# Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean.

1. Introduction

For p [member of] R and a, b > 0 with a [not equal to] b, the pth generalized logarithmic mean [L.sub.p](a, b) is defined by

[mathematical expression not reproducible]. (1)

It is well known that [L.sub.p](a, b) is continuous and strictly increasing with respect to p [member of] R for fixed a, b > 0 with a [not equal to] b. Many classical bivariate means are the special case of the generalized logarithmic mean. For example, G(a, b) = [square root of (ab)] = [L.sub.-2](a, b) is the geometric mean, L(a, b) = (a - b)/(log a - log b) = [L.sub.-1](a, b) is the logarithmic mean, I(a, b) = [([a.sup.a]/[b.sup.b]).sup.1/(a-b)]/e = [L.sub.0](a, b) is the identric mean, and A(a, b) = (a + b)/2 = [L.sub.1](a, b) is the arithmetic mean. Recently, the generalized logarithmic mean has been the subject of intensive research.

Stolarsky  proved that the inequality

[L.sub.p](a, b) < [M.sub.(2+p)/3](a, b) (2)

holds for all a, b > 0 with a [not equal to] b and p [member of] (-2, -1/2) [union] (1, [infinity]), and inequality (2) is reversed for p [member of] (-[infinity], -2) [union] (-1/2, 1), where [M.sub.r](a, b) = [[([a.sup.r] + [b.sup.r])/2].sup.1/r] (r [not equal to] 0) and [M.sub.0](a, b) = [square root of (ab)] is the rth power mean of a and b.

Yang  proved that the double inequality

A(a, b) < [L.sub.p](a, b) < [M.sub.p](a, b) (3)

holds for all a, b > 0 with a [not equal to] b if p > 1, and inequality (3) is reversed if p < 0.

In , the authors proved that the inequality

[L.sub.p](a, b) < a + b/[(p + 1).sup.1/p] (4)

holds for all a, b > 0 with a [not equal to] b and p > 1.

Li et al.  proved that the function p [??] [L.sub.p](a, b)/[L.sub.p](1 - a, 1 - b) is strictly increasing (decreasing) on R if 0 < a < b [less than or equal to] 1/2 (1/2 [less than or equal to] a < b < 1). In [5,6], the authors proved that the function q [??] [L.sub.q](a, b)/[L.sub.q](a, c) is strictly decreasing on R if 0 < a < b < c and the function r [??] [L.sub.r](d, d + [epsilon])/[L.sub.r](d + [delta], d + [epsilon] + [delta]) is strictly increasing on R for all d, [epsilon], [delta] > 0.

Shi and Wu  proved that the double inequality

[mathematical expression not reproducible] (5)

for all b > a > c > 0 and 0 < [lambda] < 1 if p > 1, and inequality (5) is reversed if p [member of] (-1,0) [union] (0,1).

Long and Chu  and Matejicka  presented the best possible parameters p = p([alpha]) and q = q([alpha]) such that the double inequality

[L.sub.p](a, b) < [alpha]A(a, b) + (1 - [alpha]) G (a, b) < [L.sub.q](a, b) (6)

holds for all a, b > 0 with a [not equal to] b and [alpha] [member of] (0,1/2) [union] (1/2, 1).

In , Qian and Long answered the question: what are the greatest value p and the least value q such that the double inequality

[L.sub.p](a, b) < [G.sup.[alpha]] (a, b) [H.sup.1-[alpha]](a, b) < [L.sub.q](a, b) (7)

holds for all a, b > 0 with a [not equal to] b and [alpha] [member of] (0, 1), where H(a, b) = 2ab/(a + b) is the harmonic mean of a and b.

In [11,12], the authors proved that the double inequalities

[mathematical expression not reproducible], (8)

hold for all a, b > 0 with a [not equal to] b if and only if [p.sub.1] < [p.sup.*.sub.1], [q.sub.1] [greater than or equal to] 2, [p.sub.2] [less than or equal to] 3, [q.sub.2] [greater than or equal to] [q.sup.*.sub.2], where [p.sup.*.sub.1] = 1.843 ... is the unique solution of the equation [(p + 1).sup.l/p] = 2 log(1 + [square root of (2)]) on the interval (0, [infinity]), [q.sup.*.sub.2] = 3.152 ... is the unique solution of the equation [(q + 1).sup.l/q] = [pi]/2 on the interval (0, [infinity]), M(a, b) = (a - b)/[2 [sinh.sup.-1]((a - b)/(a + b))] is the Neuman-Sandor mean, and T(a, b) = (a - b)/[2 arctan((a - b)/(a + b))] is the second Seiffert mean.

In [13,14], the authors presented the best possible parameters [p.sub.1] = [p.sub.1](q), [p.sub.2] = [p.sub.2](q), [lambda] = [lambda]([alpha]), and [mu] = [mu]([alpha]) such that the double inequalities

[mathematical expression not reproducible] (9)

hold for all a, b > 0 with a [not equal to] b, q > 0 with q [not equal to] 1 and [alpha] [member of] (0,2/3) [union] (2/3,1).

Gao et al.  provided the greatest value [alpha] and the least value [beta] such that the double inequality

[L.sub.[alpha]](a, b) < P(a, b) < [L.sub.[beta]](a, b) (10)

holds for all a, b > 0 with a [not equal to] b, where P(a, b) = (a - b)/[2 arcsin((a - b)/(a + b))] is the first Seiffert mean of a and b.

Very recently, Yang  introduced the Yang mean

U(a, b) = a - b/[square root of (2)] arctan((a - b)/[square root of (ab)]) (11)

of two distinct positive real numbers a and b and proved that the inequalities

[mathematical expression not reproducible], (12)

hold for all a, b > 0 with a [not equal to] b, where Q(a, b) = [square root of (([a.sup.2] + [b.sup.2])/2)] is the quadratic mean of a and b.

The Yang mean U(a, b) is the special case of the Seiffert type mean [T.sub.M,q](a, b) = (a - b)/[q arctan((a - b)/(qM(a, b)))] defined by Toader in , where M(a, b) is a bivariate mean and q is a positive real number. Indeed, U(a, b) = [T.sub.G,[square root of (2)]](a, b).

In [18,19], the authors proved that the double inequalities

[mathematical expression not reproducible] (13)

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 [bet] [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 ....

Zhou et al.  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] (14)

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

The main purpose of this paper is to present the best possible parameters p and q such that the double inequality [L.sub.p](a, b) < U(a, b) < [L.sub.q](a, b) holds for all a, b > 0 with a [not equal to] b. As application, we derive several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions. Some complicated computations are carried out using Mathematica computer algebra system.

2. Lemmas

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

Lemma 1. Let n [greater than or equal to] 4, -[infinity] [less than or equal to] a < b [less than or equal to] +[infinity], [f.sub.0], [f.sub.i], [f.sup.*.sub.i] : (a, b) [right arrow] R be n-times differentiable functions such that [f.sub.i](x) = [f.sup.*.sub.i](x)[f'.sub.i-1](x) and [f.sup.*.sub.i](x) > 0 for 1 [less than or equal to] i [less than or equal to] n and x [member of] (a, b). If

[mathematical expression not reproducible], (15)

[mathematical expression not reproducible] (16)

for 0 [less than or equal to] i [less than or equal to] n - 3 and

[f.sub.n](x) > 0 (17)

for x [member of] (a, b), then there exists [x.sub.0] [member of] (a, b) such that [f.sub.0](x) < 0 for x [member of] (a, [x.sub.0]) and 0 (x) > 0 for x [member of] ([x.sub.0], b).

Proof. From (15) and (17) we clearly see that there exists [x.sub.n-2] [member of] (a, b) such that [f.sub.n-2](x) < 0 for x [member of] (a, [x.sub.n-2]) and [f.sub.n-2](x) > 0 for x [member of] ([x.sub.n-2], b), which implies that [f.sub.n-3](x) is strictly decreasing on (a, [x.sub.n-2]] and strictly increasing on [[x.sub.n-2], b). Then (16) leads to the conclusion that there exists [x.sub.n-3] [member of] (a, b) such that [f.sub.n-3](x) < 0 for x [member of] (a, [x.sub.n-3]) and [f.sub.n-3](x) > 0 for x [member of] ([x.sub.n-3], b).

Making use of (16) and the same method as above we know that for 0 [less than or equal to] i [less than or equal to] n - 4 there exists [x.sub.i] [member of] (a, b) such that [f.sub.i](x) < 0 for x [member of] (a, [x.sub.i]) and [f.sub.i](x) > 0 for x [member of] ([x.sub.i], b).

Lemma 2. Let p [member of] R, and

[mathematical expression not reproducible]. (18)

Then the following statements are true:

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

(2) if [p.sub.0] = 0.5451 ... is the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]) and p = [p.sub.0], 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 = 2, then (18) becomes

f(x, p) = 2[(x - 1).sup.6] [(x + 1).sup.6] [([x.sup.2] + 1).sup.3]. (19)

Therefore, part (1) follows from (19).

For part (2), let p = [p.sub.0] = 0.5451 ... be the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infiity]), [f.sub.1](x, p) = (1/2x)[partial derivative] f(x, p)/[partial derivative]x, [f.sub.2](x, p) = (1/2x)[partial derivative] [f.sub.1](x, p)/[partial derivative]x, [f.sub.3](x, p) = (1/2x) [partial derivative][f.sub.2](x, p)/[partial derivative]x, [f.sub.4](x, p) = (1/2x) [partial derivative][f.sub.3](x, p), [f.sub.5](x, p) = ([x.sup.9-2p]/2p) [partial derivative][f.sub.4](x, p)/[partial derivative]x, [f.sub.6](x, p) = (1/2)p + 1)x)[partial derivative][f.sub.5](x, p)/[partial derivative]x, [f.sub.7](x, p) = (1/8x)[partial derivative][f.sub.6](x, p)/[partial derivative]x, [f.sub.8](x, p) = (1/2x)[partial derivative][f.sub.7](x, p)/[partial derivative]x, and [f.sub.9](x, p) = (1/2x)[partial derivative][f.sub.8](x, p)/[partial derivative]x. Then elaborated computations lead to

[mathematical expression not reproducible] (20)

for x [member of] (1, [infinity]).

Therefore, part (2) follows easily from Lemma 1 and (20).

3. Main Result

Theorem 3. The double inequality

[L.sub.p](a, b) < U(a, b) < [L.sub.q](a, b) (21)

holds for all a, b > 0 with a [not equal to] b if and only if p [less than or equal to] [p.sub.0] and q [greater than or equal to] 2, where [p.sub.0] = 0.5451 ... is the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]).

Proof. Since U(a, b) and [L.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) and (11) lead to

[mathematical expression not reproducible], (22)

[mathematical expression not reproducible], (23)

[mathematical expression not reproducible], (24)

[mathematical expression not reproducible], (25)

where

[mathematical expression not reproducible], (26)

[mathematical expression not reproducible], (27)

[mathematical expression not reproducible], (28)

[mathematical expression not reproducible], (29)

where f(x, p) is defined by (15).

We divide the proof into four cases.

Case 1 (p = 2). Then from Lemma 2(1) and (29) we clearly see that the function x [right arrow] [F.sub.1](x, p) is strictly decreasing on (1, [infinity]). Then (27) leads to the conclusion that

[F.sub.1](x, p) < 0 (30)

for all x [member of] (1, [infinity]). Therefore,

U(a, b) < [L.sub.2](a, b) (31)

follows easily from (22), (23), (25), and (30).

Case 2 (p = [p.sub.0]). Then from Lemma 2(2) and (29) we know that there exists [lambda] [member of] (1, [infinity]) such that the function x [right arrow] [F.sub.1](x, p) is strictly increasing on (1, [lambda]] and strictly decreasing on [[lambda], [infinity]).

It follows from (25)-(28) and the piecewise monotonicity of the function x [right arrow] [F.sub.1](x, p) that there exists [[lambda].sup.*] [member of] (1, [infinity]) such that the function x [right arrow] F(x, p) is strictly increasing on (1, [[lambda].sup.*]] and strictly decreasing on [[[lambda].sup.*], [infinity]).

Note that (24) becomes

[mathematical expression not reproducible]. (32)

Therefore,

[mathematical expression not reproducible] (33)

follows easily from (22), (23), and (32) together with the piecewise monotonicity of the function x [right arrow] F(x, p).

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

[mathematical expression not reproducible]. (34)

Equation (34) implies that there exists small enough [delta] [member of] (0,1) such that

U(1 + x, 1) > [L.sub.p](1 + x, 1) (35)

for all x [member of] (0, [delta]).

Case 4 (p > [p.sub.0]). Then from (24) and the fact that the function p [right arrow] log(p + 1)/p is strictly decreasing on (0, [infinity]) we get

[mathematical expression not reproducible]. (36)

Equation (22) and inequality (36) imply that there exists large enough X > 1 such that

U([x.sup.2], 1) < [L.sub.p]([x.sup.2], 1) (37)

for all x [member of] (X, [infinity]).

4. Applications

As applications of Theorem 3 in engineering problems, we present several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions in this section.

From (1) and (11) together with Theorem 3 we get Theorem 4 immediately.

Theorem 4. Let [p.sub.0] = 0.5451 ... be the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]). Then the double inequality

[mathematical expression not reproducible] (38)

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

Let t > 0, b= 1, and a = [t.sup.2] + t [square root of ([t.sup.2] + 2)] + 1. Then Theorem 4 leads to the following.

Theorem 5. Let [p.sub.0] = 0.5451 ... be the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]). Then the double inequality

[mathematical expression not reproducible] (39)

holds for all t > 0.

Let a > b > 0, x = log [square root of (a/b)] [member of] (0, [infinity]). Then (1) and (11) lead to

[mathematical expression not reproducible]. (40)

It follows from Theorem 3 and (40) that one has the following theorem.

Theorem 6. Let [p.sub.0] = 0.5451 ... be the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]). Then the double inequality

[mathematical expression not reproducible] (41)

holds for all x > 0.

Let a > b > 0, x = arcsin[(a - b)/(a + b)] [member of] (0, [pi]/2). Then (1) and (11) lead to

[mathematical expression not reproducible]. (42)

Theorem 3 and (42) lead to the following.

Theorem 7. Let [p.sub.0] = 0.5451 ... be the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]). Then the double inequality

[mathematical expression not reproducible] (43)

holds for all x [member of] (0, [pi]/2).

Let a > b > 0, x = arctan((a - b)/(a + b)) [member of] (0, [pi]/4), y = [sinh.sup.-1] ((a - b)/(a + b)) [member of] (0, log(1 + [square root of (2)])). Then from (1) and (11) we have

[mathematical expression not reproducible], (44)

[mathematical expression not reproducible]. (45)

From (44), (45), and Theorem 3 one has the following.

Theorem 8. Let [p.sub.0] = 0.5451 ... be the unique solution of the equation [(p + 1).sup.1/p] = [square root of (2)][pi]/2 on the interval (0, [infinity]). Then the double inequalities

[mathematical expression not reproducible] (46)

hold for all x [member of] (0, [pi]/4) and y [member of] (0, log(1 + [square root of (2)])).

http://dx.doi.org/10.1155/2016/8901258

Competing Interests

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

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 11371125, 61374086, and 11401191, the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT15G-17, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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Wei-Mao Qian (1) and Yu-Ming Chu (2)

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

(2) Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

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

Received 23 January 2016; Revised 16 March 2016; Accepted 28 March 2016