Necessary and Sufficient Conditions for Schur Geometrical Convexity of the Four-Parameter Homogeneous Means

Abstract

The necessary and sufficient conditions for Schur geometrical convexity of the four-parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.

1. Introduction and Main Result

Let 𝑝,𝑞∈ℝ and 𝑎,𝑏>0. For 𝑎≠𝑏 the Stolarsky means are defined as𝑆_𝑝,𝑞(𝑞(𝑎,𝑏)=𝑝𝑎^𝑝−𝑏^𝑝𝑎^𝑞−𝑏^𝑞)^{1/(𝑝−𝑞)}𝐿,𝑝𝑞(𝑝−𝑞)≠0,_1/𝑝(𝑎^𝑝,𝑏^𝑝𝐿),𝑝≠0,𝑞=0,_1/𝑞(𝑎^𝑞,𝑏^𝑞𝐼),𝑞≠0,𝑝=0,_1/𝑝(𝑎^𝑝,𝑏^𝑝√),𝑝=𝑞≠0,𝑎𝑏,𝑝=𝑞=0, (1.1) and 𝑆_𝑝,𝑞(𝑎,𝑎)=𝑎 (see [1]), where 𝐿(𝑥,𝑦)=𝑥−𝑦ln𝑥−ln𝑦,𝑥≠𝑦,𝑥𝑥=𝑦, (1.2)(𝑥𝐼(𝑥,𝑦)=𝑦^𝑥𝑦)^{1/(𝑥−𝑦)},𝑥≠𝑦,𝑥,𝑥=𝑦 (1.3) are the logarithmic mean and identric (exponential) mean of positive numbers 𝑥 and 𝑦, respectively.

Another two-parameter family of means was introduced by Gini in [2]. That are defined as𝐺_𝑝,𝑞(𝑎(𝑎,𝑏)=^𝑝+𝑏^𝑝𝑎^𝑞+𝑏^𝑞)^{1/(𝑝−𝑞)}𝑎,𝑝≠𝑞,exp(^𝑝ln𝑎+𝑏^𝑝ln𝑏𝑎^𝑝+𝑏^𝑝),𝑝=𝑞. (1.4)

Stolarsky and Gini means both are contained in the so-called four-parameter means [3], which are defined as follows.

Definition 1.1. Let (𝑎,𝑏)∈ℝ₊×ℝ₊ with 𝑎≠𝑏 and (𝑝,𝑞),(𝑟,𝑠)∈ℝ×ℝ. Then the four-parameter homogeneous means denoted by 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏) are defined as follows: 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏)=(𝐿(𝑎^𝑝𝑟,𝑏^𝑝𝑟)𝐿(𝑎^𝑝𝑠,𝑏^𝑝𝑠)𝐿(𝑎^𝑞𝑠,𝑏^𝑞𝑠)𝐿(𝑎^𝑞𝑟,𝑏^𝑞𝑟))^{1/(𝑝−𝑞)(𝑟−𝑠)}if𝑝𝑞𝑟𝑠(𝑝−𝑞)(𝑟−𝑠)≠0, (1.5) or 𝑎𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏)=(^𝑝𝑟−𝑏^𝑝𝑟𝑎^𝑝𝑠−𝑏^𝑝𝑠𝑎^𝑞𝑠−𝑏^𝑞𝑠𝑎^𝑞𝑟−𝑏^𝑞𝑟)^{1/(𝑝−𝑞)(𝑟−𝑠)}if𝑝𝑞𝑟𝑠(𝑝−𝑞)(𝑟−𝑠)≠0. (1.6)
If 𝑝𝑞𝑟𝑠(𝑝−𝑞)(𝑟−𝑠)=0, then 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏) are defined as their corresponding limits, for example: 𝐅(𝑝,𝑝;𝑟,𝑠;𝑎,𝑏)=lim_𝑞→𝑝𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏)=(𝐼(𝑎^𝑝𝑟,𝑏^𝑝𝑟)𝐼(𝑎^𝑝𝑠,𝑏^𝑝𝑠))^{1/𝑝(𝑟−𝑠)},if𝑝𝑟𝑠(𝑟−𝑠)≠0,𝑝=𝑞,𝐅(𝑝,0;𝑟,𝑠;𝑎,𝑏)=lim_𝑞→0𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏)=(𝐿(𝑎^𝑝𝑟,𝑏^𝑝𝑟)𝐿(𝑎^𝑝𝑠,𝑏^𝑝𝑠))^{1/𝑝(𝑟−𝑠)},if𝑝𝑟𝑠(𝑟−𝑠)≠0,𝑞=0,𝐅(0,0;𝑟,𝑠;𝑎,𝑏)=lim_𝑝→0𝐅(𝑝,0;𝑟,𝑠;𝑎,𝑏)=𝐺(𝑎,𝑏),if𝑟𝑠(𝑟−𝑠)≠0,𝑝=𝑞=0, (1.7) where 𝐿(𝑥,𝑦),𝐼(𝑥,𝑦) denote logarithmic mean and identric (exponential) mean, respectively, 𝐺(𝑎,𝑏)=√𝑎𝑏.

The Schur convexity of 𝑆_𝑝,𝑞(𝑎,𝑏) and 𝐺_𝑝,𝑞(𝑎,𝑏) on (0,∞)×(0,∞) with respect to (𝑎,𝑏) was investigated by Qi et al. [4], Shi et al. [5], Li and Shi [6], and Chu and Zhang [7]. Until now, they have been perfectly solved by Chu and Zhang [7], Wang and Zhang [8], respectively. Recently, Chu and Xia also proved the same result as Wang and Zhang [9].

The Schur convexity of 𝑆_𝑝,𝑞(𝑎,𝑏) and 𝐺_𝑝,𝑞(𝑎,𝑏) on [0,∞)×[0,∞) and (−∞,0]×(−∞,0] with respect to (𝑝,𝑞) was investigated by Qi [10] and Sándor [11], respectively. Now Schur convexity of a four-parameter homogeneous means family containing Stolarsky and Gini means on (−∞,∞)×(−∞,∞) with respect to (𝑝,𝑞) has been perfectly solved by Yang [12].

The Schur geometrical convexity was introduced by Zhang [13]. In [8, 14], Wand and Zhang proved that 𝐺_𝑝,𝑞(𝑎,𝑏) is Schur geometrically convex (Schur geometrically concave) on (0,∞)×(0,∞) with respect to (𝑎,𝑏) if 𝑝+𝑞≥(≤)0. Chu et al. [15] pointed out that this conclusion is also true for 𝑆_𝑝,𝑞(𝑎,𝑏). Shi et al. [5, 16], Li and Shi [6], and Gu and Shi [17] also obtained similar results.

The purpose of this paper is to present the necessary and sufficient conditions for Schur geometrical convexity of the four-parameter homogeneous means. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means with respect to (𝑎,𝑏).

Our main result is as follows.

Theorem 1.2. For fixed (𝑝,𝑞),(𝑟,𝑠)∈ℝ×ℝ the four-parameter homogeneous means 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏) are Schur geometrically convex (Schur geometrically concave) on (0,∞)×(0,∞) with respect to (𝑎,𝑏) if and only if (𝑝+𝑞)(𝑟+𝑠)>(<)0.

2. Definitions and Lemmas

Definition 2.1 (see [18, 19]). Let 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛) and 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈ℝ^𝑛(𝑛≥2).(i)𝑥 is said to by majorized by 𝑦 (in symbol 𝐱≺𝐲) if ^𝑘_𝑖=1𝑥_[𝑖]≤^𝑘_𝑖=1𝑦_[𝑖]for1≤𝑘≤𝑛−1,^𝑛_𝑖=1𝑥_[𝑖]=^𝑛_𝑖=1𝑦_[𝑖], (2.1) where 𝑥_[1]≥𝑥_[2]⋯≥𝑥_[𝑛] and 𝑦_[1]≥𝑦_[2]⋯≥𝑦_[𝑛] are rearrangements of 𝐱 and 𝐲 in a decreasing order.(ii)𝐱≥𝐲 means 𝑥_𝑖≥𝑦_𝑖 for all 𝑖=1,2,…,𝑛. Let Ω⊂ℝ^𝑛(𝑛≥2). The function 𝜙∶Ω→ℝ is said to be increasing if 𝐱≥𝐲 implies 𝜙(𝐱)≥𝜙(𝐲). 𝜙 is said to be decreasing if and only if −𝜙is increasing.(iii)Ω⊂ℝ^𝑛 is called a convex set if (𝛼𝑥₁+𝛽𝑦₁,…,𝛼𝑥_𝑛+𝛽𝑦_𝑛)∈Ω for all 𝐱 and 𝐲, where 𝛼,𝛽∈[0,1] with 𝛼+𝛽=1.(iv)Let Ω⊂ℝ^𝑛(𝑛≥2) be a set with nonempty interior. Then 𝜙∶Ω→ℝ is said to be Schur convex if 𝐱≺𝐲 on Ω implies 𝜙(𝐱)≤𝜙(𝐲). 𝜙 is said to be Schur concave if −𝜙 is Schur convex.

Definition 2.2 (see [13, 20]). Let 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛) and 𝐲=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈ℝ^𝑛₊(𝑛≥2). Denote ln𝐱=(ln𝑥₁,ln𝑥₂,…,ln𝑥_𝑛),ln𝐲=(ln𝑦₁,ln𝑦₂,…,ln𝑦_𝑛). (2.2)(i)Ω⊂ℝ^𝑛₊ is called a geometrically convex set if (𝑥^𝛼₁𝑦^𝛽₁,…,𝑥^𝛼_𝑛𝑦^𝛽_𝑛)∈Ω for all 𝐱 and 𝐲, where 𝛼,𝛽∈[0,1] with 𝛼+𝛽=1.(ii)Let Ω⊂ℝ^𝑛₊(𝑛≥2) be a set with nonempty interior. Then function 𝜙∶Ω→ℝ₊ is said to be Schur geometrically convex on Ω if ln𝐱≺ln𝐲 on Ω implies 𝜙(𝐱)≤𝜙(𝐲). 𝜙 is said to be Schur geometrically concave if −𝜙 is Schur geometrically convex.

Definition 2.3 (see [18]). (i) Ω⊂ℝ^𝑛(𝑛≥2) is called symmetric set if 𝐱∈Ω implies 𝑃𝐱∈Ω for every 𝑛×𝑛 permutation matrix 𝑃.
(ii) The function 𝜙∶Ω→ℝ is called symmetric if for every permutation matrix 𝑃, 𝜙(𝑃𝐱)=𝜙(𝐱) for all 𝐱∈Ω.

Lemma 2.4 (see [18, 19]). Let Ω⊂ℝ^𝑛 be a symmetric set with nonempty interior Ω⁰ and 𝜙∶Ω→ℝ be continuous on Ω and differentiable in Ω⁰. Then 𝜙 is Schur convex (Schur concave) on Ω if and only if 𝜙 is symmetric on Ω and (𝑥₁−𝑥₂)(𝜕𝜙𝜕𝑥₁−𝜕𝜙𝜕𝑥₂)≥(≤)0 (2.3) holds for any 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈Ω⁰.

Lemma 2.5 (see [13, Theorem 1.4, page 108]). Let Ω⊂ℝ^𝑛₊ be a symmetric set with a nonempty interior geometrically convex set Ω⁰. Let 𝜙∶Ω→ℝ₊ be continuous on Ω and differentiable in Ω⁰. Then 𝜙 is Schur geometrically convex (Schur geometrically concave) on Ω if and only if 𝜙 is symmetric on Ω and (ln𝑥₁−ln𝑥₂)(𝑥₁𝜕𝜙𝜕𝑥₁−𝑥₂𝜕𝜙𝜕𝑥₂)≥(≤)0 (2.4) holds for any 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈Ω⁰.

3. Schur Geometrical Convexity of Two-Parameter Homogeneous Functions

The more general form of two-parameter homogeneous means is the so-called two-parameter homogenous functions first introduced by Yang [21]. For conveniences, we record it as follows.

Definition 3.1. Assume that 𝑓: ℝ₊×ℝ₊→ℝ₊∪{0} is 𝑛-order homogeneous, continuous and exists first partial derivatives and (𝑎,𝑏)∈ℝ₊×ℝ₊, (𝑝,𝑞)∈ℝ×ℝ.
If 𝑓(𝑥,𝑦)>0for (𝑥,𝑦)∈ℝ₊×ℝ₊⧵{(𝑥,𝑥)∶𝑥∈ℝ₊} and 𝑓(𝑥,𝑥)=0 for all 𝑥∈ℝ₊, then define ℋ_𝑓(𝑝,𝑞;𝑎,𝑏)=(𝑓(𝑎^𝑝,𝑏^𝑝)𝑓(𝑎^𝑞,𝑏^𝑞))^{1(𝑝−𝑞)}ℋif𝑝≠𝑞,𝑝𝑞≠0,_𝑓(𝑝,𝑝;𝑎,𝑏)=lim_𝑞→𝑝ℋ_𝑓(𝑝,𝑞;𝑎,𝑏)=𝐺_𝑓,𝑝(𝑎,𝑏)if𝑝=𝑞≠0, (3.1) 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)), (3.2)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 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/𝑝ℋif𝑝≠0,𝑞=0;_𝑓(0,𝑞;𝑎,𝑏)=(𝑓(𝑎^𝑞,𝑏^𝑞))𝑓(1,1)^1/𝑞ℋif𝑝=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. (3.3) Since 𝑓(𝑥,𝑦) is a homogeneous function, ℋ_𝑓(𝑝,𝑞;𝑎,𝑏) is also one and called a homogeneous function with parameters 𝑝 and 𝑞 and simply denoted by ℋ_𝑓(𝑝,𝑞) or ℋ_𝑓 sometimes.

Concerning the monotonicity and log-convexity of two-parameter homogeneous functions, there have been some literatures such as [3, 21, 22], which yield some new and interesting inequalities for means.

The two-parameter homogeneous functions ℋ_𝑓(𝑝,𝑞;𝑎,𝑏) have some well properties (see [21–23]) such as the following lemma.

Lemma 3.2 (see [23]). Let 𝑓∶ℝ₊×ℝ₊→ℝ₊ be a homogenous and differentiable function and 𝑇(𝑡)=𝑇(𝑡;𝑎,𝑏)∶=ln𝑓(𝑎^𝑡,𝑏^𝑡),(𝑡;𝑎,𝑏)∈ℝ×ℝ₊×ℝ₊. (3.4) Then we have 𝜕𝑇(𝑡;𝑎,𝑏)=𝑎𝜕𝑡^𝑡𝑓_𝑥(𝑎^𝑡,𝑏^𝑡)ln𝑎+𝑏^𝑡f_𝑦(𝑎^𝑡,𝑏^𝑡)ln𝑏𝑓(𝑎^𝑡,𝑏^𝑡),lnℋ_𝑓(𝑝,𝑞;𝑎,𝑏)=¹₀𝜕𝑇(𝑡𝑝+(1−𝑡)𝑞;𝑎,𝑏)𝜕𝑡𝑑𝑡. (3.5)(3.6)

Next we give another property.

Lemma 3.3. Let 𝑓∶ℝ₊×ℝ₊→ℝ₊ be a homogenous and 𝑚-time differentiable function. Then ℋ_𝑓(𝑝,𝑞;𝑎,𝑏)∈𝐶^𝑚−1(ℝ×ℝ×ℝ₊×ℝ₊).

Proof. Since 𝑓(𝑥,𝑦) has continuous partial derivatives of 𝑚 order with respect to 𝑥,𝑦 on ℝ₊×ℝ₊, the integrand in (3.6) has continuous partial derivatives of 𝑚−1 order with respect to 𝑝,𝑞,𝑎,𝑏 on ℝ×ℝ×ℝ₊×ℝ₊, that is ℋ_𝑓(𝑝,𝑞;𝑎,𝑏)∈𝐶^𝑚−1(ℝ×ℝ×ℝ₊×ℝ₊).

For the Schur geometrical convexity, we have the following result.

Theorem 3.4. Assume that 𝑓: ℝ₊×ℝ₊→ℝ₊ is a symmetric, 𝑛-order homogeneous, continuous, and three-time differentiable function. If for any (𝑥,𝑦)∈ℝ₊×ℝ₊ with 𝑥≠𝑦𝒩(𝑥,𝑦)=(𝑥−𝑦)(𝑥(ln𝑓)_𝑥−𝑦(ln𝑓)_𝑦𝑥−2𝑥𝑦ℐln(𝑦))>(<)0,whereℐ=(ln𝑓)_𝑥𝑦, (3.7) then ℋ_𝑓(𝑝,𝑞;𝑎,𝑏) is Schur geometrically convex on (0,∞)×(0,∞) with respect to (𝑎,𝑏) if and only if 𝑝+𝑞>(<)0 and Schur geometrically concave if and only if 𝑝+𝑞<(>)0.

Proof. (1) In the case of 𝑝≠𝑞. We have lnℋ_𝑓(𝑝,𝑞;𝑎,𝑏)=ln𝑓(𝑎^𝑝,𝑏^𝑝)−ln𝑓(𝑎^𝑞,𝑏^𝑞).𝑝−𝑞 (3.8) Some simple partial derivative computations yield 𝜕lnℋ_𝑓=1𝜕𝑎ℋ_𝑓𝜕ℋ_𝑓=1𝜕𝑎(𝑝−𝑞𝑝𝑎^𝑝−1𝑓_𝑥(𝑎^𝑝,𝑏^𝑝)𝑓(𝑎^𝑝,𝑏^𝑝)−𝑞𝑎^𝑞−1𝑓_𝑥(𝑎^𝑞,𝑏^𝑞)𝑓(𝑎^𝑞,𝑏^𝑞)),𝜕lnℋ_𝑓=1𝜕𝑏ℋ_𝑓𝜕ℋ_𝑓=1𝜕𝑏(𝑝−𝑞𝑝𝑏^𝑝−1𝑓_𝑦(𝑎^𝑝,𝑏^𝑝)𝑓(𝑎^𝑝,𝑏^𝑝)−𝑞𝑏^𝑞−1𝑓_𝑦(𝑎^𝑞,𝑏^𝑞)𝑓(𝑎^𝑞,𝑏^𝑞)), (3.9) hence, 1ℋ_𝑓(𝑎𝜕ℋ_𝑓𝜕𝑎−𝑏𝜕ℋ_𝑓𝜕𝑏)=𝑔(𝑝)−𝑔(𝑞),𝑝−𝑞 (3.10) where 𝑔(𝑡)=𝑡𝑎^𝑡𝑓_𝑥(𝑎^𝑡,𝑏^𝑡)𝑓(𝑎^𝑡,𝑏^𝑡)−𝑡𝑏^𝑡𝑓_𝑦(𝑎^𝑡,𝑏^𝑡)𝑓(𝑎^𝑡,𝑏^𝑡). (3.11) It is easy to verify that 𝑔(𝑡) is even on (−∞,∞). In fact, since 𝑓(𝑥,𝑦) is 𝑛-order homogeneous and symmetric, for arbitrary 𝜆>0, we have 𝑓(𝜆𝑥,𝜆𝑦)=𝜆^𝑛𝑓(𝑥,𝑦),𝑓_𝑥(𝜆𝑥,𝜆𝑦)=𝜆^𝑛−1𝑓_𝑥(𝑥,𝑦),𝑓_𝑦(𝜆𝑥,𝜆𝑦)=𝜆^𝑛−1𝑓_𝑦(𝑥,𝑦),𝑓(𝑥,𝑦)=𝑓(𝑦,𝑥),𝑓_𝑥(𝑥,𝑦)=𝑓_𝑦(𝑦,𝑥),𝑓_𝑦(𝑥,𝑦)=𝑓_𝑥(𝑦,𝑥). (3.12) Thus, 𝑔(−𝑡)=−𝑡𝑎^−𝑡𝑓_𝑥(𝑎^−𝑡,𝑏^−𝑡)𝑓(𝑎^−𝑡,𝑏^−𝑡)−−𝑡𝑏^−𝑡𝑓_𝑦(𝑎^−𝑡,𝑏^−𝑡)𝑓(𝑎^−𝑡,𝑏^−𝑡)=−𝑡𝑎^−𝑡(𝑎^𝑡𝑏^𝑡)^{−(𝑛−1)}𝑓_𝑥(𝑏^𝑡,𝑎^𝑡)(𝑎^𝑡𝑏^𝑡)^−𝑛𝑓(𝑏^𝑡,𝑎^𝑡)−−𝑡𝑏^𝑡(𝑎^𝑡𝑏^𝑡)^{−(𝑛−1)}𝑓_𝑦(𝑏^𝑡,𝑎^𝑡)(𝑎^𝑡𝑏^𝑡)^−𝑛𝑓(𝑏^𝑡,𝑎^𝑡)=−𝑡𝑏^𝑡𝑓_𝑦(𝑎^𝑡,𝑏^𝑡)𝑓(𝑎^𝑡,𝑏^𝑡)+𝑡𝑎^𝑡𝑓_𝑥(𝑎^𝑡,𝑏^𝑡)𝑓(𝑎^𝑡,𝑏^𝑡)=𝑔(𝑡). (3.13) Let 𝑎^𝑡=𝑥, 𝑏^𝑡=𝑦. Then 𝑔^′(𝑡)=𝑥(ln𝑓)_𝑥+𝑡((𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥𝑑𝑥𝑑𝑡+(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦𝑑𝑦)𝑑𝑡−𝑦(ln𝑓)_𝑦−𝑡((𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥𝑑𝑥𝑑𝑡+(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦𝑑𝑦)𝑑𝑡=𝑥(ln𝑓)_𝑥+𝑡(𝑥(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥ln𝑎+𝑦(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦ln𝑏)−𝑦(ln𝑓)_𝑦−𝑡(𝑥(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥ln𝑎+𝑦(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦ln𝑏). (3.14) Note 𝑥𝑓_𝑥(𝑥,𝑦)/𝑓(𝑥,𝑦) and 𝑦𝑓_𝑦(𝑥,𝑦)/𝑓(𝑥,𝑦) both are 0-order homogeneous with respect to 𝑥 and 𝑦, then 𝑥(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥+𝑦(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦=0,𝑥(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥+𝑦(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦=0, (3.15) and then 𝑥(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥=−𝑦(𝑥𝑓_𝑥(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦=−𝑥𝑦ℐ,𝑦(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑦=−𝑥(𝑦𝑓_𝑦(𝑥,𝑦))𝑓(𝑥,𝑦)_𝑥=−𝑥𝑦ℐ. (3.16) Therefore, 𝑔^′(𝑡)=𝑥(ln𝑓)_𝑥+𝑡𝑥𝑦ℐ(ln𝑏−ln𝑎)−𝑦(ln𝑓)_𝑦−𝑡𝑥𝑦ℐ(ln𝑎−ln𝑏)=𝑥(ln𝑓)_𝑥−𝑦(ln𝑓)_𝑦−2𝑡𝑥𝑦ℐ(ln𝑎−ln𝑏)=𝑥(ln𝑓)_𝑥−𝑦(ln𝑓)_𝑦𝑥−2𝑥𝑦ℐln(𝑦)=𝒩(𝑥,𝑦)𝑥−𝑦for𝑥≠𝑦. (3.17) By the mean values theorem, there is a 𝜉 between |𝑝| and |𝑞| such that 𝑔(𝑝)−𝑔(𝑞)=𝑝−𝑞𝑔(|𝑝|)−𝑔(|𝑞|)=𝑝−𝑞|𝑝|−|𝑞|𝑔𝑝−𝑞^′(𝜉)=𝑝+𝑞𝑔|𝑝|+|𝑞|^′(𝜉)=𝑝+𝑞|𝑝|+|𝑞|𝒩(𝑥,𝑦)𝑥−𝑦,for𝑥≠𝑦, (3.18) where 𝑥=𝑎^𝜉, 𝑦=𝑏^𝜉. Thus we have (ln𝑎−ln𝑏)(𝑎𝜕ℋ_𝑓𝜕𝑎−𝑏𝜕ℋ_𝑓𝜕𝑏)=ℋ_𝑓𝑝+𝑞𝑎|𝑝|+|𝑞|ln(𝑏)𝒩(𝑥,𝑦)𝑥−𝑦=ℋ_𝑓𝑝+𝑞|𝑝|+|𝑞|𝒩(𝑥,𝑦)𝜉ln𝑥−ln𝑦=𝑥−𝑦>0if𝑝+𝑞>(<)0,<0if𝑝+𝑞<(>)0. (3.19) By Lemma 2.5, our required result is derived immediately.
(2) In the case of 𝑝=𝑞≠0. By Lemma 3.3 together with (3.10) and (3.17), we have 1ℋ_𝑓(𝑝,𝑝)(𝑎𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑎−𝑏𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑏)=lim_𝑞→𝑝1ℋ_𝑓(𝑝,𝑞)(𝑎𝜕ℋ_𝑓(𝑝,𝑞)𝜕𝑎−𝑏𝜕ℋ_𝑓(𝑝,𝑞))𝜕𝑏=lim_𝑞→𝑝𝑔(𝑝)−𝑔(𝑞)𝑝−𝑞=𝑔^′(𝑝)=𝒩(𝑥,𝑦),𝑥−𝑦 (3.20) where 𝑥=𝑎^𝑝, 𝑦=𝑏^𝑝. Hence we have (ln𝑎−ln𝑏)(𝑎𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑎−𝑏𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑏)=ℋ_𝑓(𝑝,𝑝)(ln𝑎−ln𝑏)𝒩(𝑥,𝑦)𝑥−𝑦=𝑝⁻¹ℋ_𝑓(𝑝,𝑝)𝒩(𝑥,𝑦)ln𝑥−ln𝑦=𝑥−𝑦>0if𝑝>(<)0,<0if𝑝<(>)0. (3.21) By Lemma 2.5, the required result holds.
(3) In the case of 𝑝=𝑞=0. By Lemma 3.3 and (3.20), we have 1ℋ_𝑓(0,0)(𝑎𝜕ℋ_𝑓(0,0)𝜕𝑎−𝑏𝜕ℋ_𝑓(0,0)𝜕𝑏)=lim_𝑝→0(𝑎𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑎−𝑏𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑏)=lim_𝑝→0𝑔^′(𝑝). (3.22) However, 𝑔^′(0)=(𝑥(ln𝑓)_𝑥−𝑦(ln𝑓)_𝑦𝑥−2𝑥𝑦ℐln(𝑦))|_{𝑥=1,𝑦=1}𝑓=1⋅^𝑥(1,1)𝑓𝑓(1,1)−1⋅^𝑦(1,1)1𝑓(1,1)−2⋅1⋅1⋅ℐ(1,1)⋅ln(1)=0, (3.23) where 𝑓_𝑥(1,1)=𝑓_𝑦(1,1) due to the symmetry of 𝑓(𝑥,𝑦). Thus (ln𝑎−ln𝑏)(𝑎𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑎−𝑏𝜕ℋ_𝑓(𝑝,𝑝)𝜕𝑏)=0. (3.24)
Summarizing the above three cases, this proof of Theorem 3.4 is complete.

4. Proof of Main Result

Establishing the Theorem 3.4, we are in a position to prove main result.

Proof of Theorem 1.2. It follows from [3, Section 1], that 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏)=ℋ_ℋ𝐿(𝑝,𝑞;𝑎,𝑏), where ℋ_𝐿=ℋ_𝐿(𝑟,𝑠)=ℋ_𝐿(𝑟,𝑠;𝑥,𝑦)=𝑆_𝑟,𝑠(𝑥,𝑦) is symmetric with respect to 𝑥 and 𝑦. From Lemma 3.3, it follows that ℋ_𝐿=ℋ_𝐿(𝑟,𝑠;𝑥,𝑦)∈𝐶^∞(ℝ×ℝ×ℝ₊×ℝ₊). Thus we have (lnℋ_𝐿(𝑟,𝑟))_𝑥=lim_𝑠→𝑟(lnℋ_𝐿(𝑟,𝑠))_𝑥,(lnℋ_𝐿(𝑟,𝑟))_𝑦=lim_𝑠→𝑟(lnℋ_𝐿(𝑟,𝑠))_𝑦,(lnℋ_𝐿(𝑟,𝑟))_𝑥𝑦=lim_𝑠→𝑟(lnℋ_𝐿(𝑟,𝑠))_𝑥𝑦,(lnℋ_𝐿(𝑟,0))_𝑥=lim_𝑠→0(lnℋ_𝐿(𝑟,𝑠))_𝑥,(lnℋ_𝐿(𝑟,0))_𝑦=lim_𝑠→0(lnℋ_𝐿(𝑟,𝑠))_𝑦,(lnℋ_𝐿(𝑟,0))_𝑥𝑦=lim_𝑠→𝑟(lnℋ_𝐿(𝑟,𝑠))_𝑥𝑦,(lnℋ_𝐿(0,0))_𝑥=lim_𝑟→0(lnℋ_𝐿(𝑟,𝑟))_𝑥,(lnℋ_𝐿(0,0))_𝑦=lim_𝑟→0(lnℋ_𝐿(𝑟,𝑟))_𝑦,(lnℋ_𝐿(0,0))_𝑥𝑦=lim_𝑟→0(lnℋ_𝐿(𝑟,𝑟))_𝑥𝑦. (4.1)(4.2)(4.3)(4.4)(4.5)(4.6)(4.7)(4.8)(4.9)(1) In the case of 𝑟𝑠(𝑟−𝑠)≠0.
Simple partial derivative calculations yield lnℋ_𝐿=1𝑟−𝑠(ln|𝑠|+ln|𝑥^𝑟−𝑦^𝑟|−ln|𝑟|−ln|𝑥^𝑠−𝑦^𝑠|),(lnℋ_𝐿)_𝑥=1(𝑟−𝑠𝑟𝑥^𝑟−1𝑥^𝑟−𝑦^𝑟−𝑠𝑥^𝑠−1𝑥^𝑠−𝑦^𝑠),(lnℋ_𝐿)_𝑦=1(𝑟−𝑠−𝑟𝑦^𝑟−1𝑥^𝑟−𝑦^𝑟+𝑠𝑦^𝑠−1𝑥^𝑠−𝑦^𝑠),ℐ=(lnℋ_𝐿)_𝑥𝑦=1(𝑟𝑥𝑦(𝑟−𝑠)²𝑥^𝑟𝑦^𝑟(𝑥^𝑟−𝑦^𝑟)²−𝑠²𝑥^𝑠𝑦^𝑠(𝑥^𝑠−𝑦^𝑠)²). (4.10) Hence, 𝒩(𝑥,𝑦)=(𝑥−𝑦)(𝑥(lnℋ_𝐿)_𝑥−𝑦(lnℋ_𝐿)_𝑦𝑥−2𝑥𝑦ℐln(𝑦=))𝑥−𝑦(𝑟−𝑠𝑟(𝑥^𝑟+𝑦^𝑟)𝑥^𝑟−𝑦^𝑟−2𝑟²𝑥^𝑟𝑦^𝑟ln(𝑥/𝑦)(𝑥^𝑟−𝑦^𝑟)²)−𝑥−𝑦(𝑟−𝑠𝑠(𝑥^𝑠+𝑦^𝑠)𝑥^𝑠−𝑦^𝑠−2𝑠²𝑥^𝑠𝑦^𝑠ln(𝑥/𝑦)(𝑥^𝑠−𝑦^𝑠)²)=(𝑥−𝑦)𝑃(𝑟)−𝑃(𝑠),𝑟−𝑠 (4.11) where 𝑥𝑃(𝑡)=𝑡(^𝑡+𝑦^𝑡𝑥^𝑡−𝑦^𝑡−2𝑥^𝑡𝑦^𝑡ln(𝑥^𝑡/𝑦^𝑡)(𝑥^𝑡−𝑦^𝑡)²). (4.12) It is easy to check that 𝑃(𝑡) is even and increasing (decreasing) on (0,∞) if 𝑥>(<)𝑦. Indeed, 𝑥𝑃(−𝑡)=−𝑡(^−𝑡+𝑦^−𝑡𝑥^−𝑡−𝑦^−𝑡−2𝑥^−𝑡𝑦^−𝑡ln(𝑥^−𝑡/𝑦^−𝑡)(𝑥^−𝑡−𝑦^−𝑡)²)=𝑃(𝑡). (4.13) With (𝑥/𝑦)^𝑡=𝑢, then 𝑡=ln𝑢/ln(𝑥/𝑦), and then 𝑃(𝑡) can be written as 1𝑃(𝑡)=(ln(𝑥/𝑦)𝑢+1𝑢−1ln𝑢−2𝑢ln²𝑢(𝑢−1)²). (4.14) Direct computation yields 𝑃^′1(𝑡)=(ln(𝑥/𝑦)𝑢+1𝑢−1ln𝑢−2𝑢ln²𝑢(𝑢−1)²)^′𝑑𝑢𝑑𝑡=𝑢((𝑢+1)(𝑢−1)/𝑢−ln𝑢(𝑢−1)²+ln𝑢−𝑢−12ln²𝑢(𝑢−1)²−4𝑢ln𝑢𝑢−1(𝑢−1)/𝑢−ln𝑢(𝑢−1)²)(𝑢−1)/ln𝑢=𝐿(𝑢+1)𝐿²−6𝑢𝐿+2𝑢(𝑢+1)(𝑢−1)𝐿²=2𝐿(((𝑢+1)/2)𝐿−𝑢)+4𝑢((𝑢+1)/2−𝐿)(𝑢−1)𝐿². (4.15) From 𝑢+12𝑢𝐿−𝑢=²−1ln𝑢²−√𝑢²>0,𝐿−𝑢+12<0, (4.16) it follows that 𝑃^′(𝑡)>0 if 𝑢−1>0, that is, 𝑥>𝑦 and 𝑃^′(𝑡)<0 if 𝑥<𝑦. Namely, (𝑥−𝑦)𝑃^′(𝑡)>0for𝑡>0with𝑥≠𝑦. (4.17)
By the mean values theorem, there is a 𝜂 between |𝑟| and |𝑠| such that 𝑃(|𝑟|)−𝑃(|𝑠|)=(|𝑟|−|𝑠|)𝑃^′(𝜂), (4.18) and then 𝒩(𝑥,𝑦)=(𝑥−𝑦)𝑃(𝑟)−𝑃(𝑠)𝑟−𝑠=(𝑥−𝑦)𝑟+𝑠|𝑟|+|𝑠|𝑃(|𝑟|)−𝑃(|𝑠|)=|𝑟|−|𝑠|𝑟+𝑠|𝑟|+|𝑠|⋅(𝑥−𝑦)𝑃^′=(𝜂)>0if𝑟+𝑠>0,<0if𝑟+𝑠<0. (4.19) Using Theorem 3.4, for fixed (𝑝,𝑞),(𝑟,𝑠)∈ℝ×ℝ with 𝑟𝑠(𝑟−𝑠)≠0, the four-parameter homogeneous means 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏) are Schur geometrically convex on (0,∞)×(0,∞) with respect to (𝑎,𝑏) if and only if (𝑝+𝑞)(𝑟+𝑠)>0 and Schur geometrically concave if and only if (𝑝+𝑞)(𝑟+𝑠)<0.(2) In the case of 𝑠=0, 𝑟≠0.
From (4.11) together with (4.4)–(4.6) and (4.19), there is a 𝜂₁ between 0 and |𝑟| such that 𝒩(𝑥,𝑦)=(𝑥−𝑦)(𝑥(lnℋ_𝐿(𝑟,0))_𝑥−𝑦(lnℋ_𝐿(𝑟,0))_𝑦−2𝑥𝑦(lnℋ_𝐿(𝑟,0))_𝑥𝑦𝑥ln(𝑦))=lim_𝑠→0((𝑥−𝑦)(𝑥(lnℋ_𝐿(𝑟,𝑠))_𝑥−𝑦(lnℋ_𝐿(𝑟,𝑠))_𝑦−2𝑥𝑦(lnℋ_𝐿(𝑟,𝑠))_𝑥𝑦𝑥ln(𝑦)))=lim_𝑠→0(𝑥−𝑦)𝑃(𝑟)−𝑃(𝑠)𝑟−𝑠=lim_𝑠→0𝑟+𝑠|𝑟|+|𝑠|⋅lim_𝑠→0(𝑥−𝑦)𝑃^′(𝜂₁)=>0if𝑟>0,<0if𝑟<0,(by(4.17)). (4.20)(3) In the case of 𝑟=0, 𝑠≠0.
Since ℋ_𝐿(𝑟,𝑠;𝑥,𝑦) is symmetric with respect to 𝑟 and 𝑠, it follows from case 2 that 𝒩(𝑥,𝑦)=(𝑥−𝑦)(𝑥(lnℋ_𝐿(0,𝑠))_𝑥−𝑦(lnℋ_𝐿(0,𝑠))_𝑦−2𝑥𝑦(lnℋ_𝐿(𝑟,𝑠))_𝑥𝑦𝑥ln(𝑦=))>0if𝑠>0,<0if𝑠<0. (4.21)(4) In the case of 𝑟=𝑠≠0.
From (4.11) together with (4.1)–(4.3), we have 𝒩(𝑥,𝑦)=(𝑥−𝑦)(𝑥(lnℋ_𝐿(𝑟,𝑟))_𝑥−𝑦(lnℋ_𝐿(𝑟,𝑟))_𝑦−2𝑥𝑦(lnℋ_𝐿(𝑟,𝑟))_𝑥𝑦𝑥ln(𝑦))=lim_𝑠→𝑟((𝑥−𝑦)(𝑥(lnℋ_𝐿(𝑟,𝑠))_𝑥−𝑦(lnℋ_𝐿(𝑟,𝑠))_𝑦−2𝑥𝑦(lnℋ_𝐿(𝑟,𝑠))_𝑥𝑦𝑥ln(𝑦)))=(𝑥−𝑦)lim_𝑠→𝑟𝑃(𝑟)−𝑃(𝑠)𝑟−𝑠=(𝑥−𝑦)𝑃^′=(𝑟)>0if𝑟>0,<0if𝑟<0.(by(4.17)) (4.22)(5) In the case of 𝑟=𝑠=0.
From (4.22) together with (4.7)–(4.9), we have 𝒩(𝑥,𝑦)=(𝑥−𝑦)(𝑥(lnℋ_𝐿(0,0))_𝑥−𝑦(lnℋ_𝐿(0,0))_𝑦−2𝑥𝑦(lnℋ_𝐿(0,0))_𝑥𝑦𝑥ln(𝑦))=lim_𝑟→0((𝑥−𝑦)(𝑥(lnℋ_𝐿(𝑟,𝑟))_𝑥−𝑦(lnℋ_𝐿(𝑟,𝑟))_𝑦−2𝑥𝑦(lnℋ_𝐿(𝑟,𝑟))_𝑥𝑦𝑥ln(𝑦)))=(𝑥−𝑦)lim_𝑟→0𝑃^′(𝑟). (4.23) But by (4.15) and some limit computations, we obtain lim_𝑡→0𝑃^′(𝑡)(𝑥/𝑦)^𝑡=𝑢lim_𝑢→1𝑢((𝑢+1)(𝑢−1)/𝑢−ln𝑢(𝑢−1)²+ln𝑢−𝑢−12ln²𝑢(𝑢−1)²−4𝑢ln𝑢𝑢−1(𝑢−1)/𝑢−ln𝑢(𝑢−1)²)=0, (4.24) which implies 𝒩(𝑥,𝑦)=0.
Summarizing the above five cases, our required results are derived.
This proof ends.

5. Other Corollaries

The four-parameter homogeneous means 𝐅(𝑝,𝑞;𝑟,𝑠;𝑎,𝑏) also contain many other two-parameter means, for instance, for the identric (exponential) mean defined by (1.3), its two-parameter means are defined as follows [21, Example 2.3]: ℋ_𝐼((𝑝,𝑞;𝑎,𝑏)=𝐼(𝑎^𝑝,𝑏^𝑝)𝐼(𝑎^𝑞,𝑏^𝑞))^{1/(𝑝−𝑞)}𝐺,𝑝≠𝑞,𝑝𝑞≠0,_𝐼,𝑝𝐼(𝑎,𝑏),𝑝=𝑞≠0,_1/𝑝(𝑎^𝑝,𝑏^𝑝𝐼),𝑝≠0,𝑞=0,_1/𝑞(𝑎^𝑞,𝑏^𝑞),𝑝=0,𝑞≠0,𝐺(𝑎,𝑏),𝑝=𝑞=0, (5.1) where 𝐺_𝐼,𝑝(𝑎,𝑏)=𝑌^1/𝑝(𝑎^𝑝,𝑏^𝑝)∶=𝑌_𝑝(𝑎,𝑏),𝑌(𝑎,𝑏)=𝐼𝑒^{1−𝐺2/𝐿2}.

By [3], we see that ℋ_𝐼(𝑝,𝑞;𝑎,𝑏)=𝐅(𝑝,𝑞;1,1;𝑎,𝑏). (5.2) And then according to Theorem 1.2, we have the following corollary.

Corollary 5.1. For fixed (𝑝,𝑞)∈ℝ×ℝ, the two-parameter identric (exponential) means ℋ_𝐼(𝑝,𝑞;𝑎,𝑏) are Schur geometrically convex on (0,∞)×(0,∞) with respect to (𝑎,𝑏) if and only if 𝑝+q>0 and Schur geometrically concave if and only if 𝑝+𝑞<0.

As another example, for Heronian mean defined by √He=𝑎+𝑎𝑏+𝑏3, (5.3) its two-parameter means are defined as follows:ℋ_He(𝑎(𝑝,𝑞;𝑎,𝑏)=^𝑝+(√𝑎𝑏)^𝑝+𝑏^𝑝𝑎^𝑞+(√𝑎𝑏)^𝑞+𝑏^𝑞)^{1/(𝑝−𝑞)}𝑎,𝑝≠𝑞,𝑝𝑞≠0,_{(𝑎𝑝√+(𝑎𝑏)𝑝/2)/(𝑎𝑝√+(𝑎𝑏)𝑝+𝑏𝑝)}𝑏^{√((𝑎𝑏)𝑝/2+𝑏𝑝)/(𝑎𝑝√+(𝑎𝑏)𝑝+𝑏𝑝)},𝑝=𝑞≠0,He^1/𝑝(𝑎^𝑝,𝑏^𝑝),𝑝≠0,𝑞=0,He^1/𝑞(𝑎^𝑞,𝑏^𝑞),𝑝=0,𝑞≠0,𝐺(𝑎,𝑏),𝑝=𝑞=0. (5.4)

By [3], we see that ℋ_𝐻𝑒(𝑝,𝑞;a,𝑏)=𝐅(𝑝,𝑞;3/2,1/2;𝑎,𝑏). (5.5) And then according to Theorem 1.2, we have the following corollary.

Corollary 5.2. For fixed (𝑝,𝑞)∈ℝ×ℝ, the two-parameter Heronian means ℋ_He(𝑝,𝑞;𝑎,𝑏) are Schur geometrically convex on (0,∞)×(0,∞) with respect to (𝑎,𝑏) if and only if 𝑝+𝑞>0 and Schur geometrically concave if and only if 𝑝+𝑞<0.