International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 540710, 13 pages

doi:10.1155/2012/540710

Research Article

The Monotonicity Results for the Ratio of Certain Mixed Means and Their Applications

Zhen-Hang Yang

Power Supply Service Center, Zhejiang Electric Power Company, Electric Power Research Institute, Hangzhou 310014, China

Received 22 July 2012; Accepted 26 September 2012

Academic Editor: Edward Neuman

Copyright © 2012 Zhen-Hang Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We continue to adopt notations and methods used in the papers illustrated by Yang (2009, 2010) to investigate the monotonicity properties of the ratio of mixed two-parameter homogeneous means. As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established.

1. Introduction

Since the Ky Fan [1] inequality was presented, inequalities of ratio of means have attracted attentions of many scholars. Some known results can be found in [2–14]. Research for the properties of ratio of bivariate means was also a hotspot at one time.

In this paper, we continue to adopt notations and methods used in the paper [13, 14] to investigate the monotonicity properties of the functions defined by

where

the , with , is the so-called two-parameter homogeneous functions defined by [15, 16]. For conveniences, we record it as follows.

Definition 1.1. Let : be a first-order homogeneous continuous function which has first partial derivatives. Then, is called a homogeneous function generated by with parameters and if is defined by for

where and denote first-order partial derivatives with respect to first and second component of ,respectively.

If exits and is positive for all , then further define

and .

Remark 1.2. Witkowski [17] proved that if the function is a symmetric and first-order homogeneous function, then for all is a mean of positive numbers and if and only if is increasing in both variables on . In fact, it is easy to see that the condition “ is symmetric” can be removed.

If is a mean of positive numbers and , then it is called two-parameter homogeneous mean generated by .

For simpleness, is also denoted by or .

The two-parameter homogeneous function generated by is very important because it can generates many well-known means. For example, substituting if with and for yields Stolarsky means defined by

where if , with , and is the identric (exponential) mean (see [18]). Substituting for yields Gini means defined by

where (see [19]).

As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established.

2. Main Results and Proofs

In [15, 16, 20], two decision functions play an important role, that are,

In [14], it is important to another key decision function defined by

Note that the function defined by

has well properties (see [15, 16]). And it has shown in [14, (3.4)], [16, Lemma 4] the relation among , and :

Moreover, it has revealed in [14, (3.5)] that

Now, we observe the monotonicities of ratio of certain mixed means defined by .

Theorem 2.1. Suppose that : is a symmetric, first-order homogenous, and three-time differentiable function, and strictly increase (decrease) with and decrease (increase) with . Then, for any with and fixed ,  , but are not equal to zero at the same time, is strictly increasing (decreasing) in on and decreasing (increasing) on .

The monotonicity of is converse if ,  , but are not equal to zero at the same time.

Proof. Since for , so is continuous on or for , then (2.13) in [13] holds. Thus we have

where

Partial derivative leads to

and then

where

Since strictly increase (decrease) with and decrease (increase) with , (2.4) and (2.6) together with yield

and therefore for . Thus, in order to prove desired result, it suffices to determine the sign of . In fact, if ,  , then for

It follows that

Clearly, the monotonicity of is converse if ,  .

This completes the proof.

Theorem 2.2. The conditions are the same as those of Theorem 2.1. Then, for any with and fixed with , , but are not equal to zero at the same time, is strictly increasing (decreasing) in on and decreasing (increasing) on .

The monotonicity of is converse if and , but are not equal to zero at the same time.

Proof. By (2.13) in [13] we have

where

Direct calculation leads to

and then

where is defined by (2.11). As shown previously, for if strictly increase (decrease) with and decrease (increase) with ; it remains to determine the sign of . It is easy to verify that if and , then

Thus, we have

Clearly, the monotonicity of is converse if and .

The proof ends.

Theorem 2.3. The conditions are the same as those of Theorem 2.1. Then, for any with and fixed , , is strictly increasing (decreasing) in on and decreasing (increasing) on .

The monotonicity of is converse if , .

Proof. From (2.13) in [13], it is derived that

where

Simple calculation yields

Hence,

where is defined by (2.11). It has shown that for if strictly increase (decrease) with and decrease (increase) with , and we have also to check the sign of . Easy calculation reveals that if , , then

which yields

It is evident that the monotonicity of is converse if , .

Thus the proof is complete.

Theorem 2.4. The conditions are the same as those of Theorem 2.1. Then, for any with and fixed with , is strictly increasing (decreasing) in on and decreasing (increasing) on if .

The monotonicity of is converse if .

Proof. By (2.13) in [13], can be expressed in integral form

The case has no interest since it can come down to the case of in Theorem 2.2. Therefore, we may assume that . We have

and then

Note that is even (see [13, (2.7)]) and so is odd, then make use of Lemma  3.3 in [13], can be expressed as

where

Hence,

where is defined by (2.11). We have shown that for if strictly increase (decrease) with and decrease (increase) with , and we also have

It follows that

This proof is accomplished.

3. Applications

As shown previously, , where is the logarithmic mean. Also, it has been proven in [14] that if and if . From the applications of Theorems 2.1–2.4, we have the following.

Corollary 3.1. Let with . Then, the following four functions are all strictly decreasing (increasing) on and increasing (decreasing) on :

(i) is defined by

for fixed ,  , but are not equal to zero at the same time,(ii) is defined by

for fixed with and , but are not equal to zero at the same time,(iii) is defined by

for fixed , .(iv) is defined by

for fixed with .

Remark 3.2. Letting in the first result of Corollary 3.1, yields Theorem  3.4 in [13] since . Letting ,   yields

Inequalities (3.5) in the case of were proved by Alzer in [21]. By letting ,   from , we have

Inequalities (3.6) in the case of are due to Alzer [22].

Remark 3.3. Letting in the second result of Corollary 3.1,,   yields Cheung and Qi’s result (see [23, Theorem 2]). And we have

When , inequalities (3.7) are changed as Alzer’s ones given in [24].

Remark 3.4. In the third result of Corollary 3.1, letting also leads to Theorem  3.4 in [13]. Put . Then from , we obtain a new inequality

Putting leads to another new inequality

Remark 3.5. Letting in the third result of Corollary 3.1, and , , , and we deduce that all the following three functions

are strictly decreasing on and increasing on , where , , and are the -order logarithmic, identric (exponential), and power mean, respectively, particularly, so are the functions , , .

4. Other Results

Let in Theorems 2.1–2.4. Then, and . From the their proofs, it is seen that the condition “ strictly increases (decreases) with and decreases (increases) with ” can be reduce to “ for ”, which is equivalent with , where , by (2.4). Thus, we obtain critical theorems for the monotonicities of , , defined as (1.2)–(1.5).

Theorem 4.1. Suppose that : is a symmetric, first-order homogenous, and three-time differentiable function and , where . Then, for with , the following four functions are strictly increasing (decreasing) in on and decreasing (increasing) on :

(i) is defined by (1.2), for fixed , but are not equal to zero at the same time;(ii) is defined by (1.3), for fixed with and , but are not equal to zero at the same time;(iii) is defined by (1.4), for fixed and ;(iv) is defined by (1.5), for fixed with .

If is defined on , then may be not continuous at , and (2.13) in [13] may not hold for but must be hold for . And then, we easily derive the following from the proofs of Theorems 2.1–2.4.

Theorem 4.2. Suppose that : is a symmetric, first-order homogenous and three-time differentiable function and , where . Then for with the following four functions are strictly increasing (decreasing) in on and decreasing (increasing) on :

(i) is defined by (1.2), for fixed ;(ii) is defined by (1.3), for fixed with and ;(iii) is defined by (1.4), for fixed and ;(iv) is defined by (1.5), for fixed .

If we substitute , , and for , where , , and denote the logarithmic, arithmetic, and identric (exponential) mean, respectively, then from Theorem 4.1, we will deduce some known and new inequalities for means. Similarly, letting in Theorem 4.2, , where with , we will obtain certain companion ones of those known and new ones. Here no longer list them.

Disclosure

This paper is in final form and no version of it will be submitted for publication elsewhere.

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