A parameter-dependent refinement of the discrete Jensen's inequality for convex and mid-convex functions

Abstract

In this paper, a new parameter-dependent refinement of the discrete Jensen's inequality is given for convex and mid-convex functions. The convergence of the introduced sequences is also studied. One of the proofs requires an interesting convergence theorem with probability theoretical background. We apply the results to define some new quasi-arithmetic and mixed symmetric means and study their monotonicity and convergence.

1 Introduction and the main results

The considerations of this paper concern

(A₁) an arbitrarily given real vector space X, a convex subset C of X, and a finite subset {x₁,..., x _n } of C, where n ≥ 1 is fixed;

(A₂) a convex function f : C → ℝ, and a discrete distribution p₁,..., p _n , which means that p _j ≥ 0 with ;

(A₃) a mid-convex function f : C → ℝ, and a discrete distribution p₁,..., p _n with rational p _j (1 ≤ j ≤ n).

The function f : C → ℝ is called convex if

(1)

and mid-convex if

For a variety of applications, the discrete Jensen's inequalities are important:

Theorem A. (see [1]) (a) If (A₁) and (A₂) are satisfied, then

(2)

(b) If (A ₁ ) and (A ₃ ) are satisfied, then (2) also holds.

Let ℕ := {0, 1, 2,...} and let ℕ₊ := {1, 2,...}.

Various attempts have been made to refine inequality (2) in the following ways: Assume either (A₁) and (A₂) or (A₁) and (A₃). Let m ≥ 2 be an integer, and let I denote either the set {1,..., m} or the set ℕ₊.

(B) Create a decreasing real sequence (B _k )_k∈Isuch that B _k = B _k (f, x _i , p _i ) (k ∈ I) is a sum whose index set is a subset of {1,..., n} ^k and

(3)

(C) Create an increasing real sequence (C _k )_k∈Isuch that C _k = C _k (f, x _i , p _i ) (k ∈ I) is a sum whose index set is a subset of {1,..., k}ⁿand

(4)

The next two typical results belong to the group of refinements of type (B).

These examples use . In [2], Pečarić and Volenec have constructed the sequence

(5)

while the other sequence

(6)

is due to Pečarić and Svrtan [3]. In a recent work, [4] Horváth and Pečarić define a lot of new sequences, they generalize and give a uniform treatment a number of well-known results from this area, especially (5) and (6) are extended. Horváth develops a method in [5] to construct decreasing real sequences satisfying (3). His paper contains some improvements of the results in [4] and gives a new approach of the topic. The description of the sequences in [4, 5] requires some work, so we do not go into the details. The problem (B) has been considered for the classical Jensen's inequality by Horváth [6].

We turn now to the group of refinements of type (C). In contrast to the previous problem, it is not easy to find such results. Recently, Xiao et al. [7] have obtained the sequence

(7)

which satisfies (4) with .

In this paper, we establish a new solution of the problem (C). The constructed sequence (C _k (λ))_k≥0depends on a parameter λ belonging to [1, ∞[, and we can use arbitrary discrete distribution p₁,..., p _n , not just the appropriate discrete uniform distribution. We give the limit of the sequence under fixed parameter. We also study the convergence of the sequence when the parameter varies and k ∈ ℕ is fixed. Finally, some applications are given which concern the theme of means.

The next theorems are the main results of this paper. We need some further hypotheses:

(A₄) Let λ ≥ 1.

(A₅) Let λ ≥ 1 be rational.

First, we give a refinement of the discrete Jensen's inequality (2).

Theorem 1 Suppose either (A ₁ ), (A ₂ ), and (A ₄ ) or (A ₁ ), (A ₃ ), and (A ₅ ). Introduce the sets

and for k ∈ ℕ define the numbers

(8)

Then,

Remark 2 (a) It follows from the definition of S _k that S _k ⊂ {0,..., k} ⁿ (k ∈ ℕ).

(b) It is easy to see that

(9)

Finally, we establish two convergence theorems.

Theorem 3 Suppose (A₁), (A₂), and (A₄). Suppose × is a normed space and f is continuous. Then,

(a) For every fixed λ > 1

(b) The function λ → C _k (λ) (λ ≥ 1) is continuous for every k ∈ ℕ.

The proof of Theorem 3(a) requires a lemma (see Lemma 15), which is interesting in its own right. Probability theoretical technique will be used to handle this problem.

Remark 4 In the previous theorem, it suffices to consider the case when (A₁), (A₂), and (A₄) are satisfied. Really, if f is mid-convex and continuous, then convex.

By (9)

We come now to the second convergence theorem.

Theorem 5 Suppose either (A₁), (A₂), and (A₄) or (A₁), (A₃), and (A₅). For each fixed k ∈ ℕ₊

2 Discussion and applications

Suppose either (A₁), (A₂), and (A₄) or (A₁), (A₃), and (A₅). First, we give three special cases of (8).

(a) k = 1, n ∈ ℕ₊:

(b) k ∈ ℕ, n = 2:

(c)

:

Assume further that f is strictly convex (strictly mid-convex) that is strict inequality holds in (1) whenever x ≠ y and 0 < α < 1. In this case, equality is satisfied in (2) if and only if x₁ = ··· = x _n , and therefore, it comes from the third part of the proof of Theorem 1 that

(10)

if not all x _i are equal.

If and f is strictly convex (strictly mid-convex), then the analysis of the proof of Theorem 1 shows that

whenever not all x _i are equal.

If the inequality (10) holds, X is a normed space and f is continuous (see Remark 4), then Theorem 3(b) and Theorem 5 insure that the range of the function λ → C _k (λ) (k ∈ ℕ₊) is the interval

Conjecture 6 Suppose either (A₁), (A₂), and (A₄) or (A₁), (A₃), and (A₅).

The function λ → C _k (λ) (λ ≥ 1) is increasing for every k ∈ ℕ.

Next, we define some new quasi-arithmetic means and study their monotonicity and convergence. About means see [8].

Definition 7 Let I ⊂ ℝ be an interval, let x _j ∈ I (1 ≤ j ≤ n), let p₁,..., p _n be a discrete distribution, and let g, h : I → ℝ be continuous and strictly monotone functions. Let λ ≥ 1. We define the quasi-arithmetic means with respect to (8) by

(11)

Some other means are also needed.

Definition 8 Let I ⊂ ℝ be an interval, let x _j ∈ I (1 ≤ j ≤ n), and let p₁,..., p _n be a discrete distribution. For a continuous and strictly monotone function z : I → ℝ, we introduce the following mean

(12)

We now prove the monotonicity of the means (11) and give limit formulas.

Proposition 9 Let I ⊂ ℝ be an interval, let x _j ∈ I (1 ≤ j ≤ n), let p₁,..., p _n be a discrete distribution, and let g, h : I → ℝ be continuous and strictly monotone functions. Let λ ≥ 1. Then,

(a)

if either h ○ g^-1is convex and h is increasing or h ○ g^-1is concave and h is decreasing.

(b)

if either h ○ g^-1is convex and h is decreasing or h ○ g^-1is concave and h is increasing.

(c) Moreover, in both cases

for each fixed λ > 1, and

for each fixed k ∈ ℕ₊.

Proof. Theorem 1 can be applied to the function h○g^-1, if it is convex (-h○g¹, if it is concave) and the n-tuples (g (x₁),..., g(x _n )), then upon taking h^-1, we get (a) and (b). (c) comes from Theorems 3(a) and 5. ■

As a special case, we consider the following example.

Example 10 If I :=]0, ∞[, h := ln, and g(x):= x (x ∈]0, ∞[), then by Proposition 9(b), we have the following inequality. for every x _j > 0 (1 ≤ j ≤ n), λ ≥ 1 , and k ∈ ℕ₊

which gives a sharpened version of the arithmetic mean - geometric mean inequality

Finally, we investigate some mixed symmetric means.

The power means of order r ∈ ℝ are defined as follows:

Definition 11 Let x _j ∈]0, ∞[ (1 ≤ j ≤ n), and let p₁,..., p _n be a discrete distribution with p _j > 0 (1 ≤ j ≤ n).

If r ≠ 0, then the power means of order r belong to the means (12) (z : ]0, ∞[→ ℝ, z(x) := x^r ), while we get the power means of order 0 by taking limit. Supported by the power means, we can introduce mixed symmetric means corresponding to (8):

Definition 12 Let x _j ∈]0, ∞[ (1 ≤ j ≤ n), and let p₁,..., p _n be a discrete distribution with p _j > 0 (1 ≤ j ≤ n). Let λ ≥ 1, and k ∈ ℕ. We define the mixed symmetric means with respect to (8) by

if s, t ∈ ℝ and s ≠ 0, and

where t ∈ ℝ.

The monotonicity and the convergence of the previous means are studied in the next result.

Proposition 13 Let x _j ∈]0, ∞[ (1 ≤ j ≤ n), and let p₁,..., p _n be a discrete distribution with p _j > 0 (1 ≤ j ≤ n). Let λ ≥ 1, and k ∈ ℕ. Suppose s, t ∈ ℝ such that s ≤ t. Then,

(a)

(b) In case of s, t ≠ 0

for each fixed λ > 1, and

for each fixed k ∈ ℕ₊.

Proof. Assume s, t ≠ 0. Then, Proposition 9 (b) can be applied with g, h :]0, ∞[→ ℝ, g(x) := x^t , and h(x) := x^s . If s = 0 or t = 0, the result follows by taking limit. ■

3 Some lemmas and the proofs of the main results

Lemma 14 Let k ∈ ℕ and (i₁,..., i _n ) ∈ S_k+1be fixed. If we set

then

Proof. The lowest common denominator is i₁!... i _n !. Combined with , the result follows. ■

The proof of Theorem 1.

Proof. (a) We separate the proof of this part of the theorem into three steps.

Let λ ≥ 1 be fixed.

I. Since S ₀ = {(0,..., 0)}

II. Next, we prove that C _k (λ) ≤ C _k+1(λ) (k ∈ ℕ).

It is easy to check that for every (i₁,..., i _n ) ∈ S _k

With the help of Theorem A, this yields that

Consequently,

(13)

By Lemma 14, it is easy to see that the right-hand side of (13) can be written in the form

which is just C_k+1(λ).

III. Finally, we prove that

(14)

It follows from Theorem A that

(15)

The multinomial theorem shows that

hence (15) implies (14). ■

The proof of Theorem 3 (a) is based on the following interesting result. The σ-algebra of Borel subsets of ℝ ⁿ is denoted by .

Lemma 15 Let p₁,..., p _n be a discrete distribution with n ≥ 2, and let λ > 1. Let l ∈ {1,..., n} be fixed. e _l denotes the vector in ℝ ⁿ that has 0s in all coordinate positions except the lth, where it has a 1. Let q₁,..., q _n be also a discrete distribution such that q _j > 0 (1 ≤ j ≤ n) and

(16)

If

is a bounded function for which

exists in ℝ and p _l > 0, then

(17)

Proof. To prove the result, we can obviously suppose that l = 1.

For the sake of clarity, we shall denote the element (i₁,..., i _n ) of S _k by (i_1k,..., i _nk ) (k ∈ ℕ₊).

Let ξ _k := (ξ_1k,..., ξ _nk ) (k ∈ ℕ₊) be a (ℝ ⁿ , )-random variable on a probability space such that ξ _k has multinomial distribution of order k and with parameters q₁,..., q _n . A fundamental theorem of the statistics (see [9], Theorem 5.4.13), which is based on the multidimensional central limit theorem and the Cochran-Fisher theorem, implies that

(18)

where F_n-1means the distribution function of the Chi-squared distribution (χ²-distribution) with n - 1 degrees of freedom.

Choose 0 < ε < 1. Since F_n-1is continuous and strictly increasing on ]0, ∞[, there exists a unique t _ε > 0 such that

Define

The definition of the set shows that

(19)

(20)

where by (18)

(21)

For j = 1,..., n construct the sequences by

(22)

We claim that

(23)

Fix 1 ≤ j ≤ n. If (23) is false, then (22) yields that we can find a positive number ρ, a strictly increasing sequence (k _u )_u≥1and points

(24)

such that

and therefore,

contrary to (24).

Let

It follows from (16) that

(25)

By (22) and (23), we can find an integer k _γ such that for each k > k _γ

Thus, for every k > k _γ

and hence, we get from (25) that

(26)

We can see that

(27)

Now, set , and consider the sequences

and

where k ∈ ℕ₊. The sum of these sequences is just the studied sequence in (17).

Since p₁> 0, we obtain from (27) that

(28)

and

(29)

According to (26), the convergence is uniform for all the possible sequences in (28) and (29); hence, for every ε₁> 0, we can find an integer that for all

(30)

Bringing in (19-20), we find that

and therefore, thanks to (19-20), (30) and the boundedness of g (|g| ≤ m)

Consequently, by (21)

and this proves the convergence claim (17).

The proof is now complete. ■

The proof of Theorem 3.

Proof. (a) We have only to observe that for every fixed 1 ≤ l ≤ n

(31)

The case p _l = 0 is trivial.

To prove the case p _l > 0, define the function

by

Consequently, the limit in (31) can be written in the form

Now, we can apply Lemma 15 with

and

(b) Elementary considerations show this part of the theorem.

The proof is complete. ■

The proof of Theorem 5.

Proof. Theorem A confirms that f is bounded on the set

where t _j (1 ≤ j ≤ n) is also rational if f is mid-convex.

It is elementary that for every (i₁,..., i _n ) ∈ S _k

By the definition of the set S _k , (0,..., 0, k, 0,..., 0) (the vector has 0s in all coordinate positions except the l th) is the only element of S _k for which i _l = k (1 ≤ l ≤ n). By using the boundedness of f on G, the previous assumptions imply the result, bringing the proof to an end. ■