A Parameter-dependent Refinement of the Discrete Jensen's Inequality for Convex and Mid-convex Functions

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.


Introduction and the main results
The considerations of this paper concern (A 1 ) an arbitrarily given real vector space X, a convex subset C of X, and a finite subset {x 1 ,..., x n } of C, where n ≥ 1 is fixed; (A 2 ) a convex function f : C ℝ, and a discrete distribution p 1 ,..., p n , which means that p j ≥ 0 with n j=1 p j = 1 ; (A 3 ) a mid-convex function f : C ℝ, and a discrete distribution p 1 ,..., p n with rational p j (1 ≤ j ≤ n). The function f : C ℝ is called convex if and mid-convex if Various attempts have been made to refine inequality (2) in the following ways: Assume either (A 1 ) and (A 2 ) or (A 1 ) and (A 3 ). 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 I such 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 f ⎛ ⎝ n j=1 p j x j ⎞ ⎠ ≤ · · · ≤ B k ≤ · · · ≤ B 1 = n j=1 p j f (x j ), k ∈ I. (3) (C) Create an increasing real sequence (C k ) k I such that C k = C k (f, x i , p i ) (k I) is a sum whose index set is a subset of {1,..., k} n and f ⎛ ⎝ n j=1 p j x j ⎞ ⎠ = C 1 ≤ · · · ≤ C k ≤ · · · ≤ n j=1 p j f (x j ), k ∈ I.
The next two typical results belong to the group of refinements of type (B).
These examples use p 1 = · · · = p n = 1 n . In [2], Pečarić and Volenec have constructed the sequence while the other sequencē 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 wellknown 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 which satisfies (4) with p 1 = · · · = p n = 1 n . In this paper, we establish a new solution of the problem (C). The constructed sequence (C k (l)) k≥0 depends on a parameter l belonging to [1, ∞[, and we can use arbitrary discrete distribution p 1 ,..., 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: (A 4 ) Let l ≥ 1. (A 5 ) Let l ≥ 1 be rational. First, we give a refinement of the discrete Jensen's inequality (2).
(b) It is easy to see that Finally, we establish two convergence theorems.  (b) The function l C k (l) (l ≥ 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 (A 1 ), (A 2 ), and (A 4 ) are satisfied. Really, if f is mid-convex and continuous, then convex. By (9) We come now to the second convergence theorem.
(a) k = 1, n N + : (c) p 1 = · · · = p n := 1 n : Assume further that f is strictly convex (strictly mid-convex) that is strict inequality holds in (1) whenever x ≠ y and 0 < a <1. In this case, equality is satisfied in (2) if and only if x 1 = ··· = x n , and therefore, it comes from the third part of the proof of Theorem 1 that if not all x i are equal.
If p 1 = · · · = p n := 1 n 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 l The function l C k (l) (l ≥ 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 p 1 ,..., p n be a discrete distribution, and let g, h : I ℝ be continuous and strictly monotone functions. Let l ≥ 1. We define the quasi-arithmetic means with respect to (8) by Some other means are also needed. Definition 8 Let I ⊂ ℝ be an interval, let x j I (1 ≤ j ≤ n), and let p 1 ,..., p n be a discrete distribution. For a continuous and strictly monotone function z : I ℝ, we introduce the following mean 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 p 1 ,..., p n be a discrete distribution, and let g, h : I ℝ be continuous and strictly monotone functions.
Let l ≥ 1. Then, if either h ○ g -1 is convex and h is increasing or h ○ g -1 is concave and h is decreasing.
if either h ○ g -1 is convex and h is decreasing or h ○ g -1 is concave and h is increasing.
which gives a sharpened version of the arithmetic meangeometric 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 p 1 ,..., 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 p 1 ,..., p n be a discrete distribution with p j >0 (1 ≤ j ≤ n). Let l ≥ 1, and k N. We define the mixed symmetric means with respect to (8) by where t ℝ. The monotonicity and the convergence of the previous means are studied in the next result.
Proof. The lowest common denominator is i 1 !... i n !. Combined with n j=1 i j = k + 1 , the result follows. ■ The proof of Theorem 1.
Proof. (a) We separate the proof of this part of the theorem into three steps. Let l ≥ 1 be fixed.
It is easy to check that for every (i 1 ,..., i n ) S k n j=1 λ i j p j x j n j=1 λ i j p j With the help of Theorem A, this yields that Consequently, By Lemma 14, it is easy to see that the right-hand side of (13) can be written in the form III. Finally, we prove that It follows from Theorem A that The multinomial theorem shows that hence (15) implies (14). ■ The proof of Theorem 3 (a) is based on the following interesting result. The s-algebra of Borel subsets of ℝ n is denoted by B n .
Lemma 15 Let p 1 ,..., p n be a discrete distribution with n ≥ 2, and let l >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 q 1 ,..., q n be also a discrete distribution such that q j >0 (1 ≤ j ≤ n) and q l > max(q 1 , . . . q l−1 , q l+1 , . . . , q n ). (16) If g : is a bounded function for which τ l := lim e l g exists in ℝ and p l >0, then Proof. To prove the result, we can obviously suppose that l = 1.
For the sake of clarity, we shall denote the element (i 1 ,..., 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 , B n )-random variable on a probability space ( , A, P) such that ξ k has multinomial distribution of order k and with parameters q 1 ,..., 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 where F n-1 means the distribution function of the Chi-squared distribution (c 2 -distribution) with n -1 degrees of freedom.
Choose 0 < ε < 1. Since F n-1 is continuous and strictly increasing on ]0, ∞[, there exists a unique t ε >0 such that The definition of the set S 1 k shows that For j = 1,..., n construct the sequences (I j k ) k≥1 by We claim that Fix 1 ≤ j ≤ n. If (23) is false, then (22) yields that we can find a positive number r, a strictly increasing sequence (k u ) u≥1 and points such that and therefore, contrary to (24). Let q := max (q 2 , . . . , q n ).
It follows from (16) that By (22) and (23), we can find an integer k g such that for each k > k g Thus, for every k > k g i 1k k > q 1 − γ and i jk k < q j + γ , 2 ≤ j ≤ n, (i 1k , . . . , i nk ) ∈ S 1 k , and hence, we get from (25) that We can see that Now, set S 2 k := S k \S 1 k (k ∈ N + ), and consider the sequences where k N + . The sum of these sequences is just the studied sequence in (17).
By the definition of the set S k , (0,..., 0, k, 0,..., 0) (the vector has 0s in all coordinate positions except the lth) 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. ■