Some Monotonicity Results for the Ratio of Two-Parameter Symmetric Homogeneous Functions

Applying well properties of homogeneous functions, 
some monotonicity results for the ratio of two-parameter symmetric homogeneous functions are presented, which give an easier access to find two-parameter symmetric homogeneous means having ratio simple monotonicity properties proposed by L. Losonczi. As an application, a chain of inequalities of ratio of bivariate means is established.


Introduction
q a p − b p p a q − b q 1/ p−q , p / q, pq / 0, Also, S p,q a, a a.In a few years, Chen and Qi 4-7 also proved equivalent results.In 8 the author has proven that inequality 1.1 is valid for power means of certain order, logarithmic, identric, and the Heronian mean of order ω.Neuman et al. 9 obtained inequalities of the form 1.1 for the Stolarsky, Gini, Schwab-Borchardt, and the lemniscatic means.
Recently Chen 10, 11 established a more general result than Pearce and Pečarić's: let a, b, c, d be fixed positive numbers with a / b, c / d and let p, q be real numbers.Then the function R p,q a, b; c, d : S p,q a, b S p,q c, d 1.8 is increasing with both p and q according to 1.2 .Soon after, Losonczi studied four monotonicity properties of the ratio R p,q a, b, c : S p,q a, b S p,q a, c p, q ∈ R, 0 < a < b < c 1.9 in the parameters p, q and completely solve the comparison problem for this ratio 12 .This generalizes Chen's result.Also, an open problem was proposed by the author.
Let M p,q p, q ∈ R be a two-parameter, symmetric, and homogeneous mean defined for positive variables and let us form the ratio For what means M p,q has this ratio simple monotonicity properties?
The more general form of two-parameter, symmetric, and homogeneous means is the so-called two-parameter homogenous functions first introduced by Yang 13 .For conveniences, we record it as follows.
Definition 1.1.Assume that f: R × R → R ∪ {0} is n-order homogeneous, and continuous and exists first partial derivatives and a, b ∈ R × R , p, q ∈ R × R.
If f x, y > 0 for x, y ∈ R × R with x / y and f x, x 0 for all x ∈ R , then define that 1.14 Since f x, y is a homogeneous function, H f a, b; p, q is also one and called a homogeneous function with parameters p and q, and simply denoted by H f p, q sometimes.The aim of this paper is to investigate the monotonicity of the ratio defined by and presents four types of monotonicity of R f p, q in the parameters p and q, which give an easier access to find two-parameter symmetric homogeneous means having ratio simple monotonicity properties mentioned by Losonczi 12 .

Properties and Lemmas
Before formulating our main results, let us recall the properties and lemmas of two-parameter homogeneous functions.
Property 2.1.H f p, q is symmetric with respect to p, q, that is, Property 2.2.Iff x, y is symmetric with respect to x and y, then where G √ ab.
Property 2.3 see 14, 1.13 .If G f,t is continuous on q, p or p, q , then ln where G f,t is defined by 1.

6 International Journal of Mathematics and Mathematical Sciences
Based on properties and lemmas above, the author has investigated the monotonicity and log-convexity of two-parameter homogeneous functions and obtained a series of valuable results in 13, 14 , which yield some new and interesting inequalities for means.Recently, two results on monotonicity and log-convexity of a four-parameter homogeneous containing Stolarsky mean and Gini mean have been presented in 15 .In the processes of proofs on 13-15 , two decision functions play an important role, which where x a t , y b t .Moreover, it is easy to verify that T 2 x, y and T 3 x, y both are zero-order homogeneous functions due to homogeneity of f x, y , and thus, 2.19

Main Results and Proofs
Next let us consider the monotonicities of ratio of two-parameter homogeneous functions defined by 1. 15 .In what follows, we always assume b/a / d/c.Theorem 3.1 first monotonicity property .Suppose that f : R × R → R is a symmetric, homogenous, and two time-differentiable function; T 2 1, u is strictly increasing (decreasing) with u > 1; 1.2 is satisfied.Then R f p, q is strictly increasing (decreasing) in either p or q unless b/a d/c.
Proof.Since H f p, q is symmetric with respect to p and q, it only needs to prove the logconvexity of R f p, q in parameter p. Direct partial derivative calculation for 2.13 leads to From 1.15 , we have Since T 2 1, u is strictly increasing decreasing with u > 1 and by 2.7 , 2.17 , and assumption 1.2 , we have always

3.3
It follows that This proof is completed.
The next monotonicity result is a direct corollary of Theorem 3.1 actually.Proof.Under the same conditions as Theorem 3.1, the function R f p, q is strictly increasing decreasing in either p or q.Hence for p 1 , p 2 ∈ R with p 1 < p 2 , we have which indicates that the function R f p, p m is strictly increasing decreasing with p.The proof ends.
To investigate the third and fourth monotonicity properties, we need a useful lemma.

3.7
According to the property of definite integral of odd functions, our required result is obtain immediately.This lemma is proved.
T s ds dt by Lemma 3.3 .

3.12
From 1.15 , we have

3.13
Since T 3 1, u is strictly increasing decreasing with u > 1, by 2.18 and 1.2 , we have always and hence

3.16
This completes the proof.

3.21
This shows that R f pr, ps is strictly increasing decreasing with p if r s > 0 and decreasing increasing if r s < 0. 2 In the case of r s.Similarly, by 3.18 , 2.7 , 1.2 , and 3.3 we have

3.22
Combining two cases above, the proof is accomplished.
International Journal of Mathematics and Mathematical Sciences 11

Applications
As applications of main results in this paper, next let us prove the monotonicity of ratio of Stolarsky means.We will see that the methods provided by this paper are simple and effective.
It is easy to verify that the two-parameter logarithmic mean is just Stolarsky mean, that is, H L p, q; a, b S p,q a, b .Consequently, the monotonicities of ratio of Stolarsky means depend on the monotonicities of T 2 1, u and T 3 1, u defined by 2.15 and 2.16 .Some simple calculations yield
Applying our main results, we can obtain all theorems involving monotonicity of ratio of Stolarsky means in Section 2 of 12 .Here we have no longer list.
Lastly, as concrete applications of the monotonicity of ratio of Stolarsky means, we now show a refined chain of inequalities of ratio of means involving logarithmic mean, exponential mean identric mean , arithmetic mean, geometric mean, and Heronian mean, which is a generalization of inequalities in 14, 5.5 and contains 1.4 .
For convenience of statement in the following theorem, corresponding to 1.3 let us define further that Let Φ a, b and Ψ c, d be bivariate means.For what means Φ and Ψ does the following inequality p M p a, b : M 1/p a p , b p , M A, H, L, I, 1.3 where A, H, L, and I stand for arithmetic mean, Heronian mean, logarithmic mean, and exponential mean identric mean of two positive numbers a and b, respectively.In 1988 Wang et al. 1 proved that for a, b, c, d > 0 with b/a ≥ d/c ≥ 1 the following inequalities of ratio of bivariate means equalities if and only if b/a d/c.That same year, Chen et al. 2 presented second inequalities of ratio of bivariate means: 1/2 and 2/3 both are best possible.In 1994, Pearce et al. 3 proved that the function p −→ L p a, b L p c, d p ∈ R 1.6 is nondecreasing, provided that a, b, c, d > 0 with b/a ≥ d/c.Here L p a, b : S p 1,1 a, b is the generalized logarithmic mean and S p,q a, b is the Stolarsky mean of a, b > 0 with parameters p, q ∈ R defined by S p,q a, b

Theorem 3 . 2
second monotonicity property .The conditions are the same as those of Theorem 3.1.Then for fixed m ∈ R, the function R f p, p m is strictly increasing (decreasing) with p unless b/a d/c.

Lemma 3 . 3 .
Let f x be odd and continuous on −m, m m > 0 .Then for arbitrary r, s ∈ −m, m .Proof.By the additivity of definite integral we have M p : M 1/p c p , d p , M A, H, L, I, 4.2where A,H,L, and I stand for arithmetic mean, Heronian mean, logarithmic mean, and exponential mean identric mean of two positive numbers c and d, respectively.

Theorem 4 . 1 .A 1 / 3 G 2 / 3 A 1 / 3 G 2
Suppose that a, b, c, d satisfy assumption 1.2 .Then the following inequalities Theorem 3.5 fourth monotonicity property .The conditions are the same as those of Theorem 3.1.Then for fixed r, s ∈ R, the function R f pr, ps is strictly increasing (decreasing) with p if r s > 0 and decreasing (increasing) if r s < 0.
Proof.By the third monotonicity property, we see that R p,1−p a, b; c, d is strictly decreasing in p on 1/2, ∞ .Put p 2, 3/2, 4/3, 1, 4/5, 3/4, 2/3, 3/5, 1/2 in R p,1−p a, b; c, d and by some calculations, the chain of inequalities 4.3 is derived immediately, with equalities if and only if b/a d/c because the monotonicity of R p,1−p a, b; c, d is strict.The proof is finished.