Abstract
In this article, we answer the question: For p, ω ∈ ℝ with ω > 0 and p(ω - 2) ≠ 0, what are the greatest value r1 = r1(p, ω) and the least value r2 = r2(p, ω) such that the double inequality holds for all a, b > 0 with a ≠ b? Here Hp,ω(a, b) and M r (a, b) denote the generalized Heronian mean and r th power mean of two positive numbers a and b, respectively.
2010 Mathematics Subject Classification: 26E60.
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1 Introduction
In the recent past, the bivariate means have been the subject of intensive research. In particular, many remarkable inequalities can be found in the literature [1–26].
The power mean M r (a, b) of order r of two positive numbers a and b is defined by
It is well-known that M r (a, b) is continuous and strictly increasing with respect to r ∈ ℝ for fixed a, b > 0 with a ≠ b. Let A(a, b) = (a + b)/2, , H(a, b) = 2ab/(a + b), I(a, b) = 1/e(bb/aa )1/(b-a)(b ≠ a), I(a, b) = a (b = a), and L(a, b) = (b-a)/(log b- log a) (b ≠ a), L(a, b) = a (b = a) be the arithmetic, geometric, harmonic, identric, and logarithmic means of two positive numbers a and b, respectively. Then
for all a, b > 0, and each inequality becomes equality if and only if a = b.
The classical Heronian mean He(a, b) of two positive numbers a and b is defined by ([27], see also [28])
In [27], Alzer and Janous established the following sharp double inequality (see also [[28], p. 350]):
for all a, b > 0 with a ≠ b.
Mao [29] proved that
for all a, b > 0 with a ≠ b, and M1/ 3(a, b) is the best possible lower power mean bound for the sum .
For any α ∈ (0, 1), Janous [30] found the greatest value p and the least value q such that M p (a, b) < αA(a, b) + (1 - α)G(a, b) < M q (a, b) for all a, b > 0 with a ≠ b.
The following sharp bounds for L, I, (LI)1/ 2and (L + I)/ 2 in terms of power mean are given in [10, 21–25, 31, 32]:
for all a, b > 0 with a ≠ b.
In [6, 7] the authors established the following sharp inequalities:
for all for all a, b > 0 with a ≠ b and α ∈ (0, 1).
For ω ≥ 0 and p ∈ ℝ the generalized Heronian mean Hp,ω(a, b) of two positive numbers a and b was introduced in [33] as follows:
It is not difficult to verify that Hp,ω(a, b) is continuous with respect to p ∈ ℝ for fixed a, b > 0 and ω ≥ 0, strictly increasing with respect to p ∈ ℝ for fixed a, b > 0 with a ≠ b and ω ≥ 0, strictly decreasing with respect to ω ≥ 0 for fixed a, b > 0 with a ≠ b and p > 0 and strictly increasing with respect to ω ≥ 0 for fixed a, b > 0 with a ≠ b and p < 0.
From (1.1) and (1.3) together with (1.4) we clearly see that Hp,0(a, b) = M p (a, b), , H0,ω(a, b) = M0(a, b) and H1,1(a, b) = H e (a, b) for all a, b > 0 and ω ≥ 0.
The purpose of this article is to answer the question: For p, ω ∈ ℝ with ω > 0 and p(ω - 2) ≠ 0, what are the greatest value r1 = r1(p, ω) and the least value r2 = r2(p, ω) such that the double inequality holds for all a, b > 0 with a ≠ b?
2 Main result
In order to establish our main results we need the following Lemma 2.1.
Lemma 2.1. (see [30]). (ω + 2)2> 2ω+2for ω ∈ (0, 2), and (ω + 2)2< 2ω+2for ω ∈ (2, +∞).
Theorem 2.1. For all a, b > 0 with a ≠ b we have
for (p, ω) ∈ {(p, ω): p > 0, ω > 2} ∪ {(p, ω): p < 0, 0 < ω < 2} and
for (p, ω) ∈ {(p, ω): p > 0, 0 < ω < 2} ∪ {(p, ω): p < 0, ω > 2}, and the parameters and are the best possible in either case.
Proof. Without loss of generality, we can assume that a > b and put .
Firstly, we compare the value of with that of Hp,ω(a, b). From (1.1) and (1.4) we have
Let
Then simple computations lead to
We divide the comparison into two cases.
Case 1. If (p, ω) ∈ {(p, ω): p > 0, ω > 2} ∪ {(p, ω): p < 0, 0 < ω < 2}, then from (2.8) we clearly see that
for t > 1.
Therefore, follows from (2.1)-(2.7) and (2.9).
Case 2. If (p, ω) ∈ {(p, ω): p > 0, 0 < ω < 2} ∪ {(p, ω): p < 0, ω > 2}, then (2.8) leads to
for t > 1.
Therefore, follows from (2.1)-(2.7) and (2.10).
Secondly, we compare the value of with that of Hp,ω(a, b). From (1.1) and (1.4) we have
Let
Then simple computations lead to
We divide the comparison into four cases.
Case A. If p > 0 and ω > 2, then from (2.15) and (2.18)-(2.20) together with Lemma 2.1 we clearly see that
and there exists a1> 1 such that
for t ∈ [1, a1) and
for t ∈ (a1, +∞).
From (2.24) and (2.25) we know that H(t) is strictly increasing in [1, a1] and strictly decreasing in [a1, +∞). Then (2.22) and (2.23) together with the monotonicity of H(t) imply that there exists a2> 1 such that H(t) > 0 for t ∈ [1, a2) and H(t) < 0 for t ∈ (a2, +∞). It follows from (2.17) that G(t) is strictly increasing in [1, a2] and strictly decreasing in [a2, +∞).
From (2.16) and (2.21) together with the monotonicity of G(t) we know that there exists a3> 1 such that G(t) > 0 for t ∈ (1, a3) and G(t) < 0 for t ∈ (a3, +∞). Then (2.14) leads to that F (t) is strictly increasing in [1, a3] and strictly decreasing in [a3, +∞).
Therefore, follows from (2.11)-(2.13) and the monotonicity of F (t).
Case B. If p > 0 and 0 < ω < 2, then (2.15) and (2.18)-(2.20) together with Lemma 2.1 lead to
and there exists b1> 1 such that
for t ∈ [1, b1) and
for t ∈ (b1, +∞).
From (2.27)-(2.30) we clearly see that there exists b2> 1 such that H(t) < 0 for t ∈ [1, b2) and H(t) > 0 for t ∈ (b2, +∞). Then (2.17) implies that G(t) is strictly decreasing in [1, b2] and strictly increasing in [b2, +∞). It follows from (2.16) and (2.26) together with the monotonicity of G(t) that there exists b3> 1 such that G(t) < 0 for t ∈ (1, b3) and G(t) > 0 for t ∈ (b3, +∞). Then (2.14) leads to that F(t) is strictly decreasing in [1, b3] and strictly increasing in [b3, +∞).
Therefore, follows from (2.11)-(2.13) and the monotonicity of F (t).
Case C. If p < 0 and ω > 2, then it follows from (2.15) and (2.18)-(2.20) together with Lemma 2.1 that
for t ∈ [1, +∞).
From (2.32)-(2.34) we clearly see that there exists c1> 1 such that H(t) > 0 for t ∈ [1, c1) and H(t) < 0 for t ∈ (c1, +∞). Then (2.17) implies that G(t) is strictly decreasing in [1, c1] and strictly increasing in [c1, +∞).
It follows from (2.16) and (2.31) together with the monotonicity of G(t) that there exists c2> 1 such that G(t) < 0 for t ∈ (1, c2) and G(t) > 0 for t ∈ (c2, +∞). Then (2.14) leads to that F (t) is strictly decreasing in [1, c2] and strictly increasing in [c2, +∞).
Therefore, follows from (2.11)-(2.13) and the monotonicity of F (t).
Case D. If p < 0 and 0 < ω < 2, then (2.15) and (2.18)-(2.20) together with Lemma 2.1 lead to
for t > 1.
From (2.17) and (2.36)-(2.38) we clearly see that there exists d1> 1 such that G(t) is strictly increasing in [1, d1] and strictly decreasing in [d1, +∞). It follows from (2.14), (2.16), (2.35) and the monotonicity of G(t) that there exists d2> 1 such that F (t) is strictly increasing in [1, d2] and strictly decreasing in [d2, +∞).
Therefore, follows from (2.11)-(2.13) and the monotonicity of F(t).
Thirdly, we prove that the parameter is the best possible in either case.
For any p, r ∈ ℝ with pr ≠ 0, ω ≥ 0 and x > 0, one has
Let x → 0, then the Taylor expansion leads to
If (p, ω) ∈{(p, ω): p > 0, ω > 2} ∪ {(p, ω): p < 0, 0 < ω < 2}, then equations (2.39) and (2.40) imply that for any there exists δ1 = δ1(r, p, ω) > 0 such that M r (1, 1 + x) > Hp,ω(1, 1 + x) for x ∈ (0, δ1).
If (p, ω) {(p, ω): p > 0, 0 < ω < 2} ∪ {(p, ω): p < 0, ω > 2}, then from (2.39) and (2.40) we know that for any there exists δ2 = δ2(r, p, ω) > 0 such that M r (1, 1 + x) < Hp, ω(1, 1 + x) for x ∈ (0, δ2).
Finally, we prove that the parameter is the optimal parameter in either case.
For any p, r ∈ ℝ with pr > 0, ω ≥ 0 and x > 0 we have
If (p, ω) ∈ {(p, ω): p > 0, ω > 2} ∪ {(p, ω): p < 0, 0 < ω < 2}, then equation (2.41) implies that for any there exists X1 = X1(r, p, ω) > 1 such that M r (1, x) < Hp, ω(1, x) for x ∈ (X1, +∞).
If (p, ω) ω ∈ {(p, ω): p > 0, 0 < ω < 2} ∪ {(p, ω): p < 0, ω > 2}, then equation (2.41) leads to that for any there exists X2 = X2(r, p, ω) > 1 such that M r (1, x) > Hp, ω(1, x) for x ∈ (X2, +∞).
References
Wang M-K, Chu Y-M, Qiu Y-F: Some comparison inequalities for generalized Muirhead and identric means. J Inequal Appl 2010, 10.
Long B-Y, Chu Y-M: Optimal inequalities for generalized logarithmic, arithmetic and geometric means. J Inequal Appl 2010, 10.
Chu Y-M, Long B-Y: Best possible inequalities between generalized logarithmic mean and classical means. Abstr Appl Anal 2010, 14.
Xia W-F, Chu Y-M, Wang G-D: The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means. Abstr Appl Anal 2010, 10.
Long B-Y, Chu Y-M: Optimal power mean bounds for the weighted geometric mean of classical means. J Inequal Appl 2010, 6.
Chu Y-M, Xia W-F: Two sharp inequalities for power mean, geometric mean and harmonic mean. J Inequal Appl 2009, 6.
Shi M-Y, Chu Y-M, Jiang Y-P: Optimal inequalities among various means of two arguments. Abstr Appl Anal 2009, 10.
Chu Y-M, Xia W-F: Inequalities for generalized logarithmic means. J Inequal Appl 2009, 7.
Wang M-K, Chu Y-M, Qiu S-L, Jiang Y-P: Bounds for the perimeter of an ellipse. J Approx Theory 2012, 164(7):928–937. 10.1016/j.jat.2012.03.011
Alzer H, Qiu S-L: Inequalities for means in two variables. Arch Math (Basel) 2003, 80(2):201–215. 10.1007/s00013-003-0456-2
Wang M-K, Chu Y-M, Qiu S-L, Jiang Y-P: Convexity of the complete elliptic integrals of the first kind with respect to Hölder means. J Math Anal Appl 2012, 388(2):1141–1146. 10.1016/j.jmaa.2011.10.063
Wang M-K, Chu Y-M, Qiu Y-F, Qiu S-L: An optimal power mean inequality for the complete elliptic integrals. Appl Math Lett 2011, 24(6):887–890. 10.1016/j.aml.2010.12.044
Chu Y-M, Xia W-F: Two optimal double inequalities between power mean and logarithmic mean. Comput Math Appl 2010, 60(1):83–89. 10.1016/j.camwa.2010.04.032
Chu Y-M, Wang M-K, Qiu S-L: Optimal combinations bounds of root-square and arithmetic means for Toader mean. Proc Indian Acad Sci Math Sci 2012, 122(1):41–51. 10.1007/s12044-012-0062-y
Wang M-K, Wang Z-K, Chu Y-M: An optimal double inequality between geometric and identric means. Appl Math Lett 2012, 25(3):471–475. 10.1016/j.aml.2011.09.038
Chu Y-M, Zong C: Optimal convex combination bounds of Seiffert and geometric means for the arithmetic mean. J Math Inequal 2011, 5(3):429–434.
Qiu Y-F, Wang M-K, Chu Y-M, Wang G-D: Two sharp inequalities for Lehmer mean, identric mean and logarithmic mean. J Math Inequal 2011, 5(3):301–306.
Wang M-K, Qiu Y-F, Chu Y-M: Sharp bounds for Seiffert means in terms of Lehmer means. J Math Inequal 2010, 4(4):581–586.
Chu Y-M, Wang M-K, Wang Z-K: A sharp double inequality between harmonic and identric means. Abstr Appl Anal 2011, 7.
Chu Y-M, Wang M-K, Qiu S-L, Qiu Y-F: Sharp generalized Seiffert mean bounds for Toader mean. Abstr Appl Anal 2011, 8.
Burk F: The geometric, logarithmic, and arithmetic mean inequality. Am Math Monthly 1987, 94(6):527–528. 10.2307/2322844
Stolarsky KB: The power and generalized logarithmic means. Am Math Monthly 1980, 87(7):545–548. 10.2307/2321420
Pittenger AO: Inequalities between arithmetic and logarithmic means. Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz 1980, 678–715: 15–18.
Pittenger AO: The symmetric, logarithmic and power means. Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz 1980, 678–715: 19–23.
Lin TP: The power mean and the logarithmic mean. Am Math Monthly 1974, 81: 879–883. 10.2307/2319447
Carlson BC: The logarithmic mean. Am Math Monthly 1972, 79: 615–618. 10.2307/2317088
Alzer H, Janous W: Solution of problem 8*. Crux Math 1987, 13: 173–178.
Bullen PS, Mitrinović DS, Vasić PM: Means and their inequalities. D Reidel Publishing Co, Dordrecht 1988.
Mao Q-J: Power mean, logarithmic mean and Heronian dual mean of two positive numbers. J Suzhou Coll Edu 1999, 16(1–2):82–85. (Chinese)
Janous W: A note on generalized Heronian means. Math Inequal Appl 2001, 4(3):369–375.
Alzer H: Ungleichungen für ( e / a ) a ( b / e ) b . Elem Math 1985, 40: 120–123.
Alzer H: Ungleichungen für Mittelwerte. Arch Math (Basel) 1986, 47(5):422–426. 10.1007/BF01189983
Shi H-N, Bencze M, Wu Sh-H, Li D-M: Schur convexity of generalized Heronian means involving two parameters. J Inequal Appl 2008, 9.
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307, the Natural Science Foundation of Hunan Province under Grant 09JJ6003, and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.
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Y-ML provided the main idea in this article. B-YL carried out the proof of the inequalities in Theorem 2.1. Y-MC carried out the proof of the optimality in Theorem 2.1. All authors read and approved the final manuscript.
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Li, YM., Long, BY. & Chu, YM. Sharp bounds by the power mean for the generalized Heronian mean. J Inequal Appl 2012, 129 (2012). https://doi.org/10.1186/1029-242X-2012-129
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DOI: https://doi.org/10.1186/1029-242X-2012-129