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 [2ā14]. 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 ā is symmetricā 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 ,āā, 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 ,āā, 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 ,āā, then for
It follows that
Clearly, the monotonicity of is converse if ,āā.
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 Lemmaāā3.3 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 2.1ā2.4, 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 ,āā, 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 Theoremāā3.4 in [13] since . Letting ,āā yields Inequalities (3.5) in the case of were proved by Alzer in [21]. By letting ,āā 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,,āā yields Cheung and Qiās result (see [23, Theorem 2]). And we have When , inequalities (3.7) are changed as Alzerās ones given in [24].
Remark 3.4. In the third result of Corollary 3.1, letting also leads to Theoremāā3.4 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 2.1ā2.4. Then, and . From the their proofs, it is seen that the condition ā strictly increases (decreases) with and decreases (increases) with ā can be reduce to ā for ā, which is equivalent with , where , by (2.4). Thus, we obtain critical theorems for the monotonicities of , , defined as (1.2)ā(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 2.1ā2.4.
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.