The Applications of Functional Variants of Jensen's Inequality

Abstract

The paper is inspired by McShane's results on the functional form of Jensen's inequality for convex functions of several variables. The work is focused on applications and generalizations of this important result. At that, the generalizations of Jensen's inequality are obtained using the positive linear functionals.

1. Introduction

1.1. Vector Space of Real Valued Functions and Positive Linear Functionals

Given a nonempty set 𝒳 we consider a space of functions 𝑔∶𝒳→ℝ containing the unit function 𝑔=1 defined with 1(𝑥)=1 for every 𝑥∈𝒳. Let 𝕏 be a real vector space of real valued functions on 𝒳 containing the unit function. A linear functional 𝑀∶𝕏→ℝ is said to be positive (nonnegative) or monotone if 𝑀(𝑔)≥0 for every nonnegative function 𝑔∈𝕏. If 𝑀(1)=1, we say that the functional 𝑀 is unital or normalized.

1.2. Functional Forms of Jensen’s Inequality

In 1931, Jessen [1] formulated the functional form of Jensen’s inequality for convex functions of one variable.

Theorem A. Let ℐ⊆ℝ be a closed interval. Let 𝑔∈𝕏 be a function such that 𝑔(𝑥)∈ℐ for every 𝑥∈𝑋. Let 𝑓∶ℐ→ℝ be a continuous convex function such that 𝑓(𝑔)∈𝕏.
Then every unital positive linear functional 𝑀∶𝕏→ℝ satisfies the inequality 𝑓𝑀𝑔≤𝑀𝑓𝑔. (1)

The interval ℐ must be closed; otherwise, it could happen that 𝑀(𝑔)∉ℐ as mentioned in [2].

Example 1. Let 𝒳=ℐ=⟨0,1], 𝕏={𝑔∶⟨0,1]→ℝ∣lim_𝑥→0+𝑔(𝑥)∈ℝ}, and 𝑀∶𝕏→ℝ is defined by 𝑀(𝑔)=lim_𝑥→0+𝑔(𝑥). Take the identity function 𝑔(𝑥)=𝑥 for 𝑥∈ℐ. Then 𝑔∈𝕏 and 𝑔(𝑥)∈ℐ for every 𝑥∈𝒳, but 𝑀(𝑔)=0∉ℐ.
The function 𝑓 must be continuous; otherwise, it could happen that the inequality in (1) does not apply as noted in [3].

Example 2. Consider the previous example with 𝒳=ℐ=[0,1]. Take the convex function 𝑓∶[0,1]→ℝ defined by 𝑓(0)=1 and 𝑓(𝑥)=0 for 𝑥∈⟨0,1]. Then 𝑓(𝑀(𝑔))=𝑓(0)=1>0=𝑀(𝑓)=𝑀(𝑓(𝑔)).

In 1937, McShane [4, Theorems 1-2] extended the functional form of Jensen’s inequality to convex functions of several variables. He has covered the generalization in two steps, calling them the geometric (Theorem B) and analytic (Theorem C) formulation of Jensen’s inequality.

Theorem B. Let 𝒞⊆ℝ^𝑛 be a closed convex set. Let 𝑔_𝑖∈𝕏 be functions such that (𝑔₁(𝑥),…,𝑔_𝑛(𝑥))∈𝒞 for every 𝑥∈𝒳.
Then every unital positive linear functional 𝑀∶𝕏→ℝ satisfies the inclusion 𝑀𝑔₁,…,𝑀𝑔_𝑛∈𝒞. (2)

Theorem C. Let 𝒞⊆ℝ^𝑛 be a closed convex set. Let 𝑔_𝑖∈𝕏 be functions such that (𝑔₁(𝑥),…,𝑔_𝑛(𝑥))∈𝒞 for every 𝑥∈𝒳. Let 𝑓∶𝒞→ℝ be a continuous convex function such that 𝑓(𝑔₁,…,𝑔_𝑛)∈𝕏.
Then every unital positive linear functional 𝑀∶𝕏→ℝ satisfies the inequality 𝑓𝑀𝑔₁,…,𝑀𝑔_𝑛≤𝑀𝑓𝑔₁,…,𝑔_𝑛. (3)

If the function 𝑓 is concave, then the reverse inequality is valid in (3). The hyperplanes that contain the set 𝒞 were used in the proof of Theorem B. The epigraph of the function 𝑓: epi𝑓=𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1∈𝒞×ℝ∣𝑥_𝑛+1≥𝑓𝑥₁,…,𝑥_𝑛 (4) and Theorem B were applied in the proof of Theorem C.

2. The Applications and Generalizations of McShane’s Result

This section presents general applications and generalizations of Theorem C. The main generalization is Theorem 8.

Using McShane’s inequality in (3) it is very easy to get the functional form of Hölder’s and Minkowski’s inequality for 𝑛 nonnegative functions.

Corollary 3 (the functional form of Hölder’s inequality). Let 𝑔_𝑖∶𝒳→ℝ be nonnegative functions such that 𝑔₁⋯𝑔_𝑛∈𝕏. Let 𝑝_𝑖>1 be real numbers such that ∑^𝑛_𝑖=11/𝑝_𝑖=1 and all 𝑔^𝑝𝑖_𝑖∈𝕏.
Then every unital positive linear functional 𝑀∶𝕏→ℝ satisfies the inequality 𝑀𝑔₁⋯𝑔_𝑛≤𝑀𝑔^𝑝1₁^1/𝑝1⋯𝑀𝑔^𝑝𝑛_𝑛^1/𝑝𝑛. (5)

Proof. Applying the concave function 𝑓(𝑥₁,…,𝑥_𝑛)=𝑥^1/𝑝1₁⋯𝑥^1/𝑝𝑛_𝑛, where all 𝑥_𝑖≥0, together with the functions 𝑔^𝑝𝑖_𝑖 to the reverse inequality in (3), it follows that 𝑀𝑔^𝑝1₁^1/𝑝1⋯𝑀𝑔^𝑝𝑛_𝑛^1/𝑝𝑛≥𝑀𝑔₁⋯𝑔_𝑛, (6) which is the required inequality.

Corollary 4 (the functional form of Minkowski’s inequality). Let 𝑔_𝑖∶𝒳→ℝ be nonnegative functions for 𝑖=1,…,𝑛. Let 𝑝≥1 be a real number such that (𝑔₁+⋯+𝑔_𝑛)^𝑝∈𝕏 and all 𝑔^𝑝_𝑖∈𝕏.
Then every unital positive linear functional 𝑀∶𝕏→ℝ satisfies the inequality 𝑀𝑔₁+⋯+𝑔_𝑛^𝑝1/𝑝≤𝑀𝑔^𝑝₁^1/𝑝+⋯+𝑀𝑔^𝑝_𝑛^1/𝑝. (7)

Proof. Applying the concave function 𝑓(𝑥₁,…,𝑥_𝑛)=(𝑥₁^1/𝑝+⋯+𝑥_𝑛^1/𝑝)^𝑝, where all 𝑥_𝑖≥0, and the functions 𝑔^𝑝_𝑖 to the reverse inequality in (3), it follows that 𝑀𝑔^𝑝₁^1/𝑝+⋯+𝑀𝑔^𝑝_𝑛^1/𝑝𝑝≥𝑀𝑔₁+⋯+𝑔_𝑛^𝑝, (8) and rising to the power of 1/𝑝 gives the desired inequality.

The McShane inequality in (3) can be easily generalized by introducing the weighted function 𝑝∈𝕏 as follows.

Theorem 5. Let 𝒞⊆ℝ^𝑛 be a closed convex set. Let 𝑔_𝑖∶𝒳→ℝ be functions such that (𝑔₁(𝑥),…,𝑔_𝑛(𝑥))∈𝒞 for every 𝑥∈𝒳. Let 𝑝∈𝕏 be either a nonnegative or nonpositive function such that all 𝑝𝑔_𝑖∈𝕏. Let 𝑓∶𝒞→ℝ be a continuous convex function such that 𝑝𝑓(𝑔₁,…,𝑔_𝑛)∈𝕏.
Then every positive linear functional 𝑀∶𝕏→ℝ such that 𝑀(𝑝)≠0 satisfies the inequality 𝑓𝑀𝑝𝑔₁𝑀𝑝𝑀,…,𝑝𝑔_𝑛𝑀𝑝≤𝑀𝑝𝑓𝑔₁,…,𝑔_𝑛𝑀𝑝. (9)

Proof. We use the function vector space 𝕐={𝑔∶𝒳→ℝ∣𝑝𝑔∈𝕏} and the linear functional 𝐿∶𝕐→ℝ defined with 𝐿(g)=𝑀(𝑝𝑔)/𝑀(𝑝). The functional 𝐿 is positive and unital. Since all 𝑔_𝑖∈𝕐 and 𝑓(𝑔₁,…,𝑔_𝑛)∈𝕐, the functional values 𝐿(𝑔_𝑖) and 𝐿(𝑓(𝑔₁,…,𝑔_𝑛)) may be applied to the inequality in (3).

If 𝕏 is a vector space of real valued 𝜇-integrable functions on 𝒞, then using the inequality in (9) with the positive linear functional 𝑀𝑔=_𝒞𝑔𝑑𝜇, (10) we get the following integral form of Jensen’s inequality for convex functions of several variables.

Corollary 6. Let 𝜇 be a measure on a closed convex set 𝒞⊆ℝ^𝑛. Let 𝑔_𝑖∶𝒞→ℝ be 𝜇-measurable functions such that (𝑔₁(𝑥),…,𝑔_𝑛(𝑥))∈𝒞 for every 𝑥∈𝒞. Let 𝑝∶𝒞→ℝ be either a nonnegative or nonpositive 𝜇-integrable function such that ∫_𝒞𝑝𝑑𝜇≠0 and all 𝑝𝑔_𝑖 are 𝜇-integrable.
Then every continuous convex function 𝑓∶𝒞→ℝ such that 𝑝𝑓(𝑔₁,…,𝑔_𝑛) is 𝜇-integrable satisfies the integral inequality 𝑓∫_𝒞𝑝𝑔₁𝑑𝜇∫_𝒞∫𝑝𝑑𝜇,…,^𝒞𝑝𝑔_𝑛𝑑𝜇∫_𝒞≤∫𝑝𝑑𝜇^𝒞𝑝𝑓𝑔₁,…,𝑔_𝑛𝑑𝜇∫_𝒞𝑝𝑑𝜇. (11)

Choosing points (𝑥_𝑗1,…,𝑥_𝑗𝑛)∈𝒞 for 𝑗=1,…,𝑚 and applying the inequality in (9) with the positive linear functional 𝑀𝑔=^𝑚_𝑗=1𝑔𝑥_𝑗1,…,𝑥_𝑗𝑛, (12) projections 𝑔_𝑖(𝑥_𝑗1,…,𝑥_𝑗𝑛)=𝑥_𝑗𝑖, and weighted function 𝑝(𝑥_𝑗1,…,𝑥_𝑗𝑛)=𝛼_𝑗, we have the following discrete form of Jensen’s inequality for convex functions of several variables.

Corollary 7. Let 𝒞⊆ℝ^𝑛 be a closed convex set. Let (𝑥_𝑗1,…,𝑥_𝑗𝑛)∈𝒞 be points for 𝑗=1,…,𝑚. Let 𝛼_𝑗∈ℝ be either nonnegative or nonpositive coefficients such that ∑^𝑚_𝑗=1𝛼_𝑗≠0.
Then every continuous convex function 𝑓∶𝒞→ℝ satisfies the discrete inequality 𝑓∑^𝑚_𝑗=1𝛼_𝑗𝑥_𝑗1∑^𝑚_𝑗=1𝛼_𝑗∑,…,^𝑚𝑗=1𝛼_𝑗𝑥_𝑗𝑛∑^𝑚_𝑗=1𝛼_𝑗≤∑^𝑚_𝑗=1𝛼_𝑗𝑓𝑥_𝑗1,…,𝑥_𝑗𝑛∑^𝑚_𝑗=1𝛼_𝑗. (13)

Now we use 𝑚 nonempty sets 𝒳_𝑗 and associated function vector spaces 𝕏_𝑗 containing the unit function.

Theorem 8. Let 𝒞⊆ℝ^𝑛 be a closed convex set. Let 𝑔_𝑗𝑖∶𝒳_𝑗→ℝ be functions such that (𝑔_𝑗1(𝑥),…,𝑔_𝑗𝑛(𝑥))∈𝒞 for every 𝑥∈𝒳_𝑗 for 𝑗=1,…,𝑚. Let 𝑝_𝑗∈𝕏_𝑗 be either nonnegative or nonpositive functions such that 𝑝_𝑗𝑔_𝑗𝑖∈𝕏_𝑗. Let 𝑓∶𝒞→ℝ be a continuous convex function such that 𝑝_𝑗𝑓(𝑔_𝑗1,…,𝑔_𝑗𝑛)∈𝕏_𝑗.
Then every 𝑚-tuple of positive linear functionals 𝑀_𝑗∶𝕏_𝑗→ℝ such that all 𝑀_𝑗(𝑝_𝑗)≠0 satisfies the inequality 𝑓∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗𝑔_𝑗1∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗∑,…,^𝑚𝑗=1𝑀_𝑗𝑝_𝑗𝑔_𝑗𝑛∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗≤∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗𝑓𝑔_𝑗1,…,𝑔_𝑗𝑛∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗. (14)

Proof. First we apply the inequality in (13) to the numbers 𝑥_𝑗𝑖=𝑀_𝑗(𝑝_𝑗𝑔_𝑗𝑖)/𝑀_𝑗(𝑝_𝑗) and coefficients 𝛼_𝑗=𝑀_𝑗(𝑝_𝑗) then the inequality in (9) to every 𝑀_𝑗: 𝑓∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗𝑔_𝑗1∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗∑,…,^𝑚𝑗=1𝑀_𝑗𝑝_𝑗𝑔_𝑗𝑛∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗∑=𝑓^𝑚𝑗=1𝑀_𝑗𝑝_𝑗𝑀_𝑗𝑝_𝑗𝑔_𝑗1/𝑀_𝑗𝑝_𝑗∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗∑,…,_𝑚𝑗=1𝑀_𝑗𝑝_𝑗𝑀_𝑗𝑝_𝑗𝑔_𝑗𝑛/𝑀_𝑗𝑝_𝑗∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗≤^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗𝑓𝑀_𝑗𝑝_𝑗𝑔_𝑗1𝑀_𝑗𝑝_𝑗𝑀,…,^𝑗𝑝_𝑗𝑔_𝑗𝑛𝑀_𝑗𝑝_𝑗×^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗⁻¹≤∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗𝑓𝑔_𝑗1,…,𝑔_𝑗𝑛∑^𝑚_𝑗=1𝑀_𝑗𝑝_𝑗, (15) and so we get the inequality in (14).

The inequality in (14) covers all the inequalities in this section.

3. The Generalizations of Jensen’s Inequality Using Functionals

This section presents the actual results of the paper. The main results are the generalizations of Jensen’s inequality written in Theorem 12 for convex functions of one variable and Theorem 16 for convex functions of several variables.

3.1. Inequalities for Convex Functions with One Variable

In what follows, we use a bounded interval [𝑎,𝑏]⊂ℝ assuming 𝑎<𝑏. Every 𝑥∈ℝ can be uniquely presented as the affine combination 𝑥=𝛼_𝑥𝑎+𝛽_𝑥𝑏, (16) where 𝛼_𝑥=^𝑥1𝑏1_𝑎1𝑏1,𝛽_𝑥=−^𝑥1𝑎1_𝑎1𝑏1. (17)

Lemma 9. Let 𝑐∈[𝑎,𝑏] be a point. Let 𝛼,𝛽∈[0,1] and 𝛾∈[−1,1] be coefficients of the sum 𝛼+𝛽+𝛾=1.
Then the affine combination 𝛼𝑎+𝛽𝑏+𝛾𝑐 belongs to [𝑎,𝑏].

Proof. Since 𝑐∈[𝑎,𝑏], it has to be 𝑐=𝛼_𝑐𝑎+𝛽_𝑐𝑏 for coefficients 𝛼_𝑐 and 𝛽_𝑐 taken from the formulas in (17). Then we have 𝛼𝛼𝑎+𝛽𝑏+𝛾𝑐=𝛼𝑎+𝛽𝑏+1−𝛼−𝛽_𝑐𝑎+𝛽_𝑐𝑏=𝛼1−𝛼_𝑐+1−𝛽𝛼_𝑐𝑎+𝛽1−𝛽_𝑐+(1−𝛼)𝛽_𝑐𝑏. (18) The coefficients in the square brackets are nonnegative with the sum equal to 1, so the observed expression 𝛼𝑎+𝛽𝑏+𝛾𝑐 belongs to [𝑎,𝑏].

Sufficient conditions on the coefficients in Lemma 9 are 𝛼,𝛽∈[0,1] and 𝛼+𝛽+𝛾=1. Applying these conditions, it follows that 𝛾∈[−1,1].

Theorem 10. Let 𝑐_𝑗∈[𝑎,𝑏] be points for 𝑗=1,…,𝑚. Let 𝛼,𝛽,𝛾_𝑗∈[0,1] and 𝛾∈[−1,1] be coefficients of the sums 𝛼+𝛽+𝛾=∑^𝑚_𝑗=1𝛾_𝑗=1.
Then the affine combination 𝛼𝑎+𝛽𝑏+𝛾∑^𝑚_𝑗=1𝛾_𝑗𝑐_𝑗∈[𝑎,𝑏].

Proof. Since ∑^𝑚_𝑗=1𝛾_𝑗𝑐_𝑗∈[𝑎,𝑏], the affine combination 𝛼𝑎+𝛽𝑏+𝛾∑^𝑚_𝑗=1𝛾_𝑗𝑐_𝑗∈[𝑎,𝑏] by Lemma 9.

Corollary 11. Let 𝑔∶[𝑎,𝑏]→ℝ be a monotone function, and let 𝑥_𝑗∈[𝑎,𝑏] be points for 𝑗=1,…,𝑚. Let 𝛼,𝛽,𝛾_𝑗∈[0,1] and 𝛾∈[−1,1] be coefficients of the sums 𝛼+𝛽+𝛾=∑^𝑚_𝑗=1𝛾_𝑗=1.
Then the affine combination 𝛼𝑔(𝑎)+𝛽𝑔(𝑏)+𝛾∑^𝑚_𝑗=1𝛾_𝑗𝑔(𝑥_𝑗)∈conv{𝑔(𝑎),𝑔(𝑏)}.

The convex hull of a set 𝒮 will be denoted by conv𝒮.

Theorem 12. Let 𝑔∶[𝑎,𝑏]→ℝ be a monotone function, and let 𝑥_𝑗∈[𝑎,𝑏] be points for 𝑗=1,…,𝑚. Let 𝛼,𝛽,𝛾_𝑗∈[0,1] and 𝛾∈[−1,1] be coefficients of the sums 𝛼+𝛽+𝛾=∑^𝑚_𝑗=1𝛾_𝑗=1.
Then every continuous convex function 𝑓∶conv{𝑔(𝑎),𝑔(𝑏)}→ℝ satisfies the inequality 𝑓𝛼𝑔(𝑎)+𝛽𝑔(𝑏)+𝛾^𝑚_𝑗=1𝛾_𝑗𝑔𝑥_𝑗≤𝛼𝑓𝑔(𝑎)+𝛽𝑓𝑔(𝑏)+𝛾^𝑚_𝑗=1𝛾_𝑗𝑓𝑔𝑥_𝑗. (19)

Proof. If 𝑓 is monotone, then the Jensen inequality in (1) should be applied to the function space 𝕏 generated with 𝑔 and 𝑓(𝑔), that is, 𝕏=ℎ∶[𝑎,𝑏]⟶ℝ∣ℎ=𝑐₁𝑔+𝑐₂𝑓𝑔+𝑐₃;𝑐₁,𝑐₂,𝑐₃∈ℝ, (20) and linear functional 𝑀 on 𝕏 defined by 𝑀(ℎ)=𝛼ℎ(𝑎)+𝛽ℎ(𝑏)+𝛾^𝑚_𝑗=1𝛾_𝑗ℎ𝑥_𝑗. (21) The functional 𝑀 is unital since 𝛼+𝛽+𝛾∑^𝑚_𝑗=1𝛾_𝑗=1 and positive since the function ℎ=(𝑐₁+𝑐₂𝑓)(𝑔)+𝑐₃ is monotone.
If 𝑓 is not monotone, then we translate 𝑓 so that its minimum 𝑓(𝑥₀)=0 and express 𝑓=𝑓₁+𝑓₂ where 𝑓₁(𝑥)=0for𝑥∈𝑐,𝑥₀𝑓(𝑥)for𝑥∈𝑥₀,𝑓,𝑑₂(𝑥)=𝑓(𝑥)for𝑥∈𝑐,𝑥₀0for𝑥∈𝑥₀,𝑑 (22) with [𝑐,𝑑]=conv{𝑔(𝑎),𝑔(𝑏)}. The previous procedure should be applied to the monotone convex functions 𝑓₁ and 𝑓₂.

Putting the identity function 𝑔(𝑥)=𝑥 for 𝑥∈[𝑎,𝑏], the inequality in (19) is reduced to 𝑓𝛼𝑎+𝛽𝑏+𝛾^𝑚_𝑗=1𝛾_𝑗𝑥_𝑗≤𝛼𝑓(𝑎)+𝛽𝑓(𝑏)+𝛾^𝑚_𝑗=1𝛾_𝑗𝑓𝑥_𝑗. (23) If 𝛼=𝛽=0 and 𝛾=1, the above inequality represents the Jensen inequality. If 𝛼=𝛽=1 and 𝛾=−1, it represents the Jensen-Mercer inequality.

3.2. Inequalities for Convex Functions with Several Variables

We assume that ℝ^𝑛 is the real vector space treating its points as vectors with the standard coordinate addition (𝑥₁,…,𝑥_𝑛)+(𝑦₁,…,𝑦_𝑛)=(𝑥₁+𝑦₁,…,𝑥_𝑛+𝑦_𝑛) and scalar multiplication 𝛼(𝑥₁,…,𝑥_𝑛)=(𝛼𝑥₁,…,𝛼𝑥_𝑛). The aim of this subsection is to generalize Theorem 12 to simplexes.

To begin let us show how Lemma 9 can be generalized to the triangle (𝑛=2). Take the three planar points 𝐴₁(𝑥₁,𝑦₁), 𝐴₂(𝑥₂,𝑦₂), and 𝐴₃(𝑥₃,𝑦₃) that do not belong to one line. Then any point 𝑃(𝑥,𝑦)∈ℝ² is presented by the unique affine combination 𝑃=𝛼_𝑃1𝐴₁+𝛼_𝑃2𝐴₂+𝛼_𝑃3𝐴₃, (24) where 𝛼_𝑃1=^{𝑥𝑥𝑦12𝑦21𝑥3𝑦31}_{𝑥1𝑦11𝑥2𝑦21𝑥3𝑦31},𝛼_𝑃2=−^{𝑥𝑥𝑦11𝑦11𝑥3𝑦31}_{𝑥1𝑦11𝑥2𝑦21𝑥3𝑦31},𝛼_𝑃3=^{𝑥𝑥𝑦11𝑦11𝑥2𝑦21}_{𝑥1𝑦11𝑥2𝑦21𝑥3𝑦31}. (25)

So, we suppose that conv{𝐴₁,𝐴₂,𝐴₃} is the triangle with vertices 𝐴₁, 𝐴₂, and 𝐴₃.

Lemma 13. Let 𝐶∈conv{𝐴₁,𝐴₂,𝐴₃} be a point. Let 𝛼₁,𝛼₂,𝛼₃∈[0,1] and 𝛾∈[−1,1] be coefficients such that 𝛼₁,𝛼₂,𝛼₃≥−𝛾 and 𝛼₁+𝛼₂+𝛼₃+𝛾=1.
Then the affine combination 𝛼₁𝐴₁+𝛼₂𝐴₂+𝛼₃𝐴₃+𝛾𝐶∈conv{𝐴₁,𝐴₂,𝐴₃}.

Proof. Let us show that the affine combination 𝛼₁𝐴₁+𝛼₂𝐴₂+𝛼₃𝐴₃+𝛾𝐶 is actually the convex combination. Since the point 𝐶∈conv{𝐴₁,𝐴₂,𝐴₃}, the convex combination 𝐶=𝛼_𝐶1𝐴₁+𝛼_𝐶2𝐴₂+𝛼_𝐶3𝐴₃ holds with the coefficients 𝛼_𝐶1, 𝛼_𝐶2, and 𝛼_𝐶3 taken from the formulas in (25). Therefore, we have 𝛼₁𝐴₁+𝛼₂𝐴₂+𝛼₃𝐴₃=𝛼+𝛾𝐶₁1−𝛼_𝐶1+𝛼₁+𝛾𝛼_𝐶1𝐴₁+𝛼₂1−𝛼_𝐶2+𝛼₂+𝛾𝛼_𝐶2𝐴₂+𝛼₃1−𝛼_𝐶3+𝛼₃+𝛾𝛼_𝐶3𝐴₃. (26) The coefficients in the square brackets are nonnegative with the sum equal to 1, so the observed combination is convex, and its center 𝐴=𝛼₁𝐴₁+𝛼₂𝐴₂+𝛼₃𝐴₃+𝛾𝐶 belongs to conv{𝐴₁,𝐴₂,𝐴₃}.

If 𝐴₁,…,𝐴_𝑛+1∈ℝ^𝑛 are points such that the points 𝐴₁−𝐴_𝑛+1,…,𝐴_𝑛−𝐴_𝑛+1 are linearly independent, then the convex hull conv{𝐴₁,…,𝐴_𝑛+1} is called the 𝑛-simplex with vertices 𝐴₁,…,𝐴_𝑛+1. Geometrically speaking, all the simplex vertices cannot belong to the same hyperplane. If 𝐴_𝑘=𝐴_𝑘(𝑥_𝑘1,…,𝑥_𝑘𝑛), then any point 𝑃(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛 is presented by the unique affine combination 𝑃=^𝑛+1_𝑘=1𝛼_𝑃𝑘𝐴_𝑘, (27) where the numerators 𝛼^num_𝑃𝑘 and denominators 𝛼^den_𝑃𝑘 of the coefficients 𝛼_𝑃𝑘 are 𝛼^num_𝑃𝑘=(−1)^𝑘+1𝑥₁…𝑥_𝑛1𝑥₁₁…𝑥_1𝑛1𝑥⋮⋱⋮⋮_𝑘−11…𝑥_𝑘−1𝑛1𝑥_𝑘+11…𝑥_𝑘+1𝑛1𝑥⋮⋱⋮⋮_𝑛+11…𝑥_𝑛+1𝑛1,𝛼^den_𝑃𝑘=𝑥₁₁…𝑥_1𝑛1𝑥⋮⋱⋮⋮_𝑘−11…𝑥_𝑘−1𝑛1𝑥_𝑘1…𝑥_𝑘𝑛1𝑥_𝑘+11…𝑥_𝑘+1𝑛1𝑥⋮⋱⋮⋮_𝑛+11…𝑥_𝑛+1𝑛1. (28)

In what follows we assume that conv{𝐴₁,…,𝐴_𝑛+1}⊂ℝ^𝑛 is the 𝑛-simplex with vertices 𝐴₁,…,𝐴_𝑛+1.

The generalization of Lemma 13 (in 𝑛 dimensions) to several points 𝐶_𝑗 stands as follows.

Theorem 14. Let 𝐶_𝑗∈conv{𝐴₁,…,𝐴_𝑛+1} be points for 𝑗=1,…,𝑚. Let 𝛼_𝑘,𝛾_𝑗∈[0,1] and 𝛾∈[−1,1] be coefficients such that all 𝛼_𝑘≥−𝛾 and ∑^𝑛+1_𝑘=1𝛼_𝑘+𝛾=∑^𝑚_𝑗=1𝛾_𝑗=1.
Then the affine combination ^𝑛+1_𝑘=1𝛼_𝑘𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗𝐶_𝑗∈conv𝐴₁,…,𝐴_𝑛+1. (29)

Corollary 15. Let 𝑔∶conv{𝐴₁,…,𝐴_𝑛+1}→ℝ^𝑛 be a monotone mapping (either nondecreasing in each variable or nonincreasing in each variable), and let 𝑃_𝑗∈conv{𝐴₁,…,𝐴_𝑛+1} be points for 𝑗=1,…,𝑚. Let 𝛼_𝑘,𝛾_𝑗∈[0,1] and 𝛾∈[−1,1] be coefficients such that all 𝛼_𝑘≥−𝛾 and ∑^𝑛+1_𝑘=1𝛼_𝑘+𝛾=∑^𝑚_𝑗=1𝛾_𝑗=1.
Then the affine combination ^𝑛+1_𝑘=1𝛼_𝑘𝑔𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗𝑔𝑃_𝑗∈conv𝑔𝐴₁,…,𝑔𝐴_𝑛+1. (30)

Theorem 16. Let 𝑔∶conv{𝐴₁,…,𝐴_𝑛+1}→ℝ^𝑛 be a monotone mapping (either nondecreasing in each variable or non-increasing in each variable), and let 𝑃_𝑗∈conv{𝐴₁,…,𝐴_𝑛+1} be points for 𝑗=1,…,𝑚. Let 𝛼_𝑘,𝛾_𝑗∈[0,1] and 𝛾∈[−1,1] be coefficients such that all 𝛼_𝑘≥−𝛾 and ∑^𝑛+1_𝑘=1𝛼_𝑘+𝛾=∑^𝑚_𝑗=1𝛾_𝑗=1.
Then every continuous convex function 𝑓∶conv{𝑔(𝐴₁),…,𝑔(𝐴_𝑛+1)}→ℝ satisfies the inequality 𝑓^𝑛+1_𝑘=1𝛼_𝑘𝑔𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗𝑔𝑃_𝑗≤^𝑛+1_𝑘=1𝛼_𝑘𝑓𝑔𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗𝑓𝑔𝑃_𝑗. (31)

Proof. Let us prove the case 𝑛=2.
If 𝑓 is monotone (either nondecreasing in both variables or nonincreasing in both variables), then we apply the McShane inequality in (3) to the function space 𝕏=ℎ∶conv𝐴₁,𝐴₂,𝐴₃⟶ℝ∣ℎ=𝑐₁𝑔+𝑐₂𝑓𝑔+𝑐₃;𝑐₁,𝑐₂,𝑐₃∈ℝ (32) and linear functional 𝑀 on 𝕏 defined by 𝑀(ℎ)=³_𝑘=1𝛼_𝑘ℎ𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗ℎ𝑃_𝑗. (33) The functional 𝑀 is unital since ∑³_𝑘=1𝛼_𝑘+𝛾∑^𝑚_𝑗=1𝛾_𝑗=1 and positive since the function ℎ=(𝑐₁+𝑐₂𝑓)(𝑔)+𝑐₃ is monotone.
If 𝑓 is not monotone, then we translate 𝑓 so that its minimum 𝑓(𝑥₀,𝑦₀)=0 and divide the triangle conv{𝑔(𝐴₁),𝑔(𝐴₂),𝑔(𝐴₃)} into 2²=4 parts considering the lines 𝑥=𝑥₀ and 𝑦=𝑦₀. Then we express 𝑓 as the sum of the monotone convex functions, 𝑓=𝑓₁+𝑓₂+𝑓₃+𝑓₄, and apply the previous procedure to these functions.
The proof goes in the same way for any integer 𝑛.

Applying the identity mapping 𝑔(𝑃)=𝑃 for 𝑃∈conv{𝐴₁,…,𝐴_𝑛+1} to the inequality in (31), it reads as follows: 𝑓^𝑛+1_𝑘=1𝛼_𝑘𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗𝑃_𝑗≤^𝑛+1_𝑘=1𝛼_𝑘𝑓𝐴_𝑘+𝛾^𝑚_𝑗=1𝛾_𝑗𝑓𝑃_𝑗. (34)