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Sharp One-Parameter Mean Bounds for Yang Mean.

1. Introduction

Let p [member of] R and a,b > 0 with a [not equal to] b. Then the pth one-parameter mean [J.sub.p](a,b), pth power mean [M.sub.p](a,b), harmonic mean H(a,b), geometric mean G(a,b), logarithmic mean L(a, b), first Seiffert mean P(a, b), identric mean I(a, b), arithmetic mean A(a, b), Yang mean U(a, b), second Seiffert mean T(a, b), and quadratic mean Q(a, b) are, respectively, defined by

[mathematical expression not reproducible]. (1)

It is well known that both the means [J.sub.p](a, b) and [M.sub.p](a, b) are continuous and strictly increasing with respect to p [member of] R for fixed a,b > 0 with a [not equal to] b. Recently, the one-parameter mean [J.sub.p](a,b) and Yang mean U(a,b) have attracted the attention of many researchers.

Alzer [1] proved that the inequalities

G (a, b) < [square root of [J.sub.p] (a,b) [J.sub.-p] (a,b)] < L (a, b) < [J.sub.p] (a,b) + [J.sub.-p] (a,b)/2 < A(a,b) (2)

hold for all a,b > 0 with a [not equal to] b and p [not equal to] 0.

In [2, 3], the authors discussed the monotonicity and logarithmic convexity properties of the one-parameter mean [J.sub.p](a,b).

In [4, 5], the authors proved that the double inequalities

[mathematical expression not reproducible], (3)

hold for all a,b > 0 with a [not equal to] b and [alpha] [member of] (0,1) if and only if [p.sub.1] [less than or equal to] [alpha]/(2 - [alpha]), [q.sub.1] [greater than or equal to] [alpha], [p.sub.2] [less than or equal to] 3[alpha] - 2, and [q.sub.2] [greater than or equal to] [alpha]/(2 - [alpha]).

Xia et al. [6] proved that the double inequality

[J.sub.(3[alpha] - 1)/2] (a,b) < [alpha]A(a,b) + (1 - [alpha])G (a, b) < [J.sub.(3[alpha] - 1)/2] (a,b) (4)

holds for all a,b > 0 with a [not equal to] b if a [member of] (0,2/3), and inequality (4) is reversed if [alpha] [member of] (2/3,1).

Gao and Niu [7] presented the best possible parameters p and q such that the double inequality [J.sub.p](a,b) < [A.sup.[alpha]] (a,b)[G.sup.[beta]](a,b) [H.sup.1-[alpha]-[beta]] (a,b) < [J.sub.q](a,b) holds for all a,b > 0 with a [not equal to] b and [alpha] + [beta] [member of] (0,1).

In [8, 9], the authors proved that the double inequalities

[mathematical expression not reproducible] (5)

hold for all a,b > 0 with a [not equal to] b if and only if [[lambda].sub.1] [less than or equal to] 2/(2 - [pi]), [[mu].sub.1] [greater than or equal to] 2, [[lambda].sub.2] < 1/2, and [[lambda].sub.2] [greater than or equal to] 1/(e - 1).

Xia et al. [10] found that [M.sub.(1+2p)/3] (a,b) is the best possible lower power mean bound for the one-parameter mean [J.sub.p](a,b) if p [member of] (-2,-1/2) [union] (1, [infinity]) and [M.sub.(1+2p)/3](a,b) is the best possible upper power mean bound for the one-parameter mean [J.sub.p](a, b) if p [member of] (-[infinity], -1/2) [union] (-1/2,1).

For all a,b > 0 with a [not equal to] b, Yang [11] provided the bounds for the Yang mean U(a, b) in terms of other bivariate means as follows:

[mathematical expression not reproducible]. (6)

In [12,13], the authors proved that the double inequalities

[mathematical expression not reproducible], (7)

hold for all a,b > 0 with a [not equal to] b if and only if p [less than or equal to] [p.sub.0], q [greater than or equal to] 1/5, [lambda] [greater than or equal to] 1/5, [mu] [less than or equal to] [p.sub.1], [alpha] [less than or equal to] 2 log 2/(2 log [pi] - log 2), and [beta] [greater than or equal to] 4/3, where [p.sub.0] = 0.1941 ... is the unique solution of the equation p log(2/[pi]) - log(1 + [2.sup.1-p]) + log 3 = 0 on the interval (1/10, [infinity]), and [p.sub.1] = log([pi] - 2)/log 2 = 0.1910 ....

Very recently, Zhou et al. [14] proved that [alpha] = 1/2 and [beta] = log 3/(1 + log 2) = 0.6488 ... are the best possible parameters such that the double inequality

[mathematical expression not reproducible] (8)

holds for all a,b > 0 with a [not equal to] b.

The aim of this paper is to present the best possible parameters [alpha] and [beta] such that the double inequality [J.sub.[alpha]](a, b) < U(a, b) < [J.sub.[beta]](a, b) holds for all a,b > 0 with a [not equal to] b.

2. Main Result

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

Lemma 1. Let p [member of] R, and

[mathematical expression not reproducible]. (9)

Then the following statements are true:

(1) if p = 3/2, then f(x, p) > 0 for all x [member of] (1, [infinity]);

(2) if p = [square root of 2]/([pi] - [square root of 2]) = 0.8187 ..., then there exists [lambda] [member of] (1, [infinity]) such that f(x, p) < 0 for x [member of] (1, [lambda]) and f(x, p) > 0 for x [member of] ([lambda], [infinity]).

Proof. For part (1), if p = 3/2, then (9) becomes

f (x, p) = 1/4[(x - 1).sup.6] ([x.sup.2] + 2x + 2) (2[x.sup.2] + 2x + 1) x (3[x.sup.2] + 4x + 3).

Therefore, part (1) follows from (10).

For part (2), let [mathematical expression not reproducible]. Then elaborated computations lead to

[mathematical expression not reproducible] (11)

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

[mathematical expression not reproducible] (15)

[mathematical expression not reproducible] (16)

[mathematical expression not reproducible] (17)

[mathematical expression not reproducible] (18)

[mathematical expression not reproducible] (19)

[mathematical expression not reproducible] (20)

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible], (23)

[mathematical expression not reproducible]. (24)

[mathematical expression not reproducible] (25)

Note that

[mathematical expression not reproducible]. (25)

It follows from (24) and (25) that

[mathematical expression not reproducible]. (26)

From (23) and (26) we clearly see that [f.sub.8](x, p) is strictly increasing with respect to x on the interval (1, [infinity]). Then (21) and (22) lead to the conclusion that there exists [[lambda].sub.1] > 1 such that the function x [right arrow] [f.sub.7] (x, p) is strictly decreasing on (1, [[lambda].sub.1]] and strictly increasing on [Ax, ot).

It follows from (19) and (20) together with the piecewise monotonicity of the function x [right arrow] [f.sub.7] (x, p) that there exists [[lambda].sub.2] > 1 such that the function x [right arrow] [f.sub.6] (x, p) is strictly decreasing on (1, [[lambda].sub.2]] and strictly increasing on [[[lambda].sub.2], [infinity]).

Making use of (13)-(18) and the same method as theabove we know that there exists [[lambda].sub.i] > 1 (i = 3,4,5,6,7) such that the function x [right arrow] [f.sub.8-i] (x, p) is strictly decreasing on (1, [[lambda].sub.i]] and strictly increasing on [[[lambda].sub.i], [infinity]).

It follows from (12) and the piecewise monotonicity of the function x [right arrow] [f.sub.1] (x, p) that there exists [[lambda].sup.*] > 1 such that the function x [right arrow] f(x, p) is strictly decreasing on (1, [[lambda].sup.*]] and strictly increasing on [[[lambda].sup.*]], [infinity]).

Therefore, part (2) follows easily from (11) and the piecewise monotonicity of the function x [right arrow] f(x, p).

Theorem 2. The double inequality

[J.sub.[alpha]] (a, b) < U(a, b) < [J.sub.[beta]] (a,b) (27)

holds for all a, b > 0 with a [not equal to] b if and only if [alpha] [less than or equal to] [square root of 2]/([pi] - [square root of 2]) = 0.8187 ... and [beta] [greater than or equal to] 3/2.

Proof. Since U(a, b) and [J.sub.p] (a, b) are symmetric and homogeneous of degree one, without loss of generality, we assume that a = [x.sup.2] > 1 and b = 1. Let p [member of] R and p [not equal to] 0, -1. Then (1) lead to

[mathematical expression not reproducible], (28)

where

F (x, p) = arctan ([x.sup.2] - 1/[square root of 2x]) - (p + 1) ([x.sup.2] - 1) ([x.sup.2p] - 1)/[square root of 2]p ([x.sup.2p+2] - 1), (29)

[mathematical expression not reproducible], (30)

[partial derivative]F (x, p)/[partial derivative]x = [square root of 2]/p([x.sup.4] + 1) [([x.sup.2p+2] - 1).sup.2] f (x, p), (31)

where f(x, p) is defined by (9).

We divide the proof into four cases.

Case 1 (p = [square root of 2]/([pi] - [square root of 2])). Then it follows from Lemma 1(2), (29), and (31) that there exists [lambda] > 1 such that the function x [right arrow] F(x, p) is strictly decreasing on (1, [lambda]] and strictly increasing on [[lambda], [infinity]), and

[mathematical expression not reproducible]. (32)

Therefore,

[J.sub.[square root of 2]/([pi] - [square root of 2]) (a, b) < U(a, b) (33)

follows easily from (28), (30), and (32) together with the piecewise monotonicity of the function x [right arrow] F(x, p).

Case 2 (p > [square root of 2]/([pi] - [square root of 2])). Then (1) leads to

[mathematical expression not reproducible]. (34)

Inequality (34) implies that there exists large enough X = X(p) > 1 such that U(a, b) < [J.sub.p] (a, b) for all a, b > 0 with a/b [member of] (0,1/X) [union] (X, [infinity]).

Case 3 (p = 3/2). Then from Lemma 1(1) and (31) we know that the function x [right arrow] F(x, p) is strictly increasing on the interval (1, [infinity]). Therefore,

U(a, b) < [J.sub.3/2] (a, b) (35)

follows from (28) and (30) together with the monotonicity of the function x [right arrow] F(x, p).

Case 4 (0 < p < 3/2). Let x > 0 and x [right arrow] 0; then making use of Taylor expansion we get

[mathematical expression not reproducible]. (36)

Equation (36) implies that there exists small enough [delta] > 0 such that U(1, 1 + x) > [J.sub.p] (1,1 + x) for all x [member of] (0, [delta]).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11401191 and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-15G17.

References

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[4] M.-K. Wang, Y.-F. Qiu, and Y.-M. Chu, "An optimal double inequality among the oneparameter, arithmetic and harmonic means," Revue d'Analyse Numerique et de Theorie de l'Approximation, vol. 39, no. 2, pp. 169-175, 2010.

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[10] W.-F. Xia, G.-D. Wang, Y.-M. Chu, and S.-W. Hou, "Sharp inequalities between one-parameter and power means," Advances in Mathematics, vol. 42, no. 5, pp. 713-722, 2013.

[11] Z.-H. Yang, "Three families of two-parameter means constructed by trigonometric functions," Journal of Inequalities and Applications, vol. 2013, article 541, 27 pages, 2013.

[12] Z.-H. Yang, Y.-M. Chu, Y.-Q. Song, and Y.-M. Li, "A sharp double inequality for trigonometric functions and its applications," Abstract and Applied Analysis, vol. 2014, Article ID 592085, 9 pages, 2014.

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[14] S.-S. Zhou, W.-M. Qian, Y.-M. Chu, and X.-H. Zhang, "Sharp power-type Heronian mean bounds for the Sandor and Yang means," Journal of Inequalities and Applications, vol. 2015, article 159, 10 pages, 2015.

http://dx.doi.org/10.1155/2016/1579468

Wei-Mao Qian, (1) Yu-Ming Chu, (2) and Xiao-Hui Zhang (2)

(1) School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China

(2) School of Mathematics and Computation Sciences, Hunan City University, Yiyang 413000, China

Correspondence should be addressed to Yu-Ming Chu; chuyuming2005@126.com

Received 19 July 2015; Accepted 9 February 2016

Academic Editor: Yann Favennec
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Title Annotation:Research Article
Author:Qian, Wei-Mao; Chu, Yu-Ming; Zhang, Xiao-Hui
Publication:Mathematical Problems in Engineering
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2016
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