Continuous refinements of some Jensen-type inequalities via strong convexity with applications

In this paper we prove new continuous refinements of some Jensen type inequalities in both direct and reversed forms. As applications we also derive some continuous refinements of Hermite–Hadamard, Hölder, and Popoviciu type inequalities. As particular cases we point out the corresponding results for sums and integrals showing that our results contain both several well-known but also some new results for these special cases.

The main aim of this paper is to further complement and develop 1. and 2. by first proving some new continuous versions of Jensen type inequalities in both direct and reversed form by using the concept of strong convexity. Moreover, as applications we derive some corresponding continuous Hermite-Hadamard, Hölder, and Popoviciu type inequalities. For another interesting use of strongly convex functions, we also refer to [11].
In this paper we use some usual notations for measure spaces. Let (X, μ) and (Z, λ) be two probability measure spaces. Let α : X × Z → [0, ∞ be a measurable mapping such that X α(x, z) dμ(x) = 1 for each z ∈ Z (1) and Z α(x, z) dλ(z) = 1 for each x ∈ X.
In [20] a continuous refinement of the Jensen inequality is given.
Theorem 1.1 Let (X, μ) and (Z, λ) be two probability measure spaces, and let α : X × Z → [0, ∞ be a measurable function on X ×Z satisfying (1) and (2). If ϕ is a real convex function on the interval I ⊆ R, then for the function f : If ϕ is concave, then the reversed signs of the inequalities hold in (3).
If λ is a discrete measure, the refinement of the Jensen inequality has been rediscovered recently, see e.g. [7], while similar results can be found in [5,6,17] for some particular cases of α.
One of our objectives is to give results for strongly convex functions. So, let us evoke a definition and some useful facts about that class of functions. Definition 1.2 Let I be an interval of the real line. A function ϕ : I → R is called a strongly convex function with modulus c > 0 if The theory of strongly convex functions is vast, but here we point out only a very useful characterization of it. Namely, a function ϕ is strongly convex with modulus c > 0 if and only if the function ψ(x) = ϕ(x)cx 2 is convex [11]. The Jensen inequality for strongly convex functions is given in its discrete and integral form in [9]. A slightly modified result is the following theorem. Theorem 1.3 Let (X, μ) be a probability measure space, I be an interval in R. Let ϕ : I → R be a strongly convex function with modulus c > 0, and let f : X → I be a function such that wheref = X f dμ.
The paper is organized as follows: in Sect. 2 we derive the announced refinement of the Jensen inequality (see Theorem 2.1). In order to be able to see that our results generalize and unify some other recent results in the literature (see [5,7,17], and [21]), we point out some more or less direct consequences of Popoviciu inequalities are discussed and proved (see Theorem 5.1). Finally, in order to put more light on the "gaps" in some of our inequalities, we use Sect. 6 to derive some important properties of the functionals describing these gaps (see Theorems 6.1 and 6.2).

Refinements of the Jensen inequality for strongly convex functions
The main result in this section reads as follows.
Proof Since the function ϕ is strongly convex, the function ϕc(·) 2 is convex, and for it the refinement (3) holds. Therefore, after adding the term c( X f dμ) 2 = cf 2 on each side of the refinement of the Jensen inequality, we get Let us transform the term X f 2 dμ -f 2 as follows: We denote F(z) := X f (x)α(x, z) dμ(x). Using the Fubini theorem and the properties of the weight α, we obtain Under this notation and using the same method as previously, we find that the second term in the middle expression of (6) is equal to By combining (6)-(8) we obtain (5), and the proof is complete.
Remark 2.2 Since c > 0, the chain of inequalities in (5) can be followed by ≤ X (ϕ • f ) dμ. So Theorem 2.1 is indeed a genuine refinement of the Jensen inequality.
It is interesting to state the corresponding refinements for some particular cases such as for discrete and for integral Jensen's inequality with finitely many functions.
(ii) By applying Theorem 2.1 for the same substitutions for Z and λ as we did in the proof of the first part, together with the following: the inequalities in (5) and trivial arguments give (10). The proof is complete.
Also, we state a refinement with finitely many functions with partition of the space X. Namely, we get the following.

Corollary 2.4
Let the assumptions of Theorem 1.3 hold. Let X 1 , . . . , X n be a partition of the set X. Let μ have the additional property that X i dμ = 0, i = 1, 2, . . . , n.
Then, for any strongly convex function ϕ : I → R with modulus c, the following refinement of the Jensen inequality holds: wheref = X f dμ.
Proof Let us use the same partition of the space Z as in the proof of Corollary 2.3, where the functions α i are defined as Here, χ S denotes the characteristic function of the set S. Then the assumptions of Theorem 2.1 are satisfied. Hence, (5) and a trivial estimate show that the inequalities in (11) hold. The proof is complete.
If in Corollary 2.4 we put X = [a, b], a = a 0 < a 1 < a 2 < · · · < a n = b and X i = [a i-1 , a i for i = 1, 2, . . . , n, dμ = w(x) W dx, then the inequalities in (11) become as follows: where Remark 2.5 When the function ϕ is convex i.e. when c = 0, some of the above-mentioned results are already known. The refinement via two functions α 1 , α 2 , α 1 + α 2 = 1 involving integrals for convex function ϕ has been published very recently in paper [7] together with applications in the information theory. The results for finite sequences for convex function ϕ are given in [21].
The result of Corollary 2.4 for n = 2 and c = 0 is the main result in paper [5]. If c = 0 i.e. if ϕ is a convex function, then the corresponding result of (12) is given in the paper [17].

Refinements of the reverse Jensen inequality for convex and strongly convex functions
The simplest form of the reverse Jensen inequality is the following inequality where one weight is positive while the second one is negative: Let ϕ be a real convex function on I. If p and q are positive numbers such that pq > 0, then for all a, b ∈ I such that pa-qb p-q ∈ I. This follows from the definition of a convex function: The reverse Jensen inequality for integrals follows from Lemma 4.25 in the book [18, p. 124] and has the following form.
If ϕ is concave, then the reversed inequality holds.
The most known consequences of the previous inequality are the Popoviciu inequality and the Bellman inequality, which are reversed inequalities of the Hölder and the Minkowski inequalities, respectively. Here we give a proof of it since we will use one step of that proof in our further investigation.
Proof By putting in (13) we obtain that where in the last inequality we use the Jensen inequality for integral.
The following theorem is a continuous refinement of the previously mentioned reverse Jensen inequality for integrals. Theorem 3.2 (Continuous refinement of the reverse Jensen inequality for convex function) Let the assumptions of Theorem 1.1 hold. Additionally, let u 0 ∈ R be such that u 0 > 1.

Let ϕ be a real convex function on an interval I and f 0 ∈ I. Let f be a function on X such that f and ϕ • f are integrable and
If ϕ is concave, then the reversed signs of the inequalities hold.
Proof Using the first inequality in (15) and the result of Theorem 1.1, we get (17) and the proof is complete.
By using the previous theorem, it is easy to obtain a continuous refinement of the reverse Jensen inequality for a strongly convex function.

Theorem 3.3 (Refinement of the reverse Jensen inequality for a strongly convex function)
Let the assumptions of Theorem 1.1 and Theorem 3.2 hold. Then, for the strongly convex function ϕ, the following holds: where f = X f dμ.
Proof By applying the result of Theorem 3.2 for the convex function ϕc(·) 2 , we get the desired result.

Refinements of some Hermite-Hadamard inequalities for strongly convex functions
As a first application of Theorem 2.1 we note the following: a particular choice of the functions w and f gives a continuous refinement of the left-hand side of the well-known Hermite-Hadamard inequality. In fact, by putting in (5) where α satisfies (1) and (2). The inequality between the first and the third term in chain (18) is already known. It is the left-hand side of the Hermite-Hadamard inequality for a strongly convex function, and it is given in [9]. Hence, (18) is a continuous refinement of this result.
A discrete refinement of the left-hand side of the Hermite-Hadamard inequality for a convex function is given in [17]. Here we give a generalization of it, namely, a discrete refinement of the left-hand side of the Hermite-Hadamard inequality for a strongly convex function. It follows from (12) applied with w(x) = 1 and f (x) = x for x ∈ [a, b]: The particular case of (19) for n = 2, a 0 = a, a 1 = a+b 2 , a 2 = b is given in [4]. The refinement of the right-hand side of the Hermite-Hadamard inequality is based on the Lah-Ribarič inequality, and we cannot directly obtain a continuous refinement. A discrete refinement of the right-hand side of the Hermite-Hadamard inequality for a convex function is given in [17]. Here we derive a refinement of the Lah-Ribarič inequality for a strongly convex function, which follows from the result for a convex function applied with the convex function ϕc(·) 2 .
Proof (i) If ϕ is a strongly convex function, then the function ϕc(·) 2 is convex. Putting in Theorem 2.3 from paper [17] ϕc(·) 2 instead of f , after simple calculations, we get the statement of this theorem.
As we can see, the chain of inequalities (20) is a refinement of the right-hand side of the Hermite-Hadamard inequality for a strongly convex function. If n = 2, a 0 = a, a 1 = a+b 2 , a 2 = b, we get the result from [4, Theorem 5].

Refinements of the Hölder and Popoviciu inequalities
It is known that the Hölder inequality is a consequence of the Jensen inequality for an appropriate function ϕ. In the further text we use a version of (3) given for general measure μ not only for a probability measure. In that case (3) has the following form: where ϕ is convex, w : X → [0, ∞ is a measurable function such that W := X w dμ = 0 and If ϕ is concave, then the reversed signs of the inequalities hold in (21). If r, s > 1 are numbers such that 1 r + 1 s = 1, then (21) for the concave function ϕ(x) = x 1/r with the substitutions w = wg s , f = f r g -s , where α satisfies assumption (22) and First we state the following complementary information about the "gaps" in inequalities (21). Moreover, if p, q ∈ K ϕ,f ,α satisfy p ≤ q, then J 1 (p) ≤ J 1 (q) and J 2 (p) ≤ J 2 (q).
Proof Since ϕ is a convex function, we get L J (p + q) = (P + Q)ϕ 1 P + Q X (p + q)f dμ = (P + Q)ϕ P P + Q where P = X p dμ and Q = X q dμ. Let us denote S(w) := W · ϕ( 1 W X f (x)α(x, z)w(x) dμ(x)), where W = X w dμ. Using the same method as in the first part of the proof, we get that S(p + q) ≤ S(p) + S(q). By integrating the terms in this inequality over Z, we get the subadditivity of M J .
In literature, the functional J 1 is called the Jensen functional, and its superadditivity is already known. Also, a lot of results connected with J 1 are given in [8]. The superadditivity of J 2 follows from the linearity of R J and the subadditivity of M J .
By using the results of Sect. 3, we see that the functionals J 1 and J 2 are nonnegative on K ϕ,f ,α . If p ≤ q, then from the superadditivity of J i , i = 1, 2, we get J i (q) = J i p + (qp) ≥ J i (p) + J i (qp) ≥ J i (p) i.e. J i , i = 1, 2, are nondecreasing functionals. The proof is complete. Now we define the following functionals, which are connected with the reverse of the Jensen inequality: C(w 0 ) := w 0 ϕ(f 0 ), P 1 (w 0 , w) := C(w 0 ) -R J (w) -K J (w 0 , w),