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

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 [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 𝑄_1𝑓𝑝𝑔∶=^1𝑓𝑝;𝑎,𝑏𝑔_1𝑓,𝑄𝑝;𝑐,𝑑_2𝑓𝑝𝑔∶=^2𝑓𝑝;𝑎,𝑏𝑔_2𝑓,𝑄𝑝;𝑐,𝑑_3𝑓𝑝𝑔∶=^3𝑓𝑝;𝑎,𝑏𝑔_3𝑓,𝑄𝑝;𝑐,𝑑_4𝑓𝑝𝑔∶=^4𝑓𝑝;𝑎,𝑏𝑔_4𝑓,𝑝;𝑐,𝑑 (1.1) where 𝑔_1𝑓𝑝=𝑔_1𝑓𝑝;𝑎,𝑏∶=ℋ_𝑓𝑝,𝑞ℋ_𝑓,𝑔2𝑘−𝑝,𝑞_2𝑓𝑝=𝑔_2𝑓𝑝;𝑎,𝑏∶=ℋ_𝑓𝑝,𝑝+𝑚ℋ_𝑓,𝑔2𝑘−𝑝,2𝑘−𝑝+𝑚_3𝑓𝑝=𝑔_3𝑓𝑝;𝑎,𝑏∶=ℋ_𝑓𝑝,2𝑚−𝑝ℋ_𝑓,𝑔2𝑘−𝑝,2𝑚−2𝑘+𝑝_4𝑓𝑝=𝑔_4𝑓𝑝;𝑎,𝑏∶=ℋ_𝑓𝑝𝑟,𝑝𝑠ℋ_𝑓2𝑘−𝑝𝑟,2𝑘−𝑝𝑠, (1.2)(1.3)(1.4)(1.5) 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: ℝ²₊⧵{(𝑥,𝑥),𝑥∈ℝ₊}→ℝ₊ be a first-order homogeneous continuous function which has first partial derivatives. Then, ℋ_𝑓∶ℝ²×ℝ²₊→ℝ₊ is called a homogeneous function generated by 𝑓 with parameters 𝑝 and 𝑞 if ℋ_𝑓 is defined by for 𝑎≠𝑏ℋ_𝑓=𝑓𝑎𝑝,𝑞;𝑎,𝑏^𝑝,𝑏^𝑝𝑓𝑎^𝑞,𝑏^{𝑞1(𝑝−𝑞)}ℋ,if𝑝𝑞𝑝−𝑞≠0,_𝑓𝑎𝑝,𝑝;𝑎,𝑏=exp^𝑝𝑓_𝑥𝑎^𝑝,𝑏^𝑝ln𝑎+𝑏^𝑝𝑓_𝑦𝑎^𝑝,𝑏^𝑝ln𝑏𝑓𝑎^𝑝,𝑏^𝑝,if𝑝=𝑞≠0, (1.6) 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→𝑥𝑓(𝑥,𝑦) exits and is positive for all 𝑥∈ℝ₊, then further define ℋ_𝑓=𝑓𝑎𝑝,0;𝑎,𝑏^𝑝,𝑏^𝑝𝑓(1,1)^1/𝑝ℋ,if𝑝≠0,𝑞=0,_𝑓=𝑓𝑎0,𝑞;𝑎,𝑏^𝑞,𝑏^𝑞𝑓(1,1)^1/𝑞ℋ,if𝑝=0,𝑞≠0,_𝑓(0,0;𝑎,𝑏)=𝑎^{𝑓𝑥(1,1)/𝑓(1,1)}𝑏^{𝑓𝑦(1,1)/𝑓(1,1)},if𝑝=𝑞=0, (1.7) 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,𝑞ℋ_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 𝐿=𝐿(𝑥,𝑦)=(𝑥−𝑦)/(ln𝑥−ln𝑦) 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 𝑆_𝑝,𝑞()=𝑞𝑎,𝑏𝑝𝑎^𝑝−𝑏^𝑝𝑎^𝑞−𝑏^{𝑞1/(𝑝−𝑞)}𝐿,if𝑝𝑞𝑝−𝑞≠0,_1/𝑝𝑎^𝑝,𝑏^𝑝𝐿,if𝑝≠0,𝑞=0,_1/𝑞𝑎^𝑞,𝑏^𝑞𝐼,if𝑞≠0,𝑝=0,_1/𝑝𝑎^𝑝,𝑏^𝑝√,if𝑝=𝑞≠0,𝑎𝑏,if𝑝=𝑞=0, (1.8) 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 𝐺_𝑝,𝑞()=𝑎𝑎,𝑏^𝑝+𝑏^𝑝𝑎^𝑞+𝑏^{𝑞1/(𝑝−𝑞)}𝑍,if𝑝≠𝑞,_1/𝑝𝑎^𝑝,𝑏^𝑝,if𝑝=𝑞, (1.9) 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.

2. Main Results and Proofs

In [15, 16, 20], two decision functions play an important role, that are, =𝜕ℐ=ℐ𝑥,𝑦²ln𝑓𝑥,𝑦=𝜕𝑥𝜕𝑦ln𝑓𝑥,𝑦_𝑥𝑦=ln𝑓_𝑥𝑦,=𝜕()𝒥=𝒥𝑥,𝑦𝑥−𝑦𝑥ℐ=()𝜕𝑥𝑥−𝑦𝑥ℐ_𝑥. (2.1) In [14], it is important to another key decision function defined by 𝒯₃𝑥,𝑦∶=−𝑥𝑦(𝑥ℐ)_𝑥ln³𝑥𝑦,whereℐ=ln𝑓_𝑥𝑦,𝑥=𝑎^𝑡,𝑦=𝑏^𝑡. (2.2) Note that the function 𝑇 defined by 𝑇(𝑡)∶=ln𝑓𝑎^𝑡,𝑏^𝑡,𝑡≠0 (2.3) has well properties (see [15, 16]). And it has shown in [14, (3.4)], [16, Lemma 4] the relation among 𝑇^′′′(𝑡), 𝒥(𝑥,𝑦) and 𝒯₃(𝑥,𝑦):𝑇^′′′(𝑡)=𝑡⁻³𝒯₃𝑥,𝑦,where𝑥=𝑎^𝑡,𝑦=𝑏^𝑡,𝑇^′′′(𝑡)=−𝐶𝑡⁻³𝒥𝑥,𝑦,where𝐶=𝑥𝑦𝑥−𝑦⁻¹ln𝑥−ln𝑦³>0. (2.4)(2.5) Moreover, it has revealed in [14, (3.5)] that 𝒯₃𝑥,𝑦=𝒯₃𝑥𝑦,1=𝒯₃𝑦1,𝑥. (2.6)

Now, we observe the monotonicities of ratio of certain mixed means defined by .

Theorem 2.1. Suppose that 𝑓: ℝ₊×ℝ₊→ℝ₊ is a symmetric, first-order homogenous, and three-time differentiable function, and 𝒯₃(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 ln𝑔_1𝑓𝑝=12lnℋ_𝑓+1𝑝,𝑞2lnℋ_𝑓=12𝑘−𝑝,𝑞2¹₀𝑇^′𝑡₁₁1𝑑𝑡+2¹₀𝑇^′𝑡₁₂𝑑𝑡, (2.7) where 𝑡₁₂=𝑡𝑝+(1−𝑡)𝑞,𝑡₁₁=𝑡2𝑘−𝑝+(1−𝑡)𝑞. (2.8) Partial derivative leads to ln𝑔_1𝑓𝑝^′=12¹₀𝑡𝑇^′′𝑡₁₂1𝑑𝑡−2¹₀𝑡𝑇^′′𝑡₁₁=1𝑑𝑡2¹₀𝑡𝑇^′′𝑡₁₂1𝑑𝑡−2¹₀𝑡𝑇^′′𝑡₁₁[],(=1𝑑𝑡by132.7)2¹₀𝑡^|𝑡12|_𝑡11𝑇^′′′(𝑣)𝑑𝑣𝑑𝑡, (2.9) and then ln𝑄_1𝑓𝑝^′=ln𝑔_1𝑓𝑝;𝑎,𝑏^′−ln𝑔_1𝑓𝑝;𝑐,𝑑^′=12¹₀𝑡^|𝑡12|_𝑡11𝑇^′′′(𝑣)1𝑑𝑣𝑑𝑡−2¹₀𝑡^|𝑡12|_𝑡11𝑇^′′′()=𝑣;𝑐,𝑑𝑑𝑣𝑑𝑡₁₀𝑡𝑡₁₂−𝑡₁₁∫^|𝑡12|_𝑡11𝑇^′′′(𝑣;𝑎,𝑏)−𝑇^′′′(𝑣;𝑐,𝑑)𝑑𝑣𝑡₁₂−𝑡₁₁𝑑𝑡∶=¹₀𝑡𝑡₁₂−𝑡₁₁ℎ𝑡₁₁,𝑡₁₂𝑑𝑡, (2.10) where ℎ∫𝑥,𝑦∶=^𝑦𝑥𝑇^′′′(𝑣;𝑎,𝑏)−𝑇^′′′(𝑣;𝑐,𝑑)𝑑𝑣𝑇𝑦−𝑥,if𝑥≠𝑦,_′′′(𝑥;𝑎,𝑏)−𝑇^′′′(𝑥;𝑐,𝑑),if𝑥=𝑦. (2.11) Since 𝒯₃(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 𝑇^′′′(𝑣;𝑎,𝑏)−𝑇^′′′(𝑣;𝑐,𝑑)=𝑣⁻³𝒯₃𝑎^𝑣,𝑏^𝑣−𝒯₃𝑐^𝑣,𝑑^𝑣=𝑣⁻³𝒯₃𝑏1,𝑎^𝑣−𝒯₃𝑑1,𝑐^𝑣>(<)0,for𝑣>0, (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 (|𝑡₁₂|−|𝑡₁₁|). In fact, if 𝑞≥0,  𝑘≥0, then for 𝑡∈[0,1]𝑡₁₂−𝑡₁₁=𝑡²₁₂−𝑡²₁₁𝑡₁₂+𝑡₁₁𝑞()=4𝑡1−𝑡+𝑘𝑡𝑡₁₂+𝑡₁₁𝑝−𝑘=>0,if𝑝>𝑘,<0,if𝑝<𝑘. (2.13) It follows that ln𝑄_1𝑓𝑝^′=>(<)<(>)0,if𝑝>𝑘,0,if𝑝<𝑘. (2.14) Clearly, the monotonicity of 𝑄_1𝑓 is converse if 𝑞≤0,  𝑘≤0.
This completes the proof.

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).
The 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.

Proof. By (2.13) in [13] we have ln𝑔_2𝑓𝑝=12lnℋ_𝑓+1𝑝,𝑝+𝑚2lnℋ_𝑓=12𝑘−𝑝,2𝑘−𝑝+𝑚2¹₀𝑇^′𝑡₂₂1𝑑𝑡+2¹₀𝑇^′𝑡₂₁𝑑𝑡, (2.15) where 𝑡₂₂=𝑡𝑝+(1−𝑡)𝑝+𝑚,𝑡₂₁=𝑡2𝑘−𝑝+(1−𝑡)2𝑘−𝑝+𝑚. (2.16) Direct calculation leads to ln𝑔_2𝑓𝑝^′=12¹₀𝑇^′′𝑡₂₂1𝑑𝑡−2¹₀𝑇^′′𝑡₂₁1𝑑𝑡=2¹₀^|𝑡22|_𝑡21𝑇^′′′(𝑣)𝑑𝑣𝑑𝑡, (2.17) and then ln𝑄_2𝑓𝑝^′=ln𝑔_2𝑓𝑝;𝑎,𝑏^′−ln𝑔_2𝑓𝑝;𝑐,𝑑^′=12¹₀^|𝑡22|_𝑡21𝑇^′′′()1𝑣;𝑎,𝑏𝑑𝑣𝑑𝑡−2¹₀^|𝑡22|_𝑡21𝑇^′′′()=1𝑣;𝑐,𝑑𝑑𝑣𝑑𝑡2¹₀𝑡₂₂−𝑡₂₁ℎ𝑡₂₁,𝑡₂₂𝑑𝑡, (2.18) where ℎ(𝑥,𝑦) is defined by (2.11). As shown previously, ℎ(𝑥,𝑦)>(<)0 for 𝑥,𝑦>0 if 𝒯₃(1,𝑢) strictly increase (decrease) with 𝑢>1 and decrease (increase) with 0<𝑢<1; it remains to determine the sign of (|𝑡₂₂|−|𝑡₂₁|). It is easy to verify that if 𝑘≥0 and 𝑘+𝑚≥0, then 𝑡₂₂−𝑡₂₁=𝑡²₂₂−𝑡²₂₁𝑡₂₂+𝑡₂₁()=4𝑘+𝑚1−𝑡𝑡₂₂+𝑡₂₁𝑝−𝑘=>0,if𝑝>𝑘,<0,if𝑝<𝑘. (2.19)
Thus, we have ln𝑄_2𝑓𝑝^′=>(<)<(>)0,if𝑝>𝑘,0,if𝑝<𝑘. (2.20) Clearly, the monotonicity of 𝑄_2𝑓 is converse if 𝑘≤0 and 𝑘+𝑚≤0.
The proof ends.

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.

Proof. From (2.13) in [13], it is derived that ln𝑔_3𝑓𝑝=12lnℋ_𝑓+1𝑝,2𝑚−𝑝2lnℋ_𝑓=12𝑘−𝑝,2𝑚−2𝑘+𝑝2¹₀𝑇^′𝑡₃₂1𝑑𝑡+2¹₀𝑇^′𝑡₃₁𝑑𝑡, (2.21) where 𝑡₃₂=𝑡𝑝+(1−𝑡)2𝑚−𝑝,𝑡₃₁=𝑡2𝑘−𝑝+(1−𝑡)2𝑚−2𝑘+𝑝. (2.22) Simple calculation yields ln𝑔_3𝑓𝑝^′=12¹₀(2𝑡−1)𝑇^′′𝑡₃₂−𝑇^′′𝑡₃₁1𝑑𝑡=2¹₀(2𝑡−1)^|𝑡32|_𝑡31𝑇^′′′(𝑣;𝑎,𝑏)𝑑𝑣𝑑𝑡. (2.23) Hence, ln𝑄_3𝑓𝑝^′=ln𝑔_3𝑓𝑝;𝑎,𝑏^′−ln𝑔_3𝑓𝑝;𝑐,𝑑^′=12¹₀(2𝑡−1)^|𝑡32|_𝑡31𝑇^′′′(𝑣;𝑎,𝑏)−𝑇^′′′()=1𝑣;𝑐,𝑑𝑑𝑣𝑑𝑡2¹₀(2𝑡−1)𝑡₃₂−𝑡₃₁ℎ𝑡₃₁,𝑡₃₂𝑑𝑡, (2.24) where ℎ(𝑥,𝑦) is defined by (2.11). It has shown that ℎ(𝑥,𝑦)>(<)0 for 𝑥,𝑦>0 if 𝒯₃(1,𝑢) strictly increase (decrease) with 𝑢>1 and decrease (increase) with 0<𝑢<1, and we have also to check the sign of (2𝑡−1)(|𝑡₃₂|−|𝑡₃₁|). Easy calculation reveals that if 𝑚>0, 0≤𝑘≤2𝑚, then (2𝑡−1)𝑡₃₂−𝑡₃₁=()𝑡2𝑡−1²³²−𝑡²₃₁𝑡₃₂+𝑡₃₁=4(2𝑡−1)²𝑡𝑘+(1−𝑡)(2𝑚−𝑘)𝑡₃₂+𝑡₃₁=𝑝−𝑘>0,if𝑝>𝑘,<0,if𝑝<𝑘, (2.25) which yields ln𝑄_3𝑓𝑝^′=>(<)<(>)0,if𝑝>𝑘,0,if𝑝<𝑘. (2.26) It is evident that the monotonicity of 𝑄_3𝑓 is converse if 𝑚<0, 2𝑚≤𝑘≤0.
Thus the proof is complete.

Theorem 2.4. 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 𝑘,𝑟,𝑠∈ℝ with r+𝑠≠0, 𝑄_4𝑓 is strictly increasing (decreasing) in 𝑝 on (𝑘,∞) and decreasing (increasing) on (−∞,𝑘) if 𝑘(𝑟+𝑠)>0.
The monotonicity of 𝑄_4𝑓 is converse if 𝑘(𝑟+𝑠)<0.

Proof. By (2.13) in [13], lnℋ_𝑓(𝑝𝑟,𝑝𝑠) can be expressed in integral form lnℋ_𝑓=1𝑝𝑟,𝑝𝑠∫𝑟−𝑠^𝑟𝑠𝑇^′𝑇𝑝𝑡𝑑𝑡,if𝑟≠𝑠,_′𝑝𝑟,if𝑟=𝑠. (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 ln𝑔_4𝑓𝑝=lnℋ_𝑓𝑝𝑟,𝑝𝑠ℋ_𝑓𝑠=12𝑘−𝑝𝑟,2𝑘−𝑝21𝑟−𝑠^𝑟_𝑠𝑇^′1𝑝𝑡𝑑𝑡+21𝑟−𝑠^𝑟_𝑠𝑇^′2𝑘−𝑝𝑡𝑑𝑡, (2.28) and then ln𝑔_4𝑓𝑝^′=121𝑟−𝑠^𝑟_𝑠𝑡𝑇^′′1𝑝𝑡𝑑𝑡−21𝑟−𝑠^𝑟_𝑠𝑡𝑇^′′𝑡=12𝑘−𝑝𝑑𝑡21𝑟−𝑠^𝑟_𝑠𝑡𝑇^′′𝑝𝑡−𝑇^′′2𝑘−𝑝𝑡. (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 in [13], (ln𝑔_4𝑓(𝑝))^′ can be expressed as ln𝑔_4𝑓𝑝^′=12𝑟+𝑠|𝑟|−|𝑠|_|𝑠|^|𝑟|𝑡𝑇^′′𝑝𝑡−𝑇^′′𝑡=12𝑘−𝑝𝑑𝑡2𝑟+𝑠|𝑟|−|𝑠|_|𝑠|^|𝑟|𝑡^|𝑡42|_𝑡41𝑇^′′′(𝑣)𝑑𝑣𝑑𝑡, (2.30) where 𝑡₄₂=𝑝𝑡,𝑡₄₁=2𝑘−𝑝𝑡. (2.31) Hence, ln𝑄_4𝑓𝑝^′=ln𝑔_4𝑓𝑝;𝑎,𝑏^′−ln𝑔_4𝑓𝑝;𝑐,𝑑^′=12𝑟+𝑠|𝑟|−|𝑠|_|𝑠|^|𝑟|𝑡^|𝑡42|_𝑡41𝑇^′′′(𝑣;𝑎,𝑏)−𝑇^′′′()=1𝑣;𝑐,𝑑𝑑𝑣𝑑𝑡2𝑟+𝑠|𝑟|−|𝑠|_|𝑠|^|𝑟|𝑡𝑡₄₂−𝑡₄₁ℎ𝑡₄₁,𝑡₄₂𝑑𝑡, (2.32) where ℎ(𝑥,𝑦) is defined by (2.11). We have shown that ℎ(𝑥,𝑦)>(<)0 for 𝑥,𝑦>0 if 𝒯₃(1,𝑢) strictly increase (decrease) with 𝑢>1 and decrease (increase) with 0<𝑢<1, and we also have sgn𝑡₄₂−𝑡₄₁=sgn𝑡²₄₂−𝑡²₄₁=sgn(𝑘)sgn𝑝−𝑘. (2.33) It follows that sgn𝑄^′_4𝑓𝑝=sgn(𝑟+𝑠)sgn(𝑘)sgn𝑝−𝑘sgnℎ𝑡₄₁,𝑡₄₂=>(<)()<(>)()<(>)()>(<)()0,if𝑘𝑟+𝑠>0,𝑝>𝑘,0,if𝑘𝑟+𝑠>0,𝑝<𝑘,0,if𝑘𝑟+𝑠<0,𝑝>𝑘,0,if𝑘𝑟+𝑠<0,𝑝<𝑘. (2.34)
This proof is accomplished.

3. 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.

Corollary 3.1. Let 𝑎,𝑏,𝑐,𝑑>0 with 𝑏/𝑎>𝑑/𝑐≥1. Then, the following four functions are all strictly decreasing (increasing) on (𝑘,∞) and increasing (decreasing) on (−∞,𝑘):(i)𝑄_1𝐿 is defined by 𝑄_1𝐿𝑝=𝑆_𝑝,𝑞(𝑎,𝑏)𝑆_{2𝑘−p,𝑞}(𝑎,𝑏)𝑆_𝑝,𝑞(𝑐,𝑑)𝑆_{2𝑘−𝑝,𝑞}(𝑐,𝑑), (3.1) for fixed 𝑞≥(≤)0,  𝑘≥(≤)0, but 𝑞,𝑘 are not equal to zero at the same time,(ii)𝑄_2𝐿 is defined by 𝑄_2𝐿𝑝=𝑆_{𝑝,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) 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 𝑄_3𝐿𝑝=𝑆_{𝑝,2𝑚−𝑝}(𝑎,𝑏)𝑆_{2𝑘−𝑝,2𝑚−2𝑘+𝑝}(𝑎,𝑏)𝑆_{𝑝,2𝑚−𝑝}(𝑐,𝑑)𝑆_{2𝑘−𝑝,2𝑚−2𝑘+𝑝}(𝑐,𝑑), (3.3) for fixed 𝑚>(<)0, 𝑘∈[0,2m]([2𝑚,0]).(iv)𝑄_4𝐿 is defined by 𝑄_4𝐿𝑝=𝑆_{𝑝𝑟,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), (3.4) for fixed 𝑘,𝑟,𝑠∈ℝ with 𝑘(𝑟+𝑠)>(<)0.

Remark 3.2. Letting in the first result of Corollary 3.1,𝑞=𝑘 yields Theorem  3.4 in [13] since 𝑆_𝑝,𝑘𝑆_{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)<𝑄_1𝐿(0)=𝐿(𝑎,𝑏)𝐿(𝑐,𝑑). (3.5) Inequalities (3.5) in the case of 𝑑=𝑐 were proved by Alzer in [21]. By letting 𝑞=1,  𝑘=1/2 from 𝑄_1𝐿(1/2)>𝑄_1𝐿(1)>𝑄_1𝐿(2), we have 𝐴(𝑎,𝑏)+𝐺(𝑎,𝑏)𝐴()()>𝑐,𝑑+𝐺𝑐,𝑑𝐿(𝑎,𝑏)𝐼(𝑎,𝑏)𝐿()𝐼()>𝑐,𝑑𝑐,𝑑𝐴(𝑎,𝑏)𝐺(𝑎,𝑏)𝐴()𝐺().𝑐,𝑑𝑐,𝑑 (3.6) Inequalities (3.6) in the case of 𝑑=𝑐 are due to Alzer [22].

Remark 3.3. 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)<𝑄_2𝐿(0)=𝐿(𝑎,𝑏)𝐿(𝑐,𝑑). (3.7) When 𝑑=𝑐, inequalities (3.7) are changed as Alzer’s ones given in [24].

Remark 3.4. 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,𝑑𝐿(𝑎,𝑏)𝐼_1/2(𝑎,𝑏)𝐿(𝑐,𝑑)𝐼_1/2(𝑐,𝑑). (3.8) Putting 𝑚=1/2,𝑘=1/3 leads to another new inequality 𝐴_1/3(𝑎,𝑏)𝐴_1/3()>𝑐,𝑑𝑆_1/6,5/6(𝑎,𝑏)𝐼_1/2(𝑎,𝑏)𝑆_1/6,5/6(𝑐,𝑑)𝐼_1/2(𝑐,𝑑). (3.9)

Remark 3.5. Letting 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), (3.10) are strictly decreasing on (1/2,∞) 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_𝑝𝐿_1−𝑝, 𝐼_𝑝𝐼_1−𝑝, 𝐴_𝑝𝐴_1−𝑝.

4. 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 “𝒯₃(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=(ln𝑓)_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=(ln𝑓)_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 ℝ²₊⧵{(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: ℝ²₊⧵{(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=(ln𝑓)_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 4.2𝑓(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.