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

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.


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 Q if i 1, 2, 3, 4 defined by the q, r, s, k, m ∈ R, a, b, c, d ∈ R with b/a > d/c ≥ 1, H f p, q is the so-called two-parameter homogeneous functions defined by 15, 16 .For conveniences, we record it as follows.
Definition 1.1.Let f: R 2 \ { x, x , x ∈ R } → R be a first-order homogeneous continuous function which has first partial derivatives.Then, H f : R 2 ×R 2 → R is called a homogeneous function generated by f with parameters p and q if H f is defined by for a / b H f p, q; a, b f a p , b p f a q , b q 1 p−q , if pq p − q / 0, where f x x, y and f y x, y denote first-order partial derivatives with respect to first and second component of f x, y , respectively.If lim y → x f x, y exits and is positive for all x ∈ R , then further define and H f p, q; a, a a.
Remark 1.2.Witkowski 17 proved that if the function x, y → f x, y is a symmetric and first-order homogeneous function, then for all p, q H f p, q; a, b is a mean of positive numbers a and b if and only if f is increasing in both variables on R .In fact, it is easy to see that the condition "f x, y is symmetric" can be removed.If H f p, q; a, b is a mean of positive numbers a and b, then it is called two-parameter homogeneous mean generated by f.
For simpleness, H f p, q; a, b is also denoted by H f p, q or H f a, b .The two-parameter homogeneous function H f p, q; a, b generated by f is very important because it can generates many well-known means.For example, substituting International Journal of Mathematics and Mathematical Sciences 3 L L x, y x − y / ln x − ln y if x, y > 0 with x / y and L x, x x for f yields Stolarsky means H L p, q; a, b S p,q a, b defined by , if pq p − q / 0, L 1/p a p , b p , if p / 0, q 0, L 1/q a q , b q , if q / 0, p 0, where I x, y e −1 x x /y y 1/ x−y if x, y > 0, with x / y, and I x, x x is the identric exponential mean see 18 .Substituting A A x, y x y /2 for f yields Gini means H A p, q; a, b G p,q a, b defined by where Z a, b a a/ a b b b/ a b 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.

Main Results and Proofs
In 15, 16, 20 , two decision functions play an important role, that are, Moreover, it has revealed in 14, 3.5 that Now, we observe the monotonicities of ratio of certain mixed means defined by 1.1 .
Theorem 2.1.Suppose that f: R × R → R is a symmetric, first-order homogenous, and threetime differentiable function, and T 3 1, u strictly increase (decrease) with u > 1 and decrease (increase) with 0 < u < 1.Then, for any a, b, c, d > 0 with b/a > d/c ≥ 1 and fixed q ≥ 0, k ≥ 0, but q, k are not equal to zero at the same time, Q 1f is strictly increasing (decreasing) in p on k, ∞ and decreasing (increasing) on −∞, k .The monotonicity of Q 1f is converse if q ≤ 0, k ≤ 0, but q, k are not equal to zero at the same time.
Proof.Since f x, y > 0 for x, y ∈ R × R , so T t is continuous on p, q or q, p for p, q ∈ R, then 2.13 in 13 holds.Thus we have where

2.11
Since T 3 1, u strictly increase decrease with u > 1 and decrease increase with 0 < u < 1, 2.4 and 2.6 together with b/a > d/c ≥ 1 yield 2.12 and therefore h x, y > < 0 for x, y > 0. Thus, in order to prove desired result, it suffices to determine the sign of |t 12 | − |t 11 | .In fact, if q ≥ 0, k ≥ 0, then for t ∈ 0, 1 Clearly, the monotonicity of Q 1f is converse if q ≤ 0, k ≤ 0. This completes the proof.
Thus, we have Clearly, the monotonicity of Q 2f is converse if k ≤ 0 and k m ≤ 0. The proof ends.

2.27
The case r s / 0 has no interest since it can come down to the case of m 0 in Theorem 2.2.Therefore, we may assume that r / s.We have

2.29
Note that T t is even see 13, 2.7 and so t T pt − T 2k − p t is odd, then make use of Lemma 3.3

2.32
where h x, y is defined by 2.11 .We have shown that h x, y > < 0 for x, y > 0 if T 3 1, u strictly increase decrease with u > 1 and decrease increase with 0 < u < 1, and we also have

2.34
This proof is accomplished.

Applications
As shown previously, S p,q a, b H L p, q; a, b , where L L x, y is the logarithmic mean.Also, it has been proven in 14 that T 3 1, u < 0 if u > 1 and T 3 1, u > 0 if 0 < u < 1.From the applications of Theorems 2.1-2.4,we have the following.
Then, the following four functions are all strictly decreasing (increasing) on k, ∞ and increasing (decreasing) on −∞, k : for fixed q ≥ ≤ 0, k ≥ ≤ 0, but q, k are not equal to zero at the same time,

International Journal of Mathematics and Mathematical Sciences
ii Q 2L is defined by for fixed m, k with k ≥ ≤ 0 and k m ≥ ≤ 0, but m, k are not equal to zero at the same time, iii Q 3L is defined by

Other Results
Let d c in Theorems 2.1-2.4.Then, H f p, q; c, d c and T t; c, c 0. From the their proofs, it is seen that the condition "T 3 1, u strictly increases decreases with u > 1 and decreases increases with 0 < u < 1" can be reduce to "T v > < 0 for v > 0", which is equivalent with J x − y xI x < > 0, where I ln f xy , by 2.4 .Thus, we obtain critical theorems for the monotonicities of g if , i 1 − 4, defined as 1.2 -1.5 .Theorem 4.1.Suppose that f: R ×R → R is a symmetric, first-order homogenous, and three-time differentiable function and J x − y xI x < > 0, where I ln f xy .Then, for a, b > 0 with a / b, the following four functions are strictly increasing (decreasing) in p on k, ∞ and decreasing (increasing) on −∞, k : i g 1f is defined by 1.2 , for fixed q, k ≥ 0, but q, k are not equal to zero at the same time; ii g 2f is defined by 1.3 , for fixed m, k with k ≥ 0 and k m ≥ 0, but m, k are not equal to zero at the same time; iii g 3f is defined by 1.4 , for fixed m > 0 and 0 ≤ k ≤ 2m; iv g 4f is defined by 1.5 , for fixed k, r, s ∈ R with k r s > 0.
If f is defined on R 2 \ { x, x , x ∈ R }, then T t may be not continuous at t 0, and 2.13 in 13 may not hold for p, q ∈ R but must be hold for p, q ∈ R .And then, we easily derive the following from the proofs of Theorems 2.1-2.4.Theorem 4.2.Suppose that f: R 2 \ { x, x , x ∈ R } → R is a symmetric, first-order homogenous and three-time differentiable function and J x − y xI x < > 0, where I ln f xy .Then for a, b > 0 with a / b the following four functions are strictly increasing (decreasing) in p on k, 2k and decreasing (increasing) on 0, k : i g 1f is defined by 1.2 , for fixed q, k > 0; ii g 2f is defined by 1.3 , for fixed m, k with k > 0 and k m > 0; iii g 3f is defined by 1.4 , for fixed m > 0 and 0 ≤ k ≤ 2m; iv g 4f is defined by 1.5 , for fixed k, r, s > 0.
If we substitute L, A, and I for f, where L, A, and I 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 f x, y D x, y |x−y|, K x, y x y | ln x/y |, where x, y > 0 with x / y, 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.
,p m a, b S 2k−p,2k−p m a, b S p,p m c, d S 2k−p,2k−p m c, d , 3.2

, 3 . 4 forRemark 3 . 2 .<
ps a, b S 2k−p r, 2k−p s a, b S pr,ps c, d S 2k−p r, 2k−p s c, d fixed k, r, s ∈ R with k r s > < 0. Letting in the first result of Corollary 3.1, q k yields Theorem 3.4 in 13 since S p,k S 2k−p,k S p,2k−p .Letting q 1, k 0 yields G a, b G c, d Q 1L ∞ < S p,1 a, b S −p,1 a, b S p,1 c, d S −p,1 c, d the case of d c were proved by Alzer in 21 .By letting q

Remark 3 . 3 .<Remark 3 . 4 .. 8 Putting m 1 / 2 ,. 9 Remark 3 . 5 ., 3 . 10 are strictly decreasing on 1 / 2 ,
the case of d c are due to Alzer 22 .Letting in the second result of Corollary 3.1, m 1, k 0 yields Cheung and Qi's result see 23, Theorem 2 .And we have G a, b G c, d Q 2L ∞ < S p,p 1 a, b S −p,−p 1 a, b S p,p 1 c, d S −p,−p 1 c, d When d c, inequalities 3.7 are changed as Alzer's ones given in 24 .In the third result of Corollary 3.1, letting k m also leads to Theorem 3.4 in 13 .Put m 1/2, k 1/4.Then from Q 3L 1/4 > Q 3L 1/2 , we obtain a new inequality He 1/2 a, b He 1/2 c, d > L a, b I 1/2 a, b L c, d I 1/2 c, d .3k 1/3 leads to another new inequality A 1/3 a, b A 1/3 c, d > S 1/6,5/6 a, b I 1/2 a, b S 1/6,5/6 c, d I 1/2 c, d .3Letting in the third result of Corollary 3.1, k 1/2 and r, s 1, 0 , 1, 1 , 2, 1 , and we deduce that all the following three functions p −→ L p a, b L 1−p a, b L p c, d L 1−p c, d , p−→ I p a, b I 1−p a, b I p c, d I 1−p c, d , p−→ A p a, b A 1−p a, b A p c, d A 1−p c, d ∞ and increasing on −∞, 1/2 , where L p L 1/p a p , b p , I p I 1/p a p , b p , and A p A 1/p a p , b p are the p-order logarithmic, identric exponential , and power mean, respectively, particularly, so are the functions L p L 1−p , I p I 1−p , A p A 1−p .
|t 12 | − |t 11 | h |t 11 |, |t 12 | dt, International Journal of Mathematics and Mathematical SciencesThe monotonicity of Q 2f is converse if k ≤ 0 and k m ≤ 0, but m, k are not equal to zero at the same time.
Theorem 2.2.The conditions are the same as those of Theorem 2.1.Then, for any a, b, c, d > 0 with b/a > d/c ≥ 1 and fixed m, k with k ≥ 0, k m ≥ 0, but m, k are not equal to zero at the same time, Q 2f is strictly increasing (decreasing) in p on k, ∞ and decreasing (increasing) on −∞, k .22|− |t 21 | h |t 21 |, |t 22 | dt,2.18whereh x, y is defined by 2.11 .As shown previously, h x, y > < 0 for x, y > 0 if T 3 1, u strictly increase decrease with u > 1 and decrease increase with 0 < u < 1; it remains to determine the sign of |t 22 | − |t 21 | .It is easy to verify that if k ≥ 0 and k m ≥ 0, then Theorem 2.3.The conditions are the same as those of Theorem 2.1.Then, for any a, b, c, d > 0 with b/a > d/c ≥ 1 and fixed m > 0, 0 ≤ k ≤ 2m, Q 3f is strictly increasing (decreasing) in p on k, ∞ and decreasing (increasing) on −∞, k .The monotonicity of Q 2f is converse if m < 0, 2m ≤ k ≤ 0.