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.
Let and be bivariate means. For what means and does the following inequality
hold true? where
where , , , and stand for arithmetic mean, Heronian mean, logarithmic mean, and exponential mean (identric mean) of two positive numbers and , respectively.
In 1988 Wang et al.  proved that for with the following inequalities of ratio of bivariate means
hold, with equalities if and only if. That same year, Chen et al.  presented second inequalities of ratio of bivariate means:
where the constant and both are best possible.
In 1994, Pearce et al.  proved that the function
is nondecreasing, provided that with . Here is the generalized logarithmic mean and is the Stolarsky mean of with parameters defined by
In  the author has proven that inequality (1.1) is valid for power means of certain order, logarithmic, identric, and the Heronian mean of order . Neuman et al.  obtained inequalities of the form (1.1) for the Stolarsky, Gini, Schwab-Borchardt, and the lemniscatic means.
is increasing with both and according to (1.2). Soon after, Losonczi studied four monotonicity properties of the ratio
in the parameters and completely solve the comparison problem
for this ratio . This generalizes Chen's result. Also, an open problem was proposed by the author.
Let be a two-parameter, symmetric, and homogeneous mean defined for positive variables and let us form the ratio
For what means 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 . For conveniences, we record it as follows.
Definition 1.1Assume that : is -order homogeneous, and continuous and exists first partial derivatives and , .
If for with and for all , then define that
where andand denote first-order partial derivative for first and second variables of respectively.
If for all , then define further
Since is a homogeneous function, is also one and called a homogeneous function with parameters and , and simply denoted by sometimes.
The aim of this paper is to investigate the monotonicity of the ratio defined by
and presents four types of monotonicity of in the parameters and , which give an easier access to find two-parameter symmetric homogeneous means having ratio simple monotonicity properties mentioned by Losonczi .
2. Properties and Lemmas
Before formulating our main results, let us recall the properties and lemmas of two-parameter homogeneous functions.
Property 2.1is symmetric with respect to , that is,
Property 2.2If is symmetric with respect to and , then where .
It is worth mentioning that the following function
is well behaved, whose properties as useful lemmas read as follows.
Lemma 2.4 (see [14, (), (), (), ()]) Suppose that is a symmetric, -order homogenous and three-time differentiable function. Then where .
Remark 2.5If , then can be extended continuously by defining , with the result that is also three times derivable at . Particularly, . Thus (2.6) can be written as
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 .
In next section we will encounter other two key decision functions defined by
Moreover, it is easy to verify that and both are zero-order homogeneous functions due to homogeneity of , and thus,
3. 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 .
Theorem 3.1 (first monotonicity property) Suppose that is a symmetric, homogenous, and two time-differentiable function; is strictly increasing (decreasing) with ; (1.2) is satisfied. Then is strictly increasing (decreasing) in either or unless .
ProofSince is symmetric with respect to and , it only needs to prove the log-convexity of in parameter .
Direct partial derivative calculation for (2.13) leads to
From (1.15), we have Since is strictly increasing (decreasing) with and by (2.7), (2.17), and assumption (1.2), we have always It follows that
This proof is completed.
The next monotonicity result is a direct corollary of Theorem 3.1 actually.
Theorem 3.2 (second monotonicity property) The conditions are the same as those of Theorem 3.1. Then for fixed , the function is strictly increasing (decreasing) with unless .
ProofUnder the same conditions as Theorem 3.1, the function is strictly increasing (decreasing) in either or . Hence for with , we have
which indicates that the function is strictly increasing (decreasing) with .
The proof ends.
To investigate the third and fourth monotonicity properties, we need a useful lemma.
Lemma 3.3 Let be odd and continuous on . Then is always true for arbitrary .
ProofBy the additivity of definite integral we have
According to the property of definite integral of odd functions, our required result is obtain immediately.
This lemma is proved.
Theorem 3.4 (third monotonicity property) Suppose that is a symmetric, homogenous, and three-time differentiable function; is strictly increasing (decreasing) with ; (1.2) is satisfied. Then for fixed , the function is(1)strictly decreasing (increasing) with on and increasing (decreasing) with on if unless ;(2)strictly increasing (decreasing) with on and decreasing (increasing) with on if unless .
ProofBy (2.13), can be expressed in integral form as
where . Direct partial derivative calculation leads to
which can be spiltted into a sum of two integrals:
Substituting in the first integral above yields
where . Hence
From (1.15), we have
Since is strictly increasing (decreasing) with , by (2.18) and (1.2), we have always It follows from that is positive (negative) if , zero if , and negative (positive) if . However, and hence
This completes the proof.
Theorem 3.5 (fourth monotonicity property) The conditions are the same as those of Theorem 3.1. Then for fixed , the function is strictly increasing (decreasing) with if and decreasing (increasing) if .
ProofBy (2.13), can be expressed in integral form as
A partial derivative calculation yields
() In the case of (2.7) implies that is odd and makes use of Lemma 3.3 and (3.18) can be written as
and then Since strictly increasing (decreasing) with and by assumption (1.2), so (3.3) is true, which indicates that . It follows that This shows that is strictly increasing (decreasing) with if and decreasing (increasing) if .
() In the case of . Similarly, by (3.18), (2.7), (1.2), and (3.3) we have
Combining two cases above, the proof is accomplished.
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, . Consequently, the monotonicities of ratio of Stolarsky means depend on the monotonicities of and defined by (2.15) and (2.16).
Some simple calculations yield
Making use of the well-known inequalities () and ( ) , we see that if and if .
Applying our main results, we can obtain all theorems involving monotonicity of ratio of Stolarsky means in Section 2 of . 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
where ,,, and stand for arithmetic mean, Heronian mean, logarithmic mean, and exponential mean (identric mean) of two positive numbers and , respectively.
Theorem 4.1 Suppose that satisfy assumption (1.2). Then the following inequalities hold, with equalities if and only if .
ProofBy the third monotonicity property, we see that is strictly decreasing in on . Put in and by some calculations, the chain of inequalities (4.3) is derived immediately, with equalities if and only if because the monotonicity of is strict.
The proof is finished.
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