Some Generalizations of the Jensen-Type Inequalities with Applications

: Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure d λ which is not necessarily positive using several Green functions. Utilizing these results we deﬁne some new mean value theorems of Lagrange and Cauchy types, and derive some new Cauchy-type means


Introduction
First papers about convex functions date from the end of the nineteenth century, but the real meaning of the convex functions has been developed with the works of the Danish mathematician J. L. W. V. Jensen (1859Jensen ( -1925) ) from 1905 and 1906 [1], and his famous inequality.In the many years after that, the Jensen inequality was considered under the weakened conditions, further improved, generalized, refined, reversed, etc., and it has been (and it is still) a constant inspiration for further investigation.By weakening of the conditions of the Jensen inequality, the Jensen-Stefensen inequality was derived.H. D. Brunk generalized it further in [2] and this result is known as the Jensen-Brunk inequality.Another generalization is the Jensen-Boas inequality (see [3]).These and many more results can be found in [4], an excellent book about convex functions.Also, many other famous inequalities are derived using the Jensen inequality, like the Cauchy inequality, the Hölder inequality, the Young inequality, inequalities between means, to name just a few.The applications of all of these inequalities are widely spread in different mathematical areas and so the influence and importance of the Jensen inequality are immeasurable.An interested reader can examine several old and brand new papers which use this inequality (for example [5][6][7][8][9][10][11][12][13]).
Though, the results that consider the Jensen inequality, its variants, reverses, converses, and refinements, consider the case when the measure is positive.Therefore it is of great interest to get the results where it is allowed that the measure can also be negative.In order to do that, we used the following set of the Green functions These functions have certain nice properties.Due to them, we already got some new results for the inequalities of the Jensen-type and the inequalities of the converse Jensen-type (see for example [14,15]) and there are more to come.
Taking into account the properties of the functions G p (p = 1, 2, 3, 4), and motivated by the results from S. S. Dragomir in [16,17] with reverses of the Jensen inequality in the case when the measure is positive, here in this paper we will give the generalization of some of his results on the Jensen-type inequalities and the converse Jensen-type inequalities, but now allowing that the measure can also be negative.The results that are presented here, represent the continuation of the research presented in [18,19].
This paper is structured in the following way.After this introduction, the section with the main results follows.There are given three theorems where the Jensen-type (or converse Jensen-type) inequality for the continuous convex function is connected to the similar inequality for the Green function G p (p = 1, 2, 3, 4).In the next, third, section these theorems are then used to define new Lagrange and Cauchy type mean-value theorems, again connecting our function with the Green functions G p (p = 1, 2, 3, 4).The fourth section presents applications of our results in the direction of constructing exponentially convex functions and deriving new Cauchy means.

Main Results
As we already mentioned in the introduction, the functions G p (p = 1, 2, 3, 4) have some interesting and very useful properties, and they will also be very important for deriving our results in this paper.It is not difficult to see that for every fixed value s ∈ ) can be expressed using these functions G p (p = 1, 2, 3, 4) as follows: These representations can be easily proved after a short calculation by integrating by parts, but the interested reader can also consult [14] (Lemma 1.1).
We will now give our main results, and for the sake of the clearer notation we will use .
To begin with, we give an improvement of the Jensen inequality.
if and only if for every s ∈ where the functions G p (p = 1, 2, 3, 4) are defined in (1)-( 4).Moreover, this equivalence holds also in the case if we reverse the inequality sign in both (9) and (10). Proof.
After some calculation, for every function G p (p = 1, 2, 3, 4) we obtain the following relation If φ is convex function, then for every s ∈ [α, β] we have that φ (s) ≥ 0. Further, if for every s ∈ [α, β] holds (10), then the term in the square brackets in (15) is less then or equal to zero.That means that the left hand side of (15) also has to be less then or equal to zero, which means that for every continuous convex function φ : (9).After all, also notice that the existence of the second derivative of the function φ is not necessary, because it is possible to uniformly approximate a continuous convex function by convex polynomials (see also [4], p. 172).
We will skip the proof of the last sentence of our theorem since it can be conducted in the same way.
Remark 2. The previous theorem considers the case when the function φ is convex.Concluding in a similar way, we can also get the results that hold for concave function.Under the conditions of the previous theorem, we have the following.
Our next theorem states a similar result for the converse Jensen inequality, in the literature also known as the Lah-Ribarič or Edmundson-Lah-Ribarič inequality.
if and only if for every s ∈ where the functions G p (p = 1, 2, 3, 4) are defined in (1)-( 4).Moreover, this equivalence holds also in the case if we reverse the inequality sign in both ( 16) and (17).
As in the proof of Theorem 1, the same remark about the differentiability condition here also holds.And the comment about proving the last sentence of this theorem holds also here.

Remark 3.
Note that here we don't need the condition that g ∈ [α, β].Remark 4. Also, in this case, concluding in a similar way, we can get the results that hold for the concave function.Under the conditions of the previous theorem, we have the following.
In our third theorem we have again an improvement of the Jensen inequality, but now without derivatives.
if and only if for every s ∈ where the functions G p (p = 1, 2, 3, 4) are defined in (1)-( 4).Moreover, this equivalence holds also in the case if we reverse the inequality sign in both (19) and (20).
Proof.This proof goes similarly to the previous two.The first implication is obvious.
The proof of the last sentence of this theorem is analogous.The previous three theorems and inequalities ( 9), ( 16) and (19) offer us the possibility of formulating new Lagrange-type and Cauchy-type mean value theorems, and these mean value theorems can give us a possibility to derive some new means.
Firstly, for easier notation and formulation of these results, for functions g and λ and for continuous convex function φ : [α, β] → R, we will define three functionals by subtraction of the right-hand from the left-hand side of inequalities ( 9), ( 16) and ( 19): with a remark that for A 1 (g, λ, φ) and A 3 (g, λ, φ) the value of g has to be in [α, β].
From Theorems 1-3 we obtain that: For each of these functionals we will now derive the Lagrange-type mean value theorem and after that also Cauchy-type mean value theorem.
As for φ the relation ( 15) is valid, we can apply the integral mean-value theorem on it, and obtain that for that p ∈ {1, 2, 3, 4} there exists ξ ∈ Now we have to calculate the integral on the right side.Suppose that p = 1.We have that and Calculating the right side of (21) we obtain what proves the statement of our theorem.
For other p we proceed the same way.For p = 2 we have that In all these cases direct calculation brings us to the same conclusion which proves our theorem.
In the next two theorems we have the Lagrange mean-value theorems for the functionals A 2 (g, λ, φ) and A 3 (g, λ, φ).We give these results here without the proofs as these proofs are conducted analogously.
Proof.We will prove this theorem only for the functional A 1 , as the other cases can be proved analogously.
Let us define the function χ as follows: On this new function we can also apply Theorem 4, because it is the linear combination of the functions φ and ψ.After short calculation we obtain that there exists ξ ∈ [α, β] such that where φ 0 (t) = t 2 .The term A 1 (g, λ, φ 0 ) has to be different from zero, because, otherwise, we would have a contradiction with the condition that the denominator of the left side is not equal to zero, and consequently, we get the statement of our theorem.
Remark 6.If our functions φ and ψ are such that there exists the inverse function of φ /ψ , then we have what brings us to new Cauchy means.

Applications
In order to round off this paper, we would like to present some applications.When we speak about the mean-value theorems it is somehow most natural to derive some means.In order to get the Cauchy-type means with certain nice properties, we will use the method from the paper [20], which will firstly help us to define some new exponentially convex functions, and then to define the new means.
At the very beginning of this section, before stating our results, we have to recall some of the very basic definitions and facts about exponential convexity.Throughout this section, with I we will denote an open interval in R. Remark 8.Here we mention also some examples of exponentially convex functions from [20], as we will need them in the process of constructing our means: (i) f : I → R defined by f (x) = ce rx , where c ≥ 0 and r ∈ R.
Remark 9. A positive function f : I → R + is log-convex in the Jensen sense on I if and only if it is 2−exponentially convex in the Jensen sense on I, i.e., if and only if for every ρ 1 , ρ 2 ∈ R and x 1 , x 2 ∈ I holds If such function is also continuous on I, it follows that it is log-convex on I.
We also recall the following two lemmas from [4], p. 2.
We have to recall also the definition of the divided difference of the second order.
Definition 3. The divided difference of the second order of a function f : I → R at mutually different points x 0 , x 1 , x 2 ∈ I is defined recursively by Remark 10.The value [x 0 , x 1 , x 2 ] f is independent of the order of the points x 0 , x 1 and x 2 .Also, we can extended this definition to include the case when some or all of the points are equal ([4], p. 14).Taking the limit x 1 → x 0 in (22), we obtain lim assuming that f exists.Further, taking the limits x i → x 0 , i = 1, 2, in (22), we obtain lim assuming that f exists.A function f : I → R is convex if and only if for every choice of three mutually different points Now, we can start with our results.We will take s certain family of functions, then apply our functionals to it, and in this way, we will construct n−exponentially convex and exponentially convex functions.

Proof. Let us define the function χ by
Proof.(i) From Theorem 8 we get that the first sentence is valid, and the log-convexity follows from Remark 9. We still have to prove (25).We have that the function u → F i (g, λ, φ u ) is strictly positive, and we can apply Lemma 1 on the function f (x) = log F i (g, λ, φ x ).We have that ) and therefore inequality (23) holds.
We get the cases u = v and y = z as limit cases from (26).

Remark 11.
When two or all of the points x 0 , x 1 , x 2 ∈ [α, β] are equal, the results from Theorem 8, Corollaries 1 and 2 are also valid.The proofs for that can be obtained using Remark 10 and adequate characterization of convexity.
Now we will look at some families of functions that fulfill the assumptions of Theorem 8, Corollaries 1 and 2, and using them we will get some Cauchy-type means.

Example 1. Let us define a family of functions
It is d 2 dx 2 ψ u (x) = e ux > 0 for x ∈ R, and so for every u ∈ R the function ψ u is convex on R. Remark 8 gives us that u → d 2 dx 2 ψ u (x) is exponentially convex, and from [20] we also have that u → [x 0 , x 1 , x 2 ]ψ u is exponentially convex and therefore exponentially convex in the Jensen sense.That means that the family Γ 1 of functions ψ u fulfills the assumptions from Corollary 1, and therefore we have that for i = 1, 2, 3 functions u → F i (g, λ, ψ u ) are exponentially convex in the Jensen sense.Although u → ψ u is not continuous at u = 0, these functions are continuous, and it follows that they are exponentially convex.
Applying Corollary 2 on Γ 1 , we obtain and from (24) we conclude that they are monotone in parameters u and v.
Using the Cauchy-type theorem from Section 3, applied for φ = ψ u ∈ Γ 1 and ψ = ψ v ∈ Γ 1 , we get that for which means that then M u,v (g, F i , Γ 1 ) are means of the function g.From (24) we have that M u,v (g, F i , Γ 1 ) are also monotone.

Example 2. Let us define a family of functions
We have that d 2 dx 2 φ u (x) = x u−2 = e (u−2) log x > 0, and so for every u ∈ R the function φ u is convex for x > 0. From Remark 8 we have that u → d 2 dx 2 φ u (x) is exponentially convex, and from [20] we have that u → [x 0 , x 1 , x 2 ]φ u is exponentially convex and therefore exponentially convex in the Jensen sense.This means that Γ 2 fulfills the assumptions from Corollary 1.Now we will assume that [α, β] from Corollaries 1 and 2 is a subset of R + , and we obtain that As in the previous example, we have that functions u → F i (g, λ, φ u ) (i = 1, 2, 3) are exponentially convex, and that µ u,v (g, F i , Γ 2 ) are monotone.

Conclusions
The results of the Jensen inequality, its variants, reverses, converses and refinements, always consider the case when the measure is positive.Therefore, it is very interesting to get the results where it is allowed that the measure can also be negative, and that is done in this paper.This paper represents a further continuation of the research already published in papers [18,19].Motivated by the results from [16,17] about the reverses of the Jensen inequality and converse Jensen inequality for positive measure, in this paper we used the Green functions G p (p = 1, 2, 3, 4) (defined in (1)-( 4)) to give generalizations of these results for real Stieltjes measure dλ which does not necessarily have to be positive.These results are then used for defining new mean-value theorems of Lagrange and Cauchy-type.As an application, these theorems are then used for constructing some new Cauchy means.

Definition 1 .Remark 7 .Definition 2 .
A function f : I → R is n−exponentially convex in the Jensen sense on I if n ∑ i,j=1 ρ i ρ j f x i + x j 2 ≥ 0 for all ρ i ∈ R and x i ∈ I, i = 1, . . ., n.A function f : I → R is n−exponentially convex if it is n−exponentially convexin the Jensen sense and continuous on I.Note that 1−exponentially convex functions in the Jensen sense are non-negative functions, as we can see from the definition.Further, n−exponentially convex functions in the Jensen sense are k−exponentially convex in the Jensen sense for every k ∈ N, k ≤ n.A function f : I → R is exponentially convex in the Jensen sense on I, if it is n−exponentially convex in the Jensen sense for all n ∈ N. A function f : I → R is exponentially convex if it is exponentially convex in the Jensen sense and continuous.