Refinements of Inequalities among Difference of Means

Abstract

In this paper, for the difference of famous means discussed by Taneja in 2005, we study the Schur-geometric convexity in (0, ∞) × (0, ∞) of the difference between them. Moreover some inequalities related to the difference of those means are obtained.

1. Introduction

In 2005, Taneja [1] proved the following chain of inequalities for the binary means for (𝑎,𝑏)∈𝑅²₊=(0,∞)×(0,∞): 𝐻(𝑎,𝑏)≤𝐺(𝑎,𝑏)≤𝑁₁(𝑎,𝑏)≤𝑁₃(𝑎,𝑏)≤𝑁₂(𝑎,𝑏)≤𝐴(𝑎,𝑏)≤𝑆(𝑎,𝑏), (1.1) where 𝐴(𝑎,𝑏)=𝑎+𝑏2,𝐺(𝑎,𝑏)=√𝐻()=𝑎𝑏,𝑎,𝑏2𝑎𝑏,𝑁𝑎+𝑏₁()=√𝑎,𝑏𝑎+√𝑏2²=𝐴(𝑎,𝑏)+𝐺(𝑎,𝑏)2,𝑁₃()=√𝑎,𝑏𝑎+𝑎𝑏+𝑏3=2𝐴(𝑎,𝑏)+𝐺(𝑎,𝑏)3,𝑁₂()=√𝑎,𝑏𝑎+√𝑏2𝑎+𝑏2,𝑆(𝑎,𝑏)=𝑎²+𝑏²2. (1.3) The means 𝐴, 𝐺, 𝐻, 𝑆,  𝑁₁ and 𝑁₃ are called, respectively, the arithmetic mean, the geometric mean, the harmonic mean, the root-square mean, the square-root mean, and Heron’s mean. The 𝑁₂ one can be found in Taneja [2, 3].

Furthermore Taneja considered the following difference of means: 𝑀_𝑆𝐴()()(),𝑀𝑎,𝑏=𝑆𝑎,𝑏−𝐴𝑎,𝑏_𝑆𝑁2(𝑎,𝑏)=𝑆(𝑎,𝑏)−𝑁₂(),𝑀𝑎,𝑏_𝑆𝑁3(𝑎,𝑏)=𝑆(𝑎,𝑏)−𝑁₃(),𝑀𝑎,𝑏_𝑆𝑁1(𝑎,𝑏)=𝑆(𝑎,𝑏)−𝑁₁(),𝑀𝑎,𝑏_𝑆𝐺()()(),𝑀𝑎,𝑏=𝑆𝑎,𝑏−𝐺𝑎,𝑏_𝑆𝐻()()(),𝑀𝑎,𝑏=𝑆𝑎,𝑏−𝐻𝑎,𝑏_𝐴𝑁2(𝑎,𝑏)=𝐴(𝑎,𝑏)−𝑁₂(),𝑀𝑎,𝑏_𝐴𝐺()()(),𝑀𝑎,𝑏=𝐴𝑎,𝑏−𝐺𝑎,𝑏_𝐴𝐻()()(),𝑀𝑎,𝑏=𝐴𝑎,𝑏−𝐻𝑎,𝑏_𝑁2𝑁1(𝑎,𝑏)=𝑁₂(𝑎,𝑏)−𝑁₁(),𝑀𝑎,𝑏_𝑁2𝐺(𝑎,𝑏)=𝑁₂(𝑎,𝑏)−𝐺(𝑎,𝑏) (1.4) and established the following.

Theorem A. The difference of means given by (1.4) is nonnegative and convex in 𝑅²₊=(0,∞)×(0,∞).

Further, using Theorem A, Taneja proved several chains of inequalities; they are refinements of inequalities in (1.1).

Theorem B. The following inequalities among the mean differences hold: 𝑀_𝑆𝐴()≤1𝑎,𝑏3𝑀_𝑆𝐻()≤1𝑎,𝑏2𝑀_𝐴𝐻()≤1𝑎,𝑏2𝑀_SG(𝑎,𝑏)≤𝑀_𝐴𝐺(),1𝑎,𝑏8𝑀_𝐴𝐻(𝑎,𝑏)≤𝑀_𝑁2𝑁1()≤1𝑎,𝑏3𝑀_𝑁2𝐺()≤1𝑎,𝑏4𝑀_𝐴𝐺(𝑎,𝑏)≤𝑀_𝐴𝑁2(),𝑀𝑎,𝑏_𝑆𝐴()≤4𝑎,𝑏5𝑀_𝑆𝑁2(𝑎,𝑏)≤4𝑀_𝐴𝑁2(),𝑀𝑎,𝑏_𝑆𝐻(𝑎,𝑏)≤2𝑀_𝑆𝑁1()≤3𝑎,𝑏2𝑀_𝑆𝐺(),𝑀𝑎,𝑏_𝑆𝐴()≤3𝑎,𝑏4𝑀_𝑆𝑁3()≤2𝑎,𝑏3𝑀_𝑆𝑁1(𝑎,𝑏). (1.5)(1.6)(1.7)(1.8)(1.9)

For the difference of means given by (1.4), we study the Schur-geometric convexity of difference between these differences in order to further improve the inequalities in (1.1). The main result of this paper reads as follows.

Theorem I. The following differences are Schur-geometrically convex in 𝑅²₊=(0,∞)×(0,∞): 𝐷_{𝑆𝐻−𝑆𝐴}()=1𝑎,𝑏3𝑀_𝑆𝐻(𝑎,𝑏)−𝑀_𝑆𝐴(),𝐷𝑎,𝑏_{𝐴𝐻−𝑆𝐻}()=1𝑎,𝑏2𝑀_𝐴𝐻()−1𝑎,𝑏3𝑀_𝑆𝐻(),𝐷𝑎,𝑏_{𝑆𝐺−𝐴𝐻}(𝑎,𝑏)=𝑀_𝑆𝐺(𝑎,𝑏)−𝑀_𝐴𝐻(),𝐷𝑎,𝑏_{𝐴𝐺−𝑆𝐺}(𝑎,𝑏)=𝑀_𝐴𝐺()−1𝑎,𝑏2𝑀_𝑆𝐺(),𝐷𝑎,𝑏_{𝑁2𝑁1−𝐴𝐻}(𝑎,𝑏)=𝑀_𝑁2𝑁1()−1𝑎,𝑏8𝑀_𝐴𝐻(),𝐷𝑎,𝑏_{𝑁2𝐺−𝑁2𝑁1}()=1𝑎,𝑏3𝑀_𝑁2𝐺(𝑎,𝑏)−𝑀_𝑁2𝑁1(),𝐷𝑎,𝑏_{𝐴𝐺−𝑁2𝐺}()=1𝑎,𝑏4𝑀_𝐴𝐺()−1𝑎,𝑏3𝑀_𝑁2𝐺(),𝐷𝑎,𝑏_{𝐴𝑁2−𝐴𝐺}(𝑎,𝑏)=𝑀_𝐴𝑁2()−1𝑎,𝑏4𝑀_𝐴𝐺(),𝐷𝑎,𝑏_{𝑆𝑁2−𝑆𝐴}()=4𝑎,𝑏5𝑀_𝑆𝑁2(𝑎,𝑏)−𝑀_𝑆𝐴(),𝐷𝑎,𝑏_{𝐴𝑁2−𝑆𝑁2}(𝑎,𝑏)=4𝑀_𝐴𝑁2()−4𝑎,𝑏5𝑀_𝑆𝑁2(),𝐷𝑎,𝑏_{𝑆𝑁1−𝑆𝐻}(𝑎,𝑏)=2𝑀_𝑆𝑁1(𝑎,𝑏)−𝑀_𝑆𝐻(),𝐷𝑎,𝑏_{𝑆𝐺−𝑆𝑁1}()=3𝑎,𝑏2𝑀_𝑆𝐺(𝑎,𝑏)−2𝑀_𝑆𝑁1(),𝐷𝑎,𝑏_{𝑆𝑁3−𝑆𝐴}()=3𝑎,𝑏4𝑀_𝑆𝑁3(𝑎,𝑏)−𝑀_𝑆𝐴(),𝐷𝑎,𝑏_{𝑆𝑁1−𝑆𝑁3}()=2𝑎,𝑏3𝑀_𝑆𝑁1()−3𝑎,𝑏4𝑀_𝑆𝑁3(𝑎,𝑏). (1.11)

The proof of this theorem will be given in Section 3. Applying this result, in Section 4, we prove some inequalities related to the considered differences of means. Obtained inequalities are refinements of inequalities (1.5)–(1.9).

2. Definitions and Auxiliary Lemmas

The Schur-convex function was introduced by Schur in 1923, and it has many important applications in analytic inequalities, linear regression, graphs and matrices, combinatorial optimization, information-theoretic topics, Gamma functions, stochastic orderings, reliability, and other related fields (cf. [4–14]).

In 2003, Zhang first proposed concepts of “Schur-geometrically convex function” which is extension of “Schur-convex function” and established corresponding decision theorem [15]. Since then, Schur-geometric convexity has evoked the interest of many researchers and numerous applications and extensions have appeared in the literature (cf. [16–19]).

In order to prove the main result of this paper we need the following definitions and auxiliary lemmas.

Definition 2.1 (see [4, 20]). Let 𝐱=(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛 and 𝐲=(𝑦₁,…,𝑦_𝑛)∈ℝ^𝑛. (i)𝐱 is said to be majorized by 𝐲 (in symbols 𝐱≺𝐲) if ∑^𝑘_𝑖=1𝑥_[𝑖]≤∑^𝑘_𝑖=1𝑦_[𝑖] for 𝑘=1,2,…,𝑛−1 and ∑^𝑛_𝑖=1𝑥_𝑖=∑^𝑛_𝑖=1𝑦_𝑖, where 𝑥_[1]≥⋯≥𝑥_[𝑛] and 𝑦_[1]≥⋯≥𝑦_[𝑛] are rearrangements of 𝐱 and 𝐲 in a descending order.(ii)Ω⊆ℝ^𝑛 is called a convex set if (𝛼𝑥₁+𝛽𝑦₁,…,𝛼𝑥_𝑛+𝛽𝑦_𝑛)∈Ω for every 𝐱 and 𝐲∈Ω, where 𝛼 and 𝛽∈[0,1] with 𝛼+𝛽=1. (iii) Let Ω⊆ℝ^𝑛. The function 𝜑: Ω→ℝ is said to be a Schur-convex function on Ω if 𝐱≺𝐲 on Ω implies 𝜑(𝐱)≤𝜑(𝐲).𝜑 is said to be a Schur-concave function on Ω if and only if −𝜑 is Schur-convex.

Definition 2.2 (see [15]). Let 𝐱=(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛 and 𝐲=(𝑦₁,…,𝑦_𝑛)∈ℝ^𝑛₊. (i)Ω⊆ℝ^𝑛₊ is called a geometrically convex set if (𝑥^𝛼₁𝑦^𝛽₁,…,𝑥^𝛼_𝑛𝑦^𝛽_𝑛)∈Ω for all 𝐱,𝐲∈Ω and 𝛼,𝛽∈[0,1] such that 𝛼+𝛽=1. (ii) Let Ω⊆ℝ^𝑛₊. The function 𝜑: Ω→ℝ₊ is said to be Schur-geometrically convex function on Ω if (ln𝑥₁,…,ln𝑥_𝑛)≺(ln𝑦₁,…,ln𝑦_𝑛) on Ω implies 𝜑(𝐱)≤𝜑(𝐲). The function 𝜑 is said to be a Schur-geometrically concave on Ω if and only if −𝜑 is Schur-geometrically convex.

Definition 2.3 (see [4, 20]). (i) The set Ω⊆ℝ^𝑛 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 [15]). Let Ω⊆ℝ^𝑛₊ be a symmetric and geometrically convex set with a nonempty interior Ω⁰. Let 𝜑∶Ω→ℝ₊ be continuous on Ω and differentiable in Ω⁰. If 𝜑 is symmetric on Ω and ln𝑥₁−ln𝑥₂𝑥₁𝜕𝜑𝜕𝑥₁−𝑥₂𝜕𝜑𝜕𝑥₂≥0(≤0) (2.1) holds for any 𝐱=(𝑥₁,…,𝑥_𝑛)∈Ω⁰, then 𝜑 is a Schur-geometrically convex (Schur-geometrically concave) function.

Lemma 2.5. For (𝑎,𝑏)∈𝑅²₊=(0,∞)×(0,∞) one has 1≥𝑎+𝑏2𝑎²+𝑏²≥12+2𝑎𝑏(𝑎+𝑏)²,𝑎+𝑏2𝑎²+𝑏²−𝑎𝑏(𝑎+𝑏)²≤34,32≥√𝑎+𝑏√2√𝑎+√𝑏+√𝑎+√𝑏√2√≥5𝑎+𝑏4+𝑎𝑏(𝑎+𝑏)². (2.2)(2.3)(2.4)

Proof. It is easy to see that the left-hand inequality in (2.2) is equivalent to (𝑎−𝑏)²≥0, and the right-hand inequality in (2.2) is equivalent to 2𝑎²+𝑏²−(𝑎+𝑏)2𝑎²+𝑏²≤(𝑎+𝑏)²−4𝑎𝑏2(𝑎+𝑏)², (2.5) that is, (𝑎−𝑏)²2𝑎²+𝑏²+2𝑎²+𝑏²()≤()𝑎+𝑏𝑎−𝑏²2(𝑎+𝑏)². (2.6) Indeed, from the left-hand inequality in (2.2) we have 2𝑎²+𝑏²+2𝑎²+𝑏²(𝑎+𝑏)≥2𝑎²+𝑏²+(𝑎+𝑏)²≥2(𝑎+𝑏)², (2.7) so the right-hand inequality in (2.2) holds.
The inequality in (2.3) is equivalent to 2𝑎²+𝑏²−(𝑎+𝑏)2𝑎²+𝑏²≥(𝑎−𝑏)²4(𝑎+𝑏)². (2.8) Since 2𝑎²+𝑏²−(𝑎+𝑏)2𝑎²+𝑏²=2𝑎²+𝑏²−(𝑎+𝑏)²2𝑎²+𝑏²2𝑎²+𝑏²+()=()𝑎+𝑏𝑎−𝑏²2𝑎²+𝑏²+(𝑎+𝑏)2𝑎²+𝑏², (2.9) so it is sufficient prove that 2𝑎²+𝑏²+(𝑎+𝑏)2𝑎²+𝑏²≤4(𝑎+𝑏)², (2.10) that is, (𝑎+𝑏)2𝑎²+𝑏²≤2𝑎²+𝑏²+4𝑎𝑏, (2.11) and, from the left-hand inequalities in (2.2), we have (𝑎+𝑏)2𝑎²+𝑏²≤2𝑎²+𝑏²≤2𝑎²+𝑏²+4𝑎𝑏, (2.12) so the inequality in (2.3) holds.
Notice that the functions in the inequalities (2.4) are homogeneous. So, without loss of generality, we may assume √𝑎+√𝑏=1, and set 𝑡=√𝑎𝑏. Then 0<𝑡≤1/4 and (2.4) reduces to 32≥√1−2𝑡√2+1√2√≥51−2𝑡4+𝑡²(1−2𝑡)². (2.13) Squaring every side in the above inequalities yields 94≥1−2𝑡2+12−4𝑡+1≥25+𝑡16(1−⁴2𝑡)⁴+5𝑡²2(1−2𝑡)². (2.14) Reducing to common denominator and rearranging, the right-hand inequality in (2.14) reduces to (1−2𝑡)16𝑡²(2𝑡−1)²+(1/8)(16𝑡−7)²+(7/8)16(2𝑡−1)⁴≥0, (2.15) and the left-hand inequality in (2.14) reduces to 2(1−2𝑡)²+2−5(1−2𝑡)2(1−2𝑡)=−1+2𝑡2≤0, (2.16) so two inequalities in (2.4) hold.

Lemma 2.6 (see [16]). Let 𝑎≤𝑏,𝑢(𝑡)=𝑡𝑎+(1−𝑡)𝑏,𝑣(𝑡)=𝑡𝑏+(1−𝑡)𝑎. If 1/2≤𝑡₂≤𝑡₁≤1 or 0≤𝑡₁≤𝑡₂≤1/2, then 𝑎+𝑏2,𝑎+𝑏2≺𝑢𝑡₂,𝑣𝑡₂≺𝑢𝑡₁,𝑣𝑡₁≺(𝑎,𝑏). (2.17)

3. Proof of Main Result

Proof of Theorem I. Let (𝑎,𝑏)∈𝑅²₊.
(1) For 𝐷_{𝑆𝐻−𝑆𝐴}()=1𝑎,𝑏3𝑀_𝑆𝐻(𝑎,𝑏)−𝑀_𝑆𝐴(𝑎,𝑏)=𝑎+𝑏2−2𝑎𝑏3()−2𝑎+𝑏3𝑎²+𝑏²2, (3.1) we have 𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)=1𝜕𝑎2−2𝑏²3(𝑎+𝑏)²−23𝑎2𝑎²+𝑏²,𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)=1𝜕𝑏2−2𝑎²3(𝑎+𝑏)²−23𝑏2𝑎²+𝑏², (3.2) whence Λ∶=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)=()1𝜕𝑏𝑎−𝑏)(ln𝑎−ln𝑏2+2𝑎𝑏3(𝑎+𝑏)²−23𝑎+𝑏2𝑎²+𝑏². (3.3) From (2.3) we have 12+2𝑎𝑏3(𝑎+𝑏)²−23𝑎+𝑏2𝑎²+𝑏²≥0, (3.4) which implies Λ≥0 and, by Lemma 2.4, it follows that 𝐷_{𝑆𝐻−𝑆𝐴} is Schur-geometrically convex in 𝑅²₊.
(2) For 𝐷_{𝐴𝐻−𝑆𝐻}()=1𝑎,𝑏2𝑀_𝐴𝐻()−1𝑎,𝑏3𝑀_𝑆𝐻(𝑎,𝑏)=𝑎+𝑏4−𝑎𝑏3()−1𝑎+𝑏3𝑎²+𝑏²2. (3.5)
To prove that the function 𝐷_{𝐴𝐻−𝑆𝐻} is Schur-geometrically convex in 𝑅²₊ it is enough to notice that 𝐷_{𝐴𝐻−𝑆𝐻}(𝑎,𝑏)=(1/2)𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏).
(3) For 𝐷_{𝑆𝐺−𝐴𝐻}(𝑎,𝑏)=𝑀_𝑆𝐺(𝑎,𝑏)−𝑀_𝐴𝐻(𝑎,𝑏)=𝑎²+𝑏²2−√𝑎𝑏−𝑎+𝑏2+2𝑎𝑏,𝑎+𝑏 (3.6) we have 𝜕𝐷_{𝑆𝐺−𝐴𝐻}(𝑎,𝑏)=𝑎𝜕𝑎2𝑎²+𝑏²−𝑏2√−1𝑎𝑏2+2𝑏²(𝑎+𝑏)²,𝜕𝐷_{𝑆𝐺−𝐴𝐻}(𝑎,𝑏)=𝑏𝜕𝑏2𝑎²+𝑏²−𝑎2√−1𝑎𝑏2+2𝑎²(𝑎+𝑏)², (3.7) and then Λ∶=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)=()𝜕𝑏𝑎−𝑏)(ln𝑎−ln𝑏𝑎+𝑏2𝑎²+𝑏²−12−2𝑎𝑏(𝑎+𝑏)². (3.8)
From (2.2) we have Λ≥0, so by Lemma 2.4, it follows that 𝐷_{𝑆𝐻−𝑆𝐴} is Schur-geometrically convex in 𝑅²₊.
(4) For 𝐷_{𝐴𝐺−𝑆𝐺}(𝑎,𝑏)=𝑀_𝐴𝐺()−1𝑎,𝑏2𝑀_𝑆𝐺()=1𝑎,𝑏2𝑎+𝑏−√𝑎𝑏−𝑎²+𝑏²2, (3.9) we have 𝜕𝐷_{𝐴𝐺−𝑆𝐺}(𝑎,𝑏)=1𝜕𝑎2𝑏1−2√−𝑎𝑎𝑏2𝑎²+𝑏²,𝜕𝐷_{𝐴𝐺−𝑆𝐺}(𝑎,𝑏)=1𝜕𝑏2𝑎1−2√−𝑏𝑎𝑏2𝑎²+𝑏², (3.10) and then Λ∶=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝑆𝐻−𝑆𝐴}(𝑎,𝑏)=()𝜕𝑏𝑎−𝑏)(ln𝑎−ln𝑏1−𝑎+𝑏2𝑎²+𝑏². (3.11) By (2.2) we infer that 1−𝑎+𝑏2𝑎²+𝑏²≥0, (3.12) so Λ≥0. By Lemma 2.4, we get that 𝐷_{𝐴𝐺−𝑆𝐺} is Schur-geometrically convex in 𝑅²₊.
(5) For 𝐷_{𝑁2𝑁1−𝐴𝐻}(𝑎,𝑏)=𝑀_𝑁2𝑁1()−1𝑎,𝑏8𝑀_𝐴𝐻()=√𝑎,𝑏𝑎+√𝑏2𝑎+𝑏2−14()−1𝑎+𝑏2√𝑎𝑏−18𝑎+𝑏2−2𝑎𝑏,𝑎+𝑏 (3.13) we have 𝜕𝐷_{𝑁2𝑁1−𝐴𝐻}(𝑎,𝑏)=1𝜕𝑎4√𝑎𝑎+𝑏2+14√𝑎+√𝑏2𝑎+𝑏2^−1/2−14−𝑏4√−1𝑎𝑏812−2𝑏²(𝑎+𝑏)²,𝜕𝐷_{𝑁2𝑁1−𝐴𝐻}(𝑎,𝑏)=1𝜕𝑏4√𝑏𝑎+𝑏2+14√𝑎+√𝑏2𝑎+𝑏2^−1/2−14−𝑎4√−1𝑎𝑏812−2𝑎²(𝑎+𝑏)², (3.14) and then Λ=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝑁2𝑁1−𝐴𝐻}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝑁2𝑁1−𝐴𝐻}(𝑎,𝑏)=1𝜕𝑏4()√𝑎−𝑏)(ln𝑎−ln𝑏𝑎+𝑏√2√𝑎+√𝑏+√𝑎+√𝑏√2√−5𝑎+𝑏4−𝑎𝑏(𝑎+𝑏)². (3.15) From (2.4) we have √𝑎+𝑏√2√𝑎+√𝑏+√𝑎+√𝑏√2√−5𝑎+𝑏4−𝑎𝑏(𝑎+𝑏)²≥0, (3.16) so Λ≥0; it follows that 𝐷_{𝑁2𝑁1−𝐴𝐻} is Schur-geometrically convex in 𝑅²₊.
(6) For 𝐷_{𝑁2𝐺−𝑁2𝑁1}()=1𝑎,𝑏3𝑀_𝑁2𝐺(𝑎,𝑏)−𝑀_𝑁2𝑁1()=𝑎,𝑏𝑎+𝑏4+√𝑎𝑏6−23√𝑎+√𝑏2𝑎+𝑏2, (3.17) we have 𝜕𝐷_{𝑁2𝐺−𝑁2𝑁1}(𝑎,𝑏)=1𝜕𝑎4+𝑏12√−1𝑎𝑏6√𝑎𝑎+𝑏2−16√𝑎+√𝑏2𝑎+𝑏2^−1/2,𝜕𝐷_{𝑁2𝐺−𝑁2𝑁1}(𝑎,𝑏)=1𝜕𝑏4+𝑎12√−1𝑎𝑏6√𝑏𝑎+𝑏2−16√𝑎+√𝑏2𝑎+𝑏2^−1/2, (3.18) and then Λ=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝑁2𝐺−𝑁2𝑁1}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝑁2𝐺−𝑁2𝑁1}(𝑎,𝑏)=()1𝜕𝑏ln𝑎−ln𝑏4()−√𝑎−𝑏𝑎−√𝑏6𝑎+𝑏2−(𝑎−𝑏)√𝑎+√𝑏12𝑎+𝑏2^−1/2=16()3𝑎−𝑏)(ln𝑎−ln𝑏2−√𝑎+𝑏√2√𝑎+√𝑏−√𝑎+√𝑏√2√.𝑎+𝑏 (3.19) By (2.4) we infer that Λ≥0, which proves that 𝐷_{𝑁2𝐺−𝑁2𝑁1} is Schur-geometrically convex in 𝑅²₊.
(7) For 𝐷_{𝐴𝐺−𝑁2𝐺}()=1𝑎,𝑏4𝑀_𝐴𝐺()−1𝑎,𝑏3𝑀_𝑁2𝐺()=𝑎,𝑏𝑎+𝑏8+1√121𝑎𝑏−3√𝑎+√𝑏2𝑎+𝑏2, (3.20) we have 𝜕𝐷_{𝐴𝐺−𝑁2𝐺}(𝑎,𝑏)=1𝜕𝑎8+𝑏24√−√𝑎𝑏𝑎+𝑏12√−√2𝑎𝑎+√𝑏122(),𝑎+𝑏𝜕𝐷_{𝐴𝐺−𝑁2𝐺}(𝑎,𝑏)=1𝜕𝑏8+𝑎24√−√𝑎𝑏𝑎+𝑏12√−√2𝑏𝑎+√𝑏122(),𝑎+𝑏 (3.21) and then Λ=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝐴𝐺−𝑁2𝐺}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝐴𝐺−𝑁2𝐺}(𝑎,𝑏)=()𝜕𝑏ln𝑎−ln𝑏𝑎−𝑏8−√𝑎+𝑏√𝑎−√𝑏12√2−(𝑎−𝑏)√𝑎+√𝑏122()=()𝑎+𝑏𝑎−𝑏)(ln𝑎−ln𝑏821−3√𝑎+𝑏√2√𝑎+√𝑏+√𝑎+√𝑏√2√.𝑎+𝑏 (3.22) From (2.4) we have Λ≥0, and, consequently, by Lemma 2.4, we obtain that 𝐷_{𝐴𝐺−𝑁2𝐺} is Schur-geometrically convex in 𝑅²₊.
(8) In order to prove that the function 𝐷_{𝐴𝑁2−𝐴𝐺}(𝑎,𝑏) is Schur-geometrically convex in 𝑅²₊ it is enough to notice that 𝐷_{𝐴𝑁2−𝐴𝐺}(𝑎,𝑏)=𝑀_𝐴𝑁2()−1𝑎,𝑏4𝑀_𝐴𝐺(𝑎,𝑏)=3𝐷_{𝐴𝐺−𝑁2𝐺}(𝑎,𝑏). (3.23)
(9) For 𝐷_{𝑆𝑁2−𝑆𝐴}()=4𝑎,𝑏5𝑀_𝑆𝑁2(𝑎,𝑏)−𝑀_𝑆𝐴()=𝑎,𝑏𝑎+𝑏2−15𝑎²+𝑏²2−15√𝑎+√𝑏2(𝑎+𝑏), (3.24) we have 𝜕𝐷_{𝑆𝑁2−𝑆𝐴}(𝑎,𝑏)=1𝜕𝑎2−𝑎52𝑎²+𝑏²−15𝑎+𝑏−√2𝑎𝑎+√𝑏52(),𝑎+𝑏𝜕𝐷_{𝑆𝑁2−𝑆𝐴}(𝑎,𝑏)=1𝜕𝑏2−𝑏52𝑎²+𝑏²−15𝑎+𝑏−√2𝑏𝑎+√𝑏52(),𝑎+𝑏 (3.25) and then Λ=(ln𝑎−ln𝑏)𝜕𝐷_{𝑆𝑁2−𝑆𝐴}(𝑎,𝑏)−𝜕𝑎𝜕𝐷_{𝑆𝑁2−𝑆𝐴}(𝑎,𝑏)=()𝜕𝑏ln𝑎−ln𝑏𝑎−𝑏2−𝑎²−𝑏²52𝑎²+𝑏²−15𝑎(𝑎+𝑏)2−𝑏(𝑎+𝑏)2−√𝑎+√𝑏(𝑎−𝑏)52()=()𝑎+𝑏𝑎−𝑏)(ln𝑎−ln𝑏5√25√2−𝑎+𝑏𝑎²+𝑏²−√𝑎+𝑏√𝑎+√𝑏−√𝑎+√𝑏√.𝑎+𝑏 (3.26) From (2.2) and (2.4) we obtain that 5√2−𝑎+𝑏𝑎²+𝑏²−√𝑎+𝑏√𝑎+√𝑏−√𝑎+√𝑏√≥5𝑎+𝑏√2−√32−√2=0, (3.27) so Λ≥0, which proves that the function 𝐷_{𝑆𝑁2−𝑆𝐴}(𝑎,𝑏) is Schur-geometrically convex in 𝑅²₊.
(10) One can easily check that 𝐷_{𝐴𝑁𝐴𝑁2−𝑆𝑁2}(𝑎,𝑏)=4𝐷_{𝑆𝑁2−𝑆𝐴}(𝑎,𝑏), (3.28) and, consequently, the function 𝐷_{𝐴𝑁2−𝑆𝑁2} is Schur-geometrically convex in 𝑅²₊.
(11) To prove that the function 𝐷_{𝑆𝑁1−𝑆𝐻}(𝑎,𝑏)=2𝑀_𝑆𝑁1(𝑎,𝑏)−𝑀_𝑆𝐻(𝑎,𝑏)=𝑎²+𝑏²2−𝑎+𝑏2−√𝑎𝑏+2𝑎𝑏𝑎+𝑏 (3.29) is Schur-geometrically convex in 𝑅²₊ it is enough to notice that 𝐷_{𝑆𝑁1−𝑆𝐻}(𝑎,𝑏)=𝐷_{𝑆𝐺−𝐴𝐻}(𝑎,𝑏). (3.30)
(12) For 𝐷_{𝑆𝐺−𝑆𝑁1}()=3𝑎,𝑏2𝑀_𝑆𝐺(𝑎,𝑏)−2𝑀_𝑆𝑁1()=1𝑎,𝑏2𝑎+𝑏−√𝑎𝑏−𝑎²+𝑏²2, (3.31) we have 𝜕𝐷_{𝑆𝐺−𝑆𝑁1}(𝑎,𝑏)=1𝜕𝑎2𝑏1−2√−𝑎𝑎𝑏2𝑎²+𝑏²,𝜕𝐷_{𝑆𝐺−𝑆𝑁1}(𝑎,𝑏)=1𝜕𝑏2𝑎1−2√−𝑏𝑎𝑏2𝑎²+𝑏², (3.32) and then Λ=(ln𝑎−ln𝑏)𝑎𝜕𝐷_{𝑆𝐺−𝑆𝑁1}(𝑎,𝑏)𝜕𝑎−𝑏𝜕𝐷_{𝑆𝐺−𝑆𝑁1}(𝑎,𝑏)=()𝜕𝑏𝑎−𝑏)(ln𝑎−ln𝑏21−𝑎+𝑏2𝑎²+𝑏². (3.33)
By the inequality (2.2) we get that Λ≥0, which proves that 𝐷_{𝑆𝐺−𝑆𝑁1} is Schur-geometrically convex in 𝑅²₊.
(13) It is easy to check that 𝐷_{𝑆𝑁3−𝑆𝐴}()=1𝑎,𝑏2𝐷_{𝐴𝐺−𝑆𝐺}(𝑎,𝑏), (3.34) which means that the function 𝐷_{𝑆𝑁3−𝑆𝐴} is Schur-geometrically convex in 𝑅²₊.
(14) To prove that the function 𝐷_{𝑆𝑁1−𝑆𝑁3} is Schur-geometrically convex in 𝑅²₊ it is enough to notice that 𝐷_{𝑆𝑁1−𝑆𝑁3}()=1𝑎,𝑏6𝐷_{𝐴𝐺−𝑆𝐺}(𝑎,𝑏). (3.35)
The proof of Theorem I is complete.

4. Applications

Applying Theorem I, Lemma 2.6, and Definition 2.2 one can easily prove the following.

Theorem II. Let 0<𝑎≤𝑏. 1/2≤𝑡≤1 or 0≤𝑡≤1/2, 𝑢=𝑎^𝑡𝑏^1−𝑡 and 𝑣=𝑏^𝑡𝑎^1−𝑡. Then 𝑀_𝑆𝐴()≤1𝑎,𝑏3𝑀_𝑆𝐻()−1𝑎,𝑏3𝑀_𝑆𝐻(𝑢,𝑣)−𝑀_𝑆𝐴(≤1𝑢,𝑣)3𝑀_𝑆𝐻()≤1𝑎,𝑏2𝑀_𝐴𝐻()−1𝑎,𝑏2𝑀_𝐴𝐻()−1𝑢,𝑣3𝑀_𝑆𝐻()≤1𝑢,𝑣2𝑀_𝐴𝐻()≤1𝑎,𝑏2𝑀_𝑆𝐺()−1𝑎,𝑏2𝑀_𝑆𝐺()−1𝑢,𝑣2𝑀_𝐴𝐻()≤1𝑢,𝑣2𝑀_𝑆𝐺(𝑎,𝑏)≤𝑀_𝐴𝐺(𝑎,𝑏)−𝑀_𝐴𝐺()−1𝑢,𝑣2𝑀_𝑆𝐺(𝑢,𝑣)≤𝑀_𝐴𝐺(),1𝑎,𝑏8𝑀_𝐴𝐻(𝑎,𝑏)≤𝑀_𝑁2𝑁1(𝑎,𝑏)−𝑀_𝑁2𝑁1()−1𝑢,𝑣8𝑀_𝐴𝐻(𝑢,𝑣)≤𝑀_𝑁2𝑁1()≤1𝑎,𝑏3𝑀_𝑁2𝐺()−1𝑎,𝑏3𝑀_𝑁2𝐺(𝑢,𝑣)−𝑀_𝑁2𝑁1()≤1𝑢,𝑣3𝑀_𝑁2𝐺()≤1𝑎,𝑏4𝑀_𝐴𝐺()−1𝑎,𝑏4𝑀_𝐴𝐺()−1𝑢,𝑣3𝑀_𝑁2𝐺()≤1𝑢,𝑣4𝑀_𝐴𝐺(𝑎,𝑏)≤𝑀_𝐴𝑁2(𝑎,𝑏)−𝑀_𝐴𝑁2()−1𝑢,𝑣4𝑀_𝐴𝐺(𝑢,𝑣)≤𝑀_𝐴𝑁2(),𝑀𝑎,𝑏_𝑆𝐴()≤4𝑎,𝑏5𝑀_𝑆𝑁2()−4𝑎,𝑏5𝑀_𝑆𝑁2()−4𝑢,𝑣5𝑀_𝑆𝑁2()≤4𝑢,𝑣5𝑀_𝑆𝑁2(𝑎,𝑏)≤4𝑀_𝐴𝑁2(𝑎,𝑏)−4𝑀_𝐴𝑁2()−4𝑢,𝑣5𝑀_𝑆𝑁2(𝑢,𝑣)≤4𝑀_𝐴𝑁2(),𝑀𝑎,𝑏_𝑆𝐻(𝑎,𝑏)≤2𝑀_𝑆𝑁1(𝑎,𝑏)−2𝑀_𝑆𝑁1(𝑢,𝑣)−𝑀_𝑆𝐻(𝑢,𝑣)≤2𝑀_𝑆𝑁1()≤3𝑎,𝑏2𝑀_𝑆𝐺()−3𝑎,𝑏2𝑀_𝑆𝐺()−3𝑢,𝑣2𝑀_𝑆𝐺()≤3𝑢,𝑣2𝑀_𝑆𝐺(),𝑀𝑎,𝑏_𝑆𝐴()≤3𝑎,𝑏4𝑀_𝑆𝑁3()−3𝑎,𝑏4𝑀_𝑆𝑁3(𝑢,𝑣)−𝑀_𝑆𝐴()≤3𝑢,𝑣4𝑀_𝑆𝑁3()≤2𝑎,𝑏3𝑀_𝑆𝑁1()−2𝑎,𝑏3𝑀_𝑆𝑁1()−3𝑢,𝑣4𝑀_𝑆𝑁3()≤2𝑢,𝑣3𝑀_𝑆𝑁1(𝑎,𝑏). (4.1)(4.2)(4.3)(4.4)(4.5)

Remark 4.1. Equation (4.1), (4.2), (4.3), (4.4), and (4.5) are a refinement of (1.5), (1.6), (1.7), (1.8), and (1.9), respectively.

Acknowledgments

The authors are grateful to the referees for their helpful comments and suggestions. The first author was supported in part by the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201011417013).