Lebesgue Integral Inequalities of Jensen Type for λ-Convex Functions

Some Lebesgue integral inequalities of Jensen type for λ-convex functions defined on real intervals are given.


Introduction 0.1 h-Convex Functions
We recall here some concepts of convexity that are well known in the literature.
Let I be an interval in R.
Definition 1 ( [42]) We say that f : I → R is a Godunova-Levin function or that f belongs to the class Q (I) if f is non-negative and for all x, y ∈ I and t ∈ (0, 1) we have Some further properties of this class of functions can be found in [32], [33], [35], [48], [51] and [52].Among others, its has been noted that nonnegative monotone and non-negative convex functions belong to this class of functions.
The above concept can be extended for functions f : C ⊆ X → [0, ∞) where C is a convex subset of the real or complex linear space X and the inequality (1) is satisfied for any vectors x, y ∈ C and t ∈ (0, 1) .If the function f : C ⊆ X → R is non-negative and convex, then it is of Godunova-Levin type.
Definition 2 ( [35]) We say that a function f : I → R belongs to the class P (I) if it is nonnegative and for all x, y ∈ I and t ∈ [0, 1] we have f (tx + (1 − t) y) ≤ f (x) + f (y) . (2) Obviously Q (I) contains P (I) and for applications it is important to note that also P (I) contains all nonnegative monotone, convex and quasi convex functions, i. e. nonnegative functions satisfying for all x, y ∈ I and t ∈ [0, 1] .
For some results on P -functions see [35] and [49], while for quasi convex functions the reader can consult [34].
If f : C ⊆ X → [0, ∞), where C is a convex subset of the real or complex linear space X, then we say that it is of P -type (or quasi-convex) if the inequality (2) (or (3)) holds true for x, y ∈ C and t ∈ [0, 1] .
The concept of Breckner s-convexity can be similarly extended for functions defined on convex subsets of linear spaces.
It is well known that if (X, • ) is a normed linear space, then the function Utilising the elementary inequality (a + b) s ≤ a s + b s that holds for any a, b ≥ 0 and s ∈ (0, 1], we have for the function g for any x, y ∈ X and t ∈ [0, 1] , which shows that g is Breckner s-convex on X.
In order to unify the above concepts for functions of real variable, S. Varošanec introduced the concept of h-convex function as follows.
Assume that I and J are intervals in R, (0, 1) ⊆ J and functions h and f are real non-negative functions defined in J and I, respectively.
This concept can be extended for functions defined on convex subsets of linear spaces in the same way as above replacing the interval I be the corresponding convex subset C of the linear space X.
Now we can introduce another class of functions.

Definition 5
We say that the function f : for all t ∈ (0, 1) and x, y ∈ C.
We observe that for s = 0 we obtain the class of P -functions while for s = 1 we obtain the class of Godunova-Levin.If we denote by Q s (C) the class of s-Godunova-Levin functions defined on C, then we obviously have For different inequalities related to these classes of functions, see [1]- [4], [6], [9]- [41], [44]- [46] and [49]- [57].
A function h : J → R is said to be supermultiplicative if h (ts) ≥ h (t) h (s) for any t, s ∈ J.
If the inequality (6) is reversed, then h is said to be submultiplicative.If the equality holds in (6) then h is said to be a multiplicative function on J.
We observe that, if h, g are nonnegative and supermultiplicative, then their product is alike.In particular, if h is supermultiplicative then its product with a power function r (t) = t r is also supermultiplicative.
We recall the following Hermite-Hadamard type inequality for h-convex functions from [53]:
We observe that if f : Indeed, if α, β ≥ 0 with α + β > 0 and x, y ∈ C then The following proposition contains some properties of λ-convex functions [26].

We have the following result providing many examples of subadditive functions
is nonnegative, increasing and subadditive on [0, ∞) .
We have the following fundamental examples of power series with positive coefficients Other important examples of functions as power series representations with positive coefficients are: where Γ is Gamma function.
Corollary 1 ( [26]) Let h (z) = ∞ n=0 a n z n be a power series with nonnegative coefficients a n ≥ 0 for all n ∈ N and convergent on the open disk D (0, R) with R > 0 or R = ∞ and r ∈ (0, R) .For a mapping f : C → R defined on convex subset C of a linear space X, the following statements are equivalent: (ii) We have the inequality (14) for any α, β ≥ 0 with α + β > 0 and x, y ∈ C.
Remark 2 We observe that, in the case when for any α, β ≥ 0 with α + β > 0 and x, y ∈ C. We observe that this definition is independent on r > 0.
We have the following Jensen inequality for the Riemann integral [28]: then The following weighted version of Jensen inequality for the Riemann integral [28] also holds. then Motivated by the above results in this paper we establish some Jensen type inequalities for the general Lebesgue integral.

Some Results for Differentiable Functions
If we assume that the function f : I → [0, ∞) is differentiable on the interior of I, denoted by I, then we have the following "gradient inequality" that will play an essential role in the following.
Lemma 1 Let λ : (0, ∞) → (0, ∞) be a function such that the right limit exists and is finite, and the left derivative in 1 denoted by λ − (1) exists and is finite.
for any x, y ∈ I with x = y.
Proof.Since f is λ-convex on I, then for any t ∈ (0, 1) and for any x, y ∈ I, which is equivalent to and by dividing by t > 0 we get for any t ∈ (0, 1) .Now, since f is differentiable on y ∈ I, then we have for any x ∈ I with x = y.
Also we have lim Taking the limit over t → 0+ in (24) and utilizing ( 25) and ( 26) we get the desired result (23).
Remark 3 If we assume that then the inequality (23) also holds for x = y.
In this situation the inequality ( 23) becomes for λ + (0) > 0 for any x, y ∈ I with x = y.
If the function λ is subadditive on [0, ∞) and has finite lateral derivatives with λ + (0) > 0, then Taking the limit over t → 0+ in (29) we get therefore the inequality (28) also holds for x = y.
We have the following result.
As examples of such functions we have: a n z n a power series with nonnegative coefficients a n ≥ 0 for all n ∈ N and convergent on the open disk D (0, R) with R > 0 or R = ∞ and r ∈ (0, R) .If the function f : for any x, y ∈ I.
for any x, y ∈ I.

Jensen Type Inequalities
Let (Ω, A, µ) be a measure space consisting of a set Ω, a σ -algebra A of parts of Ω and a countably additive positive measure µ on A with values in R ∪ {∞} .For a µ−measurable function w : Ω → R, with w (x) ≥ 0 for µa.e.(almost every) x ∈ Ω, consider the Lebesgue space For simplicity of notation we write everywhere in the sequel Ω wdµ instead of Ω w (x) dµ (x) .
The following inequality of Hermite-Hadamard type holds: The inequality (37) provides other lower bound for the integral mean than the first inequality in (7).Since for h-convexity, h (0) may not be defined, the lower bounds from ( 37) and ( 7) cannot be compared in general.
If we consider the discrete measure, then we have: Remark 5 Let h (z) = ∞ n=0 a n z n be a power series with nonnegative coefficients a n ≥ 0 for all n ∈ N and convergent on the open disk D (0, R) with R > 0 or R = ∞ and r ∈ (0, R) .Assume that the function f : I → [0, ∞) is differentiable on I and λ r -convex with λ r : [0, ∞) → [0, ∞) , λ r (t) := ln h (r) h (r exp (−t)) .
If f : I → [0, ∞) is differentiable on I and λ r -convex, then for any u : Ω → [m, M ] ⊂ I so that f • u, u ∈ L w (Ω, µ) , where w ≥ 0 µ-a.e.(µ-almost everywhere) on Ω with Ω wdµ = 1 we have Also, for any x i ∈ I and p i ≥ 0, i ∈ {1, ..., n} with n i=1 p i = 1 we have Recall Slater's inequality for differentiable convex functions [56]: Lemma 2 Let f : I → R be a nondecreasing (nonincreasing) differentiable convex function on I, x i ∈ I , p i ≥ 0 with P n = n i=1 p i > 0 and assume that n i=1 p i f (x i ) = 0. Then one has the inequality As shown in [22, pp. 129-130], the monotonicity condition in Lemma 2 can be weakened by assuming that We can state the following result that is similar to Slater's inequality: , where w ≥ 0 µ-a.e.(µ-almost everywhere) on Ω with Ω wdµ = 1 and Proof.Since the function f : I → [0, ∞) is differentiable on I and λ-convex, then by (30) we have then by taking x = x 0 in (44) we get the desired result (42).
The following Hermite-Hadamard type inequality holds: The following discrete inequality also holds: Remark 7 The interested reader can obtain some particular inequalities of interest by taking λ r -convex functions with λ r : [0, ∞) → [0, ∞) , λ r (t) := ln h (r) h (r exp (−t)) , and h is as in Theorem 2. The details are omitted.
be a function with the property that λ (t) > 0 for all t > 0 and the function f : [m, M ] → [0, ∞) is λ-convex and Riemann integrable on the interval [m, M ] .If the following limit exists