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

Abstract

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.

1. Introduction

Let Φ(𝑎,𝑏)and Ψ(𝑐,𝑑) be bivariate means. For what means Φ and Ψ does the following inequality

Φ(𝑎,𝑏)≤Φ(𝑐,𝑑)Ψ(𝑎,𝑏)Ψ(𝑐,𝑑) (1.1) hold true? where

𝑏𝑎,𝑏,𝑐,𝑑>0,𝑎≥𝑑𝑐≥1. (1.2)

Define that

𝑀_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

𝐺(𝑎,𝑏)≤𝐺(𝑐,𝑑)𝐿(𝑎,𝑏)≤𝐴𝐿(𝑐,𝑑)^1/3(𝑎,𝑏)𝐴_1/3(𝑐,𝑑) (1.4) hold, with equalities if and only if𝑏/𝑎=𝑑/𝑐. That same year, Chen et al. [2] presented second inequalities of ratio of bivariate means:

𝐴_1/2(𝑎,𝑏)𝐴_1/2≤(𝑐,𝑑)𝐻(𝑎,𝑏)≤𝐴𝐻(𝑐,𝑑)^2/3(𝑎,𝑏)𝐴_2/3(𝑐,𝑑), (1.5) where the constant 1/2 and 2/3both are best possible.

In 1994, Pearce et al. [3] proved that the function

𝐿𝑝→^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)=𝑞(𝑎^𝑝−𝑏^𝑝)𝑝(𝑎^𝑞−𝑏^𝑞))^{1/(𝑝−𝑞)}(𝑎,𝑝≠𝑞,𝑝𝑞≠0,_𝑝−𝑏^𝑝)𝑝(ln𝑎−ln𝑏)^1/𝑝(𝑎,𝑝≠0,𝑞=0,_𝑞−𝑏^𝑞)𝑞(ln𝑎−ln𝑏)^1/𝑞𝑎,𝑝=0,𝑞≠0,exp(^𝑝ln𝑎−𝑏^𝑝ln𝑏𝑎^𝑝−𝑏^𝑝−1𝑝√),𝑝=𝑞≠0,𝑎𝑏,𝑝=𝑞=0. (1.7) 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,𝑞𝑆(𝑎,𝑏;𝑐,𝑑)∶=^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,𝑞𝑆(𝑎,𝑏,𝑐)∶=^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

𝑅_𝑝,𝑞(𝑎,𝑏,𝑐)≤𝑅_𝑟,𝑠(𝑎,𝑏,𝑐)(𝑝,𝑞,𝑟,𝑠∈ℝ,0<𝑎<𝑏<𝑐) (1.10) 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

𝑅_𝑝,𝑞𝑀(𝑎,𝑏,𝑐)∶=^𝑝,𝑞(𝑎,𝑏)𝑀_𝑝,𝑞(𝑎,𝑐)(𝑝,𝑞∈ℝ,0<𝑎<𝑏<𝑐). (1.11) 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 1.1Assume 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)>0for (𝑥,𝑦)∈ℝ₊×ℝ₊ with 𝑥≠𝑦 and 𝑓(𝑥,𝑥)=0 for all 𝑥∈ℝ₊, then define that
ℋ_𝑓(𝑝,𝑞;𝑎,𝑏)∶=(𝑓(𝑎^𝑝,𝑏^𝑝)𝑓(𝑎^𝑞,𝑏^𝑞))^{1(𝑝−𝑞)}ℋ(𝑝≠𝑞,𝑝𝑞≠0),_𝑓(𝑝,𝑝;𝑎,𝑏)∶=lim_𝑞→𝑝ℋ_𝑓(𝑎,𝑏;𝑝,𝑞)=𝐺_𝑓,𝑝(𝑎,𝑏)(𝑝=𝑞≠0), (1.12) where 𝐺_𝑓,𝑝(𝑎,𝑏)=𝐺_𝑓^1/𝑝(𝑎^𝑝,𝑏^𝑝),𝐺_𝑓(𝑥,𝑦)=exp(𝑥𝑓_𝑥(𝑥,𝑦)ln𝑥+𝑦𝑓_𝑦(𝑥,𝑦)ln𝑦𝑓(𝑥,𝑦)), (1.13) and𝑓_𝑥(𝑥,𝑦)and 𝑓_𝑦(𝑥,𝑦)denote first-order partial derivative for first and second variables of 𝑓(𝑥,𝑦),respectively.
If 𝑓(𝑥,𝑦)>0for all (𝑥,𝑦)∈ℝ₊×ℝ₊, then define further
ℋ_𝑓(𝑝,0;𝑎,𝑏)∶=(𝑓(𝑎^𝑝,𝑏^𝑝))𝑓(1,1)^1/𝑝ℋ(𝑝≠0,𝑞=0),_𝑓(0,𝑞;𝑎,𝑏)∶=(𝑓(𝑎^𝑞,𝑏^𝑞))𝑓(1,1)^1/𝑞ℋ(𝑝=0,𝑞≠0),_𝑓(0,0;𝑎,𝑏)∶=lim_𝑝→0ℋ_𝑓(𝑎,𝑏;𝑝,0)=𝑎^{𝑓𝑥(1,1)/𝑓(1,1)}𝑏^{𝑓𝑦(1,1)/𝑓(1,1)}(𝑝=𝑞=0). (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

𝑅_𝑓ℋ(𝑝,𝑞)∶=^𝑓(𝑝,𝑞;𝑎,𝑏)ℋ_𝑓𝑏(𝑝,𝑞;𝑐,𝑑)(𝑝,𝑞∈ℝ,𝑎,𝑏,𝑐,𝑑>0with𝑎>𝑑𝑐≥1) (1.15) 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].

2. Properties and Lemmas

Before formulating our main results, let us recall the properties and lemmas of two-parameter homogeneous functions.

Property 2.1ℋ_𝑓(𝑝,𝑞)is symmetric with respect to 𝑝,𝑞, that is, ℋ_𝑓(𝑝,𝑞)=ℋ_𝑓(𝑞,𝑝). (2.1)

Property 2.2If𝑓(𝑥,𝑦) is symmetric with respect to 𝑥 and 𝑦, then ℋ_𝑓𝐺(−𝑝,−𝑞;𝑎,𝑏)=^2𝑛ℋ_𝑓,ℋ(𝑝,𝑞;𝑎,𝑏)_𝑓(𝑝,−𝑝;𝑎,𝑏)=𝐺^𝑛, (2.2) where 𝐺=√𝑎𝑏.

Property 2.3 (see [14, (1.13)]) If 𝐺_𝑓,𝑡 is continuous on [𝑞,𝑝] or [𝑝,𝑞], then lnℋ_𝑓1(𝑝,𝑞)=𝑝−𝑞^𝑝_𝑞ln𝐺_𝑓,𝑡𝑑𝑡, (2.3) where 𝐺_𝑓,𝑡 is defined by (1.13).

It is worth mentioning that the following function

𝑇(𝑡)=𝑇(𝑡;𝑎,𝑏)∶=ln𝑓(𝑎^𝑡,𝑏^𝑡),𝑡≠0 (2.4) is well behaved, whose properties as useful lemmas read as follows.

Lemma 2.4 (see [14, (1.14), (1.15), (2.10), (2.11)]) Suppose that 𝑓∶ℝ₊×ℝ₊⧵{(𝑥,𝑥)∶𝑥∈ℝ₊}→ℝ₊) is a symmetric, 𝑛-order homogenous and three-time differentiable function. Then 𝑇𝑇(𝑡)−𝑇(−𝑡)=2𝑛𝑡ln𝐺,_′(𝑡)+𝑇^′𝑇(−𝑡)=2𝑛ln𝐺,_′′(−𝑡)=𝑇^′′𝑇(𝑡),_′′′(−𝑡)=−𝑇^′′′(𝑡), (2.5)(2.6)(2.7)(2.8) where 𝐺=√𝑎𝑏.

Remark 2.5If 𝑓(1,1)∶=lim_𝑥→1𝑓(𝑥,1)>0, then 𝑇(𝑡) can be extended continuously by defining 𝑇(0)∶=lim_𝑡→0𝑇(𝑡)=ln𝑓(1,1), with the result that 𝑇(𝑡) is also three times derivable at 𝑡=0. Particularly, 𝑇^′(0)∶=𝑛ln𝐺. Thus (2.6) can be written as 𝑇^′(𝑡)+𝑇^′(−𝑡)=2𝑇^′(0). (2.9)

Lemma 2.6 (see [14, Lemma 3, 4]) Suppose that 𝑓∶ℝ₊×ℝ₊⧵{(𝑥,𝑥)∶𝑥∈ℝ₊}→ℝ₊is an 𝑛-order homogenous and three times differentiable function. Then 𝑇^′𝑎(𝑡)=^𝑡𝑓_𝑥(𝑎^𝑡,𝑏^𝑡)ln𝑎+𝑏^𝑡𝑓_𝑏(𝑎^𝑡,𝑏^𝑡)ln𝑏𝑓(𝑎^𝑡,𝑏^𝑡),𝑇^′′(𝑡)=−𝑥𝑦ℐln²(𝑏𝑎),ℐ=(ln𝑓)_𝑥𝑦𝑇_′′′,(𝑡)=−𝐶𝑡⁻³𝒥,𝒥=(𝑥−𝑦)(𝑥ℐ)_𝑥,𝐶=𝑥𝑦ln³(𝑥/𝑦)𝑥−𝑦>0, (2.10)(2.11)(2.12) where 𝑥=𝑎^𝑡,𝑦=𝑏^𝑡, 𝐺_𝑓(𝑥,𝑦) is defined by (1.13).

Remark 2.7Comparing (1.13) with (2.10), we see that 𝑇^′(𝑡)=ln𝐺_𝑓,𝑡(𝑎,𝑏). Thus (2.3) can be written as lnℋ_𝑓1(𝑝,𝑞)=𝑝−𝑞^𝑝_𝑞𝑇^′𝑇(𝑡)𝑑𝑡if𝑝≠𝑞_′=(𝑞)if𝑝=𝑞₁₀𝑇^′(𝑡𝑝+(1−𝑡)𝑞)𝑑𝑡. (2.13)

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

𝜕ℐ=ℐ(𝑥,𝑦)=²ln𝑓(𝑥,𝑦)𝜕𝑥𝜕𝑦=(ln𝑓(𝑥,𝑦))_𝑥𝑦=(ln𝑓)_𝑥𝑦,𝒥=𝒥(𝑥,𝑦)=(𝑥−𝑦)𝜕(𝑥ℐ)𝜕𝑥=(𝑥−𝑦)(𝑥ℐ)_𝑥. (2.14) In next section we will encounter other two key decision functions defined by

𝒯₂(𝑥,𝑦)∶=−𝑥𝑦ℐln²(𝑥𝑦𝒯),₃(𝑥,𝑦)∶=−𝑥𝑦(𝑥ℐ)_𝑥ln³(𝑥𝑦), (2.15)(2.16) where ℐ=(ln𝑓)_𝑥𝑦, 𝑥=𝑎^𝑡,𝑦=𝑏^𝑡. Combining (2.11), (2.12) with (2.15), (2.16) we have the following relations:

𝑇^′′(𝑡)=𝑡⁻²𝒯₂𝑇(𝑥,𝑦),_′′′(𝑡)=𝑡⁻³𝒯₃(𝑥,𝑦), (2.17)(2.18) where 𝑥=𝑎^𝑡,𝑦=𝑏^𝑡.

Moreover, it is easy to verify that 𝒯₂(𝑥,𝑦) and 𝒯₃(𝑥,𝑦) both are zero-order homogeneous functions due to homogeneity of 𝑓(𝑥,𝑦), and thus,

𝒯₂(𝑥,𝑦)=𝒯₂(𝑥𝑦,1)=𝒯₂𝑦(1,𝑥𝒯),₃(𝑥,𝑦)=𝒯₃(𝑥𝑦,1)=𝒯₃𝑦(1,𝑥). (2.19)

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 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; 𝒯₂(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.

ProofSince ℋ_𝑓(𝑝,𝑞) is symmetric with respect to 𝑝 and 𝑞, it only needs to prove the log-convexity of 𝑅_𝑓(𝑝,𝑞) in parameter 𝑝.
Direct partial derivative calculation for (2.13) leads to
𝜕lnℋ_𝑓(𝑝,𝑞)=𝜕𝑝¹₀𝑡𝑇^′′(𝑡𝑝+(1−𝑡)𝑞)𝑑𝑡. (3.1) From (1.15), we have 𝜕ln𝑅_𝑓(𝑝,𝑞)=𝜕𝑝¹₀𝑡(𝑇^′′(𝑡𝑝+(1−𝑡)𝑞;𝑎,𝑏)−𝑇^′′(𝑡𝑝+(1−𝑡)𝑞;𝑐,𝑑))𝑑𝑡. (3.2) Since 𝒯₂(1,𝑢) is strictly increasing (decreasing) with 𝑢>1 and by (2.7), (2.17), and assumption (1.2), we have always 𝑇^′′(𝑡;𝑎,𝑏)−𝑇^′′(𝑡;𝑐,𝑑)=𝑇^′′(|𝑡|;𝑎,𝑏)−𝑇^′′(|𝑡|;𝑐,𝑑)=𝑡⁻²𝒯₂(𝑎^|𝑡|,𝑏^|𝑡|)−𝑡⁻²𝒯₂(𝑐^|𝑡|,𝑑^|𝑡|)=𝑡⁻²(𝒯₂𝑏(1,(𝑎)^|𝑡|)−𝒯₂𝑑(1,(𝑐)^|𝑡|))>(<)0. (3.3) It follows that 𝜕ln𝑅_𝑓(𝑝,𝑞)𝜕𝑝>(<)0. (3.4)
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 𝑚∈ℝ, the function 𝑅_𝑓(𝑝,𝑝+𝑚) is strictly increasing (decreasing) with 𝑝 unless 𝑏/𝑎=𝑑/𝑐.

ProofUnder the same conditions as Theorem 3.1, the function 𝑅_𝑓(𝑝,𝑞) is strictly increasing (decreasing) in either 𝑝 or 𝑞. Hence for 𝑝₁,𝑝₂∈ℝ with 𝑝₁<𝑝₂, we have 𝑅_𝑓(𝑝₁,𝑝₁+𝑚)<𝑅_𝑓(𝑝₂,𝑝₁+𝑚)<𝑅_𝑓(𝑝₂,𝑝₂+𝑚), (3.5) 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.

Lemma 3.3 Let 𝑓(𝑥) be odd and continuous on (−𝑚,𝑚)(𝑚>0). Then ^𝑟_𝑠𝑓(𝑥)𝑑𝑥=^|𝑟|_|𝑠|𝑓(𝑥)𝑑𝑥 (3.6) is always true for arbitrary 𝑟,𝑠∈(−𝑚,𝑚).

ProofBy the additivity of definite integral we have ^𝑟_𝑠𝑓(𝑥)𝑑𝑥=_𝑠^|𝑠|𝑓(𝑥)𝑑𝑥+^|𝑟|_|𝑠|𝑓(𝑥)𝑑𝑥+^𝑟_|𝑟|𝑓(𝑥)𝑑𝑥. (3.7) 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 𝑓∶ℝ₊×ℝ₊→ℝ₊ is a symmetric, homogenous, and three-time differentiable function; 𝒯₃(1,𝑢) is strictly increasing (decreasing) with 𝑢>1; (1.2) is satisfied. Then for fixed 𝑚≠0, the function 𝑅_𝑓(𝑝,2𝑚−𝑝) is(1)strictly decreasing (increasing) with 𝑝 on (−∞,𝑚) and increasing (decreasing) with 𝑝 on (𝑚,∞) if 𝑚>0 unless 𝑏/𝑎=𝑑/𝑐;(2)strictly increasing (decreasing) with 𝑝 on (−∞,𝑚) and decreasing (increasing) with 𝑝 on (𝑚,∞) if 𝑚<0 unless 𝑏/𝑎=𝑑/𝑐.

ProofBy (2.13), lnℋ_𝑓(𝑝,2𝑚−𝑝) can be expressed in integral form as lnℋ_𝑓(𝑝,2𝑚−𝑝)=¹₀𝑇^′(𝑡₁(𝑡))𝑑𝑡, (3.8) where 𝑡₁(𝑡)=𝑡𝑝+(1−𝑡)(2𝑚−𝑝). Direct partial derivative calculation leads to 𝜕lnℋ_𝑓(𝑝,2𝑚−𝑝)=𝜕𝑝¹₀(2𝑡−1)𝑇^′′(𝑡₁(𝑡))𝑑𝑡, (3.9) which can be spiltted into a sum of two integrals: ₀^1/2(2𝑡−1)𝑇^′′(𝑡₁(𝑡))𝑑𝑡+¹_1/2(2𝑡−1)𝑇^′′(𝑡₁(𝑡))𝑑𝑡. (3.10) Substituting 𝑡=1−𝑣 in the first integral above yields ₀^1/2(2𝑡−1)𝑇^′′(𝑡₁(𝑡))𝑑𝑡=−¹_1/2(2𝑣−1)𝑇^′′(𝑡₂(𝑣))𝑑𝑣, (3.11) where 𝑡₂(𝑡)=(1−𝑡)𝑝+𝑡(2𝑚−𝑝). Hence 𝜕lnℋ_𝑓(𝑝,2𝑚−𝑝)𝜕𝑝=−¹_1/2(2𝑣−1)𝑇^′′(𝑡₂(𝑣))𝑑𝑣+¹_1/2(2𝑡−1)𝑇^′′(𝑡₁=(𝑡))𝑑𝑡_11/2(2𝑡−1)(𝑇^′′(𝑡₁(𝑡))−𝑇^′′(𝑡₂=(𝑡)))𝑑𝑡_11/2(2𝑡−1)(^𝑡1_𝑡^(𝑡)_2(𝑡)𝑇^′′′=(𝑠)𝑑𝑠)𝑑𝑡_11/2(2𝑡−1)(^{|𝑡1(𝑡)|}_{|𝑡2(𝑡)|}𝑇^′′′(𝑠)𝑑𝑠)𝑑𝑡(byLemma(3.3)). (3.12)
From (1.15), we have
𝜕ln𝑅_𝑓(𝑝,2𝑚−𝑝)=𝜕𝑝𝜕lnℋ_𝑓(𝑝,2𝑚−𝑝;𝑎,𝑏)−𝜕𝑝𝜕lnℋ_𝑓(𝑝,2𝑚−𝑝;𝑐,𝑑)=𝜕𝑝_11/2(2𝑡−1)(^{|𝑡1(𝑡)|}_{|𝑡2(𝑡)|}(𝑇^′′′(𝑠;𝑎,𝑏)−𝑇^′′′(𝑠;𝑐,𝑑))𝑑𝑠)𝑑𝑡. (3.13) Since 𝒯₃(1,𝑢) is strictly increasing (decreasing) with 𝑢>1, by (2.18) and (1.2), we have always 𝑇^′′′(𝑡;𝑎,𝑏)−𝑇^′′′(𝑡;𝑐,𝑑)=𝑡⁻³(𝒯₃(𝑎^𝑡,𝑏^𝑡)−𝒯₃(𝑐^𝑡,𝑑^𝑡))=𝑡⁻³(𝒯₃𝑏(1,(𝑎)^𝑡)−𝒯₃𝑑(1,(𝑐)^𝑡))>(<)0for𝑡>0. (3.14) It follows from 2𝑡−1≥0(𝑡∈1/2,1] that 𝜕ln𝑅_𝑓(𝑝,2𝑚−𝑝)/𝜕𝑝 is positive (negative) if |𝑡₁(𝑡)|>|𝑡₂(𝑡)|, zero if |𝑡₂(𝑡)|=|𝑡₂(𝑡)|, and negative (positive) if |𝑡₁(𝑡)|<|𝑡₂(𝑡)|. However, |𝑡₁(𝑡)|²−|𝑡₂(𝑡)|²1=8𝑚(𝑡−2)(𝑝−𝑚), (3.15) and hence 𝜕ln𝑅_𝑓(𝑝,2𝑚−𝑝)𝜕𝑝>(<)0if𝑚>0,𝑝>𝑚or𝑚<0,𝑝<𝑚,=0if𝑚=0,<(>)0if𝑚>0,𝑝<𝑚or𝑚<0,𝑝>𝑚. (3.16)
This completes the proof.

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.

ProofBy (2.13), lnℋ_𝑓(𝑝𝑟,𝑝𝑠) can be expressed in integral form as lnℋ_𝑓1(𝑝𝑟,𝑝𝑠)=𝑟−𝑠^𝑟_𝑠𝑇^′𝑇(𝑝𝑡)𝑑𝑡if𝑟≠𝑠,_′(𝑝𝑟)if𝑟=𝑠. (3.17) A partial derivative calculation yields 𝜕lnℋ_𝑓(𝑝𝑟,𝑝𝑠)=1𝜕𝑝𝑟−𝑠^𝑟_𝑠𝑡𝑇^′′(𝑝𝑡)𝑑𝑡if𝑟≠𝑠,𝑟𝑇^′′(𝑝𝑟)if𝑟=𝑠. (3.18)
(1) In the case of 𝑟≠𝑠 (2.7) implies that 𝑡𝑇^′′(𝑝𝑡)=𝑡𝑇^′′(|𝑝𝑡|) is odd and makes use of Lemma 3.3 and (3.18) can be written as
𝜕lnℋ_𝑓(𝑝𝑟,𝑝𝑠)=1𝜕𝑝𝑟−𝑠^|𝑟|_|𝑠|𝑡𝑇^′′(|𝑝𝑡|)𝑑𝑡=𝑟+𝑠|𝑟|−|𝑠|^|𝑟|_|𝑠|𝑡𝑇^′′(|𝑝𝑡|)𝑑𝑡, (3.19) and then 𝜕ln𝑅_𝑓(𝑝𝑟,𝑝𝑠)=𝜕𝑝𝑟+𝑠|𝑟|−|𝑠|^|𝑟|_|𝑠|𝑡(𝑇^′′(|𝑝𝑡|;𝑎,𝑏)−𝑇^′′(|𝑝𝑡|;𝑐,𝑑))𝑑𝑡. (3.20) Since 𝒯₂(1,𝑢) strictly increasing (decreasing) with 𝑢>1 and by assumption (1.2), so (3.3) is true, which indicates that 𝑇^′′(|𝑡|;𝑎,𝑏)−𝑇^′′(|𝑡|;𝑐,𝑑)>(<)0. It follows that 𝜕ln𝑅_𝑓(𝑝𝑟,𝑝𝑠)𝜕𝑝>(<)0if𝑟+𝑠>0,<(>)0if𝑟+𝑠<0. (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
𝜕ln𝑅_𝑓(𝑝𝑟,𝑝𝑠)𝜕𝑝=𝑟(𝑇^′′(|𝑝𝑟|;𝑎,𝑏)−𝑇^′′(|𝑝𝑟|;𝑐,𝑑))>(<)0if𝑟>0,<(>)0if𝑟<0. (3.22)
Combining two cases above, the proof is accomplished.

4. 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 𝒯₂(1,𝑢) and 𝒯₃(1,𝑢) defined by (2.15) and (2.16).

Some simple calculations yield

ℐ=(ln𝐿)_𝑥𝑦=1(𝑥−𝑦)²−1𝑥𝑦(ln𝑥−ln𝑦)²,𝒯₂(𝑥,𝑦)=−𝑥𝑦ℐln²(𝑥𝑦)=−𝑥𝑦ln²(𝑥/𝑦)(𝑥−𝑦)²+1,𝑑𝒯₂(1,𝑢)𝑑𝑢=−2(𝑢−1)⁻³ln²𝑢(𝑢−1−ln𝑢𝑢+12),(𝑥ℐ)_𝑥=−𝑥+𝑦(𝑥−𝑦)³+2𝑥𝑦(ln𝑥−ln𝑦)³,𝒯₃(𝑥,𝑦)=−𝑥𝑦(𝑥ℐ)_𝑥ln³(𝑥𝑦)=−2+𝑥𝑦(𝑥+𝑦)(𝑥−𝑦)³ln³(𝑥𝑦),𝑑𝒯₃(1,𝑢)𝑑𝑢=6𝑢(𝑢−1)⁴ln³𝑢𝑢(²−1ln𝑢²−((𝑢²+1)/2)+2√𝑢²3). (4.1) Making use of the well-known inequalities 𝐿(𝑥,𝑦)<(𝑥+𝑦)/2 (𝑥,𝑦>0) and 𝐿(𝑥,𝑦)<((𝑥+𝑦)/2+2√𝑥𝑦)/3(𝑥,𝑦>0 ) [16], we see that 𝑑𝒯₂(1,𝑢)/𝑑𝑢>0 if 𝑢>1 and 𝑑𝒯₃(1,𝑢)/𝑑𝑢<0 if 𝑢>1.

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

𝑀_𝑝∶=𝑀^1/𝑝(𝑐^𝑝,𝑑^𝑝),𝑀=𝐴,𝐻,𝐿,𝐼, (4.2) where 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 Suppose that 𝑎,𝑏,𝑐,𝑑 satisfy assumption (1.2). Then the following inequalities 𝐴^1/3𝐺^2/3𝐴^1/3𝐺^2/3≤√𝐺𝐻𝐺𝐻≤𝐺^2/5𝐴^1/5_1/3𝐴^2/5_2/3𝐺^2/5𝐴^1/5_1/3𝐴^2/5_2/3≤𝐿𝐿≤𝐴^1/3_1/5𝐴^2/3_2/5𝐴^1/3_1/5𝐴^2/3_2/5≤𝐻_1/2𝐻_1/2≤𝐴_1/3𝐴_1/3≤𝐻²_2/5𝐴⁻¹_1/5𝐻²_2/5𝐴⁻¹_1/5≤𝐼_1/2𝐼̃_1/2 (4.3) hold, with equalities if and only if 𝑏/𝑎=𝑑/𝑐.

ProofBy 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.