Bellman–Steffensen type inequalities

In this paper some Bellman–Steffensen type inequalities are generalized for positive measures. Using sublinearity of a class of convex functions and Jensen’s inequality, nonnormalized versions of Steffensen’s inequality are obtained. Further, linear functionals, from obtained Bellman–Steffensen type inequalities, are produced and their action on families of exponentially convex functions is studied.


Introduction
Since its appearance in 1918 Steffensen's inequality [1] has been a subject of investigation by many mathematicians because it plays an important role not only in the theory of inequalities but also in statistics, functional equations, time scales, special functions, etc. A comprehensive survey on generalizations and applications of Steffensen's inequality can be found in [2].
In 1959 Bellman gave an L p generalization of Steffensen's inequality (see [3]) for which Godunova, Levin and Čebaevskaya noted that it is incorrect as stated (see [4]). Further, in [5] Pečarić showed that the Bellman generalization of Steffensen's inequality is true with very simple modifications of conditions. Using some substitutions in his result from [5], Pečarić also proved the following modification of Steffensen's inequality in [6].

1)
where λ is a positive number. Then In [7] Mitrinović and Pečarić gave necessary and sufficient conditions for inequality (1.2). The purpose of this paper is to generalize the aforementioned result for positive measures, using the approach from [8] and [9], and to give some applications related to exponential convexity.
Let B ([a, b]) be the Borel σ -algebra generated on [a, b]. In [10] the authors proved the following measure theoretic generalization of Steffensen's inequality. (1. 3) The inequality

Main results
Motivated by Theorem 1.1 and necessary and sufficient conditions given in [7], we prove some generalizations of Bellman-Steffensen type inequalities for positive measures.
(Necessity) If we put the function for a ≤ x ≤ a + λ into inequality (2.1), we obtain the conditions in (2.2).
In the following lemma we recall the property of sublinearity of the class of convex functions.
and [a,b] g 1 (t) dμ(t) ≤ 1, then the following inequality is valid: where [a,a+λ] Proof Using Jensen's inequality, we have and since φ • f is nonincreasing, we only have to check conditions in (2.2). Since G is nonnegative, Using sublinearity from Lemma 2.1 and Jensen's inequality, we have Since G is a nonnegative nondecreasing function and h is a nonnegative nonincreasing function, we see that for each so along with (2.4) we proved (2.3). Hence, the proof is completed.
Remark 2.1 In Theorems 2.1 and 2.2 we proved similar results to those obtained by Liu in [11] but we only need μ to be finite and positive instead of finite continuous and strictly increasing as in [11].
We continue with some more general Bellman-Steffensen type inequalities related to the function f /k.

Theorem 2.3 Let μ be a finite, positive measure on B([a, b]), f , h and k be μ-integrable functions such that h is nonnegative, k is positive and f
Proof (Sufficiency) Let us define the function such that there exists a nonnegative function g 1 defined by the equation and [a,b] g 1 (t) dμ(t) ≤ 1, then the following inequality is valid: Proof Using Jensen's inequality, we have .
if conditions in (2.6) are satisfied. Obviously, [x,b] k(t)G(t) dμ(t) ≥ 0 since k and μ are positive and G is nonnegative. Hence, we have to show (2.10) Since G and h are nonnegative nondecreasing and k is positive, we have i.e., [a,b] So, along with (2.10), we have proved (2.9). Hence the theorem is proved.
The following example will be useful in our applications.
Example 3.1 (i) f (x) = e αx is exponentially convex on R, for any α ∈ R.
The following families of functions given in the next two lemmas will be useful in constructing exponentially convex functions.  Then x → (ϕ p /k)(x) is increasing on (0, ∞) for each p ∈ R and p → (ϕ p /k)(x) is exponentially convex on (0, ∞) for each x ∈ (0, ∞).
k(x) ) = x p-1 > 0 on (0, ∞) for each p ∈ R we have that x → (ϕ p /k)(x) is increasing on (0, ∞). From Example 3.1 the mappings p → e p log x and p → 1 p are exponentially convex, and since p → x p p = e p log x · 1 p , the second conclusion follows.
Similarly we obtain the following lemma.

Lemma 3.2 For p
Using the Bellman-Steffensen type inequality given by (2.5), under the assumptions of Theorem 2.3, we can define a linear functional L by

a+λ] k(t)h(t) dμ(t)
. (v) If p, q, r ∈ R are such that p < q < r, then Proof (i) Continuity of the function p → (p) is obvious for p ∈ R \ {0}. For p = 0 it is directly checked using Heine characterization.
Let k be a positive function and let {θ p /k : p ∈ (0, ∞)} be the family of functions defined on [0, ∞) by Similarly as in Lemma 3.1 we conclude that Let us define a linear functional M by where θ p is defined by (3.4). Then the following statements hold: (i) The function F is continuous on (0, ∞).
(ii) If n ∈ N and p 1 , . . . , p n ∈ (0, ∞) are arbitrary, then the matrix Proof (i) Continuity of the function p → F(p) is obvious.
In the following theorem we give the Lagrange-type mean value theorem. Denote Then (h 1 /k)(a) = (h 2 /k)(a) = 0 and Using the standard Cauchy type mean value theorem, we obtain the following corollary.
, (3.6) provided that the denominator on the right-hand side is nonzero.
Using the characterization of convexity by the monotonicity of the first order divided differences, it follows (see [12, p. 4]): if p, q, u, v ∈ (0, ∞) are such that p ≤ u, q ≤ v then M(p, q) ≤ M(u, v).
Using (2.7), under assumptions of Theorem 2.4, we can define a linear functional N by . (3.8) We have that the linear functional N is nonnegative on the class of increasing convex functions φ on [0, ∞) with the property φ(0) = 0.
where φ p is defined by (3.2). Then the following statements hold: Proof (i) Continuity of the function p → H(p) is obvious.
(ii) Let n ∈ N, p i ∈ (1, ∞) (i = 1, . . . , n) be arbitrary and define an auxiliary function ψ : This means that the matrix is positive semidefinite. Claims (iii), (iv), (v) are simple consequences of (i) and (ii). Similar to Corollary 3.1 we also have the following corollary. , (3.12) provided that the denominator on the right-hand side is nonzero. Remark 3.2 By (3.12) we can define various means (assuming that the inverse of ψ 1 /ψ 2 exists). That is, If we substitute ψ 1 (x) = φ p (x), ψ 2 (x) = φ q (x) in (3.13) and use the continuous extension, we obtain the following expressions: where φ 0 (x) = log x and p, q ∈ (1, ∞). Again, by the monotonicity one has: if p, q, u, v ∈ (1, ∞) are such that p ≤ u, q ≤ v then N(p, q) ≤ N(u, v).
For a fixed n ≥ 2, let us define where ψ p ∈ C n . Then the following statements hold: (i) S is n-exponentially convex in the Jensen sense on J.
(ii) If S is continuous on J, then it is n-exponentially convex on J and for p, q, r ∈ J such that p < q < r, we have S(q) r-p ≤ S(p) r-q S(r) q-p .