmonic, geometric, quadratic, and contraharmonic means, Abstract and Applied Analysis. (2012) 2012, 9, 302635, https://doi.org/10.1155/2012/302635.
10.1155/2012/302635
Google Scholar
27 Baumgartner K., Zur Unauflösbarkeit für , Elemente der Mathematik. (1985) 40, no. 5, MR881898, ZBL0579.20004.
Google Scholar
28 Alzer H., Ungleichungen für Mittelwerte, Archiv der Mathematik. (1986) 47, no. 5, 422–426, https://doi.org/10.1007/BF01189983, MR870279, ZBL0585.26014.
10.1007/BF01189983
Google Scholar
29 Burk F., Notes: the geometric, logarithmic, and arithmetic mean inequality, The American Mathematical Monthly. (1987) 94, no. 6, 527–528, https://doi.org/10.2307/2322844, MR1541119.
10.1080/00029890.1987.12000678
Web of Science® Google Scholar
30 Lin T. P., The power mean and the logarithmic mean, The American Mathematical Monthly. (1974) 81, 879–883, MR0354972, https://doi.org/10.2307/2319447, ZBL0292.26015.
10.1080/00029890.1974.11993684
Web of Science® Google Scholar
31 Pittenger A. O., Inequalities between arithmetic and logarithmic means, Univerzitet u Beogradu. (1980) no. 678–715, 15–18, MR623217, ZBL0469.26009.
Google Scholar
32 Pittenger A. O., The symmetric, logarithmic and power means, Univerzitet u Beogradu. (1980) no. 678–715, 19–23, MR623218, ZBL0469.26010.
Google Scholar
33 Jagers A. A., Solution of problem 887, Nieuw Archief voor Wiskunde. (1994) 12, 230–231.
Google Scholar
34 Sándor J., On certain identities for means, Universitatis Babeş-Bolyai. (1993) 38, no. 4, 7–14, MR1434724, ZBL0831.26013.
Google Scholar

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Bounds of the Neuman-Sándor Mean Using Power and Identric Means

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Acknowledgments

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Bounds of the Neuman-Sándor Mean Using Power and Identric Means

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