Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.


Introduction and preliminary results
The well-known Jensen inequality asserts that for a convex function ϕ : I ⊆ R → R we have where x i ∈ I for i = , . . . , n, and p i are nonnegative real numbers such that P n = n i= p i > . Steffensen [] showed that inequality (.) also holds in the case when (x  , . . . , x n ) is a monotonic n-tuple of numbers from the interval I and (p  , . . . , p n ) is an arbitrary real n-tuple such that  ≤ P k ≤ P n (k = , . . . , n), P n > , where P k = k i= p i . His result is called the Jensen-Steffensen inequality.
Boas [] gave the integral analog of the Jensen-Steffensen inequality. Then .
(  .  ) The generalization of this result is also given by Boas in []. It is the so-called Jensen-Boas inequality (see also []).

Theorem . ([]) If λ : [a, b] → R is either continuous or of bounded variation satisfying
for all x k ∈ y k- , y k , y  = a, y n = b, and λ(a) < λ (b), and if f is continuous and monotonic (either increasing or decreasing) in each of the n intervals y k- , y k , then inequality (.) holds.
The following theorem states the well-known Levinson inequality.  (ii) Pečarić [] proved that inequality (.) is valid when one weakens the previous assumption (.) to x i + x n-i+ ≤ c and p i x i + p n-i+ x n-i+ p i + p n-i+ ≤ c, for i = , , . . . , n.
(iii) Mercer [] made a significant improvement by replacing condition (.) with a weaker one, i.e. he proved that inequality (.) holds under the following conditions: max{x  , . . . , x n } ≤ max{y  , . . . , y n }, (iv) Witkowski [] showed that it is enough to assume that f is -convex in Mercer's assumptions. Furthermore, Witkowski weakened the assumption (.) and showed that equality can be replaced by inequality in a certain direction.
Furthermore, Baloch, Pečarić, and Praljak in their paper [] introduced a new class of functions K c  (a, b) that extends -convex functions and can be interpreted as functions that are '-convex at point c ∈ a, b ' . They showed that K c  (a, b) is the largest class of functions for which Levinson's inequality (.) holds under Mercer's assumptions, i.e. that f ∈ K c  (a, b) if and only if inequality (.) holds for arbitrary weights p i > , n i= p i =  and sequences x i and y i that satisfy x i ≤ c ≤ y i for i = , , . . . , n.
We give the definition of the class K c  (a, b) extended to an arbitrary interval I.
for all x k ∈ y k- , y k , y  = a  , y n = b  , and λ(a  ) < λ(b  ), and On the other hand, in [] Pečarić, Perić, and Rodić Lipanović generalized the Jensen inequality (.) for a real Stieltjes measure. They considered the Green function G defined which is convex and continuous with respect to both s and t. The function G is continuous under s and continuous under t, and it can easily be shown by integrating by parts that any function ϕ : [α, β] → R, ϕ ∈ C  ([α, β]), can be represented by Using that fact, the authors in [] gave the conditions under which inequality (.) holds for a real Stieltjes measure, which is not necessarily positive nor increasing. This result is stated in the following theorem.
Then the following two statements are equivalent: Note that for every continuous concave function ϕ : [α, β] → R inequality (.) is reversed, i.e. the following corollary holds.  The main aim of our paper is to give a Levinson type generalization of the result from Theorem .. In that way, a generalization of Theorem . for real Stieltjes measure, not necessarily positive nor increasing, will also be obtained.

Main results
In order to simplify the notation, throughout this paper we use the following notation: The following theorem states our main result. holds.
If for all s  ∈ [α, c] and for all s  ∈ [c, β] we have where the function G is defined in (.), then for every continuous function The statement also holds if we reverse all signs of inequalities in (.) and (.).
Since the function φ is continuous and concave on [α, c] and for all s  ∈ [α, c] (.) holds, from Corollary . it follows that . When we rearrange the previous inequality, we get Since the function φ is continuous and convex on [c, β] and for all s  ∈ [c, β] (.) holds, from Theorem . it follows that , and after rearranging we get Remark . It is obvious from the proof of the previous theorem that if we replace the equality (.) by a weaker condition Since the function ϕ belongs to class K c Remark . It is easy to see that Theorem . further generalizes the Levinson type generalization of the Jensen-Boas inequality given in Theorem .. Namely, if in Theorem . we set the functions f and g to be monotonic, and the functions λ and μ to satisfy for all x k ∈ y k- , y k , y  = a  , y n = b  , and λ(a  ) < λ(b  ), and , then since the function G is continuous and convex in both variables, we can apply the Jensen inequality and see that for all s  ∈ [α, c] and s  ∈ [c, β] inequalities (.) hold, so we get exactly Theorem ..

Discrete case
In this section we give the results for the discrete case. The proofs are similar to those in the integral case given in the previous section, so we will state these results without the proofs.
In Levinson's inequality (.) and its generalizations (see []) we see that p i (i = , . . . , n) are positive real numbers. Here, we will give a generalization of that result, allowing p i to also be negative, with the sum not equal to zero, but with a supplementary demand on p i and x i given by using the Green function G defined in (.).
We already know from the first section that we can represent any function ϕ : where the function G is defined in (.), and by some calculation it is easy to show that the following holds: Using that fact the authors in [] derived the analogous results of Theorem . and Corollary . for discrete case, and here, similarly as in the previous section, we get the following results.
where the function G is defined in (.), then for every continuous function Corollary . Let the conditions from the previous theorem hold.

Converses of the Jensen inequality
The Jensen inequality for convex functions implies a whole series of other classical inequalities. One of the most famous ones amongst them is the so-called Edmundson-Lah-Ribarič inequality which states that, for a positive measure μ on [, ] and a convex func- Furthermore, the statements () and () are also equivalent if we change the sign of inequality in both (.) and (.).

Corollary . ([])
Let the conditions from the previous theorem hold. Then the following two statements are equivalent: Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality in both statements ( ) and ( ).
In the following theorem we give the Levinson type generalization of the upper result, and we use a similar method to Section  of this paper.
The statement also holds if we reverse all signs of inequalities in (.), (.) and (.).
When we rearrange the previous inequality, we get and after rearranging we get . Remark . It is obvious from the proof of the previous theorem that if we replace the equality (.) by a weaker condition , if additionally ϕ is convex (resp. concave), the condition (.) can be further weakened to .

Discrete form of the converses of the Jensen inequality
In this section we give the Levinson type generalization for converses of Jensen's inequality in discrete case. The proofs are similar to those in the integral case given in the previous section, so we give these results with the proofs omitted.
As we can represent any function ϕ : where the function G is defined in (.), by some calculation it is easy to show that the following holds: Using that fact the authors in [] derived analogous results of Theorem . and Corollary . for discrete case. In [] the authors obtained the following Levinson type generalization of the discrete Edmundson-Lah-Ribarič inequality.
. . , n and j = , . . . , m are such that n i= p i = m j= q j =  and wherex = n i= p i x i andȳ = m j= q j y j , then for every f ∈ K c  (a, B) we have Our first result is a generalization of the result from [] stated above, in which it is allowed for p i , q j to also be negative, with the sum not equal to zero, but with supplementary demands on p i , q j and x i , y j given by using the Green function G defined in (.).
. . , m) be such that P n =  and Q m =  and and for all s ∈ [c, β] we have The statement also holds if we reverse all signs of the inequalities in (.), (.), and (.). Remark . We can replace the equality from the condition (.) by a weaker condition in the analogous way as in Remark . from the previous chapter.

The Hermite-Hadamard inequality
The classical Hermite-Hadamard inequality states that for a convex function ϕ : [a, b] → R the following estimation holds: Its weighted form is proved by Fejér in []: If ϕ : [a, b] → R is a convex function and p : [a, b] → R nonnegative integrable function, symmetric with respect to the middle point (a + b)/, then the following estimation holds: Fink in [] discussed the generalization of (.) by separately looking the left and right side of the inequality and considering certain signed measures. In their paper [], the authors gave a complete characterization of the right side of the Hermite-Hadamard inequality.
Rodić Lipanović, Pečarić, and Perić in [] obtained the complete characterization for the left and the right side of the generalized Hermite-Hadamard inequality for the real Stieltjes measure.
In this section a Levinson type generalization of the Hermite-Hadamard inequality for signed measures will be given as a consequence of the results given in Sections  and .
Here we use the following notation: .

Corollary . Let
and such that

If for all s  ∈ [α, c] and for all s  ∈ [c, β] the inequalities
The statement also holds if we reverse all signs of the inequalities in (.) and (.).

Remark .
where the function G is defined in (.), then for every continuous function .

(  .  )
The statement also holds if we reverse all signs of the inequalities in (.), (.) and (.).
Remark . Let the conditions from the previous theorem hold.

then for every continuous function
, then the inequalities in (i) and (ii) are reversed.

The inequalities of Giaccardi and Petrović
The

Theorem . (Giaccardi, [])
Let p = (p  , . . . , p n ) be a nonnegative n-tuple and x = (x  , . . . , x n ) be a real n-tuple such that In this section we will use an analogous technique as in the previous sections to obtain a Levinson type generalization of the Giaccardi inequality for n-tuples p of real numbers which are not necessarily nonnegative. As a simple consequence, we will obtain a Levinson type generalization of the original Giaccardi inequality (.). In order to do so, we first need to state two results from [].
The statement also holds if we reverse all signs of the inequalities in (.), (.), and (.).
Proof We follow the same idea as in the proof of Theorem . from Section . We apply Theorem . and Corollary . to the function φ( c], and then we set a = min{y  , m j= q j y j } and b = max{y  , m j= q j y j } on [c, β], as well as consider the signs of P n and Q m . We omit the details. Remark . As in the previous sections, we can replace the equality (.) by a weaker condition and then (.) becomes Since ϕ -(c) ≤ D ≤ ϕ + (c) (see []), if additionally ϕ is convex (resp. concave), the condition (.) can be further weakened to Motivated by the results obtained in previous sections, we define the following linear functionals which, respectively, represent the difference between the right and the left side of inequalities (.) and (.):

Mean-value theorems
where m  = M  and m  = M  . We have: Proof Let us define a function χ ∈ C  ([α, β]) by χ(x) = J (ψ)ϕ(x) -J (ϕ)ψ(x). Due to the linearity of J we have J (χ) = . Theorem . implies that there exist ξ  , ξ ∈ [α, β] such that whereφ(x) = x  . Now we have J (φ) = , because otherwise we would have J (ψ) = , which is a contradiction with the assumption J (ψ) = . So we have and this gives us the first claim of the theorem. The second claim is proved in an analogous manner, by observing the linear functional ELR instead of J .
Remark . Note that if in Theorem . we set the function ψ to be ψ(x) = x  , we get exactly Theorem ..
Remark . Note that if we set the functions f , g, λ, and μ from our theorems to fulfill the conditions from Jensen's integral inequality or Jensen-Steffensen's, or Jensen-Brunk's, or Jensen-Boas' inequality, then -applying that inequality on the function G which is continuous and convex in both variables -we see that in these cases for all s  ∈ [α, c], s  ∈ [c, β] inequalities in (.) hold, and so from our results we directly get the results from the paper [].
Remark . If in the definition of the functional J (resp. ELR ) we set f (x) = x and g(x) = x, then we get a functional that represents the difference between the right and the left side of the left-hand part (resp. right-hand part) of the generalized Hermite-Hadamard inequality. In the same manner, adequate results of Lagrange and Cauchy type for those functionals can be derived directly from Theorem . and Theorem ..
As before, motivated by the discrete results obtained in previous sections, we define the following linear functionals which, respectively, represent the difference between the right and the left side of inequalities (.), (.), and (.): where a  = b  and a  = b  ; where the conditions (.) hold and A  , B  , A  , B  are defined in (.).
We have: hold. The following two results are mean-value theorems of the Lagrange and Cauchy type, respectively, and they are obtained in an analogous way to the theorems of the same type in the previous sections, so we omit the proof.
As a consequence of the previous two theorems, we now give some further results in which we give explicit conditions on p i , x i (i = , . . . , n) and q j , y j (j = , . . . , m) for (.) and (.) to hold, where using the properties of the function G we can skip the supplementary conditions on that function. Corollary . Let x i ∈ [α, c], p i ∈ R + (i = , . . . , n) and y j ∈ [c, β], q j ∈ R + (j = , . . . , m), and let ϕ, ψ : [α, β] → R.
( Proof Suppose that x  ≥ x  ≥ · · · ≥ x n . We have P n (x  -x) = n i= p i (x x i ) = n j= (x j-x j )(P n -P j- ) ≥  so it follows that x  ≥x. Furthermore, sox ≥ x n . We see that we have obtained x n ≤x ≤