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New Inequalities of Steffensen's type for s-convex functions

In this work, new inequalities connected with the Steffensen's integral inequality for s-convex functions are proved


Introduction
In order to study certain inequalities between mean values, J.F. Steffensen [11] has proved the following inequality (see also [9], p. 311): Theorem 1. Let f and g be two integrable functions defined on (a, b), f is decreasing and for each t ∈ (a, b), 0 ≤ g(t) ≤ 1. Then, the following inequality Some minor generalization of Steffensen's inequality (1.1) was considered by T. Hayashi [5], using the substituting g (t)/A for g (t), where A is positive constant. For other result involving Steffensen type inequality, see [3], [5] and [8]- [11].
In the recent work [1], Alomari et. al. proved the following result: Theorem 2. Let f, g : [a, b] → R be integrable such that 0 ≤ g(t) ≤ 1, for all t ∈ [a, b] such that b a g (t) df (t) exists. If f is absolutely continuous on [a, b] with f ′ ∈ L p [a, b], 1 ≤ p ≤ ∞, then we have .
In [4], Dragomir and Fitzpatrick proved a variant of Hadamard's inequality which holds for s-convex functions in the second sense: The constant k = 1 s+1 is the best possible in the second inequality in (1.4). For another inequalities of Hermite-Hadamard type see [2] and [7].
The aim of this paper is to establish new inequalities of Steffensen's type for s-convex functions in the second sense.

The results
Lemma 1. Let f, g : [a, b] ⊂ R + → R be integrable such that 0 ≤ g(t) ≤ 1, for all t ∈ [a, b] and b a g (t) f ′ (t) dx exists. Then we have the following representation which gives the desired representation (2.1). The identity (2.2) can be also proved in a similar way, we shall omit the details.
2.1. Inequalities involving s-convexity. In the following, inequalities for absolutely continuous functions whose first derivatives are s-convex (s-concave) are given: which proves the first inequality in (2.3). In similar way and using (2.2) we may deduce the desired inequality (2.4), and we shall omit the details.

Remark 1. In the inequalities (2.3) and (2.4)
, choose λ = 0, then we have Another approach leads to the following result: on [a, b], for some fixed s ∈ (0, 1] then we have and where λ := b a g (t) dt.
Proof. From Lemma 1, we may write which proves the first inequality in (2.8). The second inequality in (2.8) follows directly, since 0 ≤ g(t) ≤ 1 for all t ∈ [a, b], then The inequalities in (2.9) may be proved in the same way using the identity (2.2), we shall omit the details.
Proof. Utilizing the triangle inequality on (2.1), and since |f ′ | is s-concave on [a, b] then by (1.4) we may state which proves the first inequality in (2.10). The second inequality in (2.10) follows directly, since 0 ≤ g(t) ≤ 1 for all t ∈ [a, b], then The inequalities in (2.11) may be proved in the same way using the identity (2.2), we shall omit the details.
Another result is incorporated in the following theorem: with |f ′ | q is s-concave on [a, b], for some fixed s ∈ (0, 1] and q > 1, then we have where λ := b a g (t) dt.
Proof. From Lemma 1 and using the Hölder inequality for q > 1, and p = q q−1 , we obtain