Steffensen Type Inequalites for Convex Functions on Borel σ -Algebra

: In the paper, we prove Steffensen type inequalities for positive ﬁnite measures by using functions which are convex in point. Further, we prove Steffensen type inequalities on Borel σ -algebra for the function of the form f / h which is convex in point. We conclude the paper by showing that these results also hold for convex functions.


Introduction
In 1918, Steffensen proved an inequality that intrigued many mathematicians. Over the years, various generalizations and refinements have been proven not only in the classical sense, but also in the theory of measures, time scales, quantum calculus, fractional calculus, special functions, functional equations and many others. Some of the results dealing with the topic can be found in [1][2][3][4][5][6][7][8][9][10][11][12]. A complete overview of the results related to Steffensen's inequality can be found in monographs [13,14].
Theorem 1 (Steffensen's inequality, [15]). Suppose that f is non-increasing and g is integrable on [a, b] with 0 ≤ g ≤ 1 and λ = Steffensen's inequality is closely related to many other well-known inequalities such as Gauss', Gauss-Steffensen's, Hölder's, Jenssen-Steffensen's, Iyengar's and other inequalities. Therefore, the Steffensen type inequalities proved in this paper can be used to obtain different generalizations and refinements of these inequalities. For more details about the connection of Steffensen's inequality to other inequalities see in [14].

Remark 1.
As noted in [16], the class of functions introduced in Definition 1 can be described as a class of functions convex in point c. Therefore, the function f is convex on [a, b] if and only if it is convex in every c ∈ (a, b).
Throughout the paper, we will use the notation B([a, b]) for the Borel σ-algebra which is generated on segment [a, b].
In the following two theorems, we recall Steffensen's inequality for positive measures on B([a, b]) proved in [9]. Theorem 3. Let µ be a positive, finite, measure on B([a, b]) and let f and g be measurable functions such that f is non-increasing and 0 ≤ g ≤ 1. If there exists λ ∈ R + such that (1) If f is non-decreasing, the reverse inequality in (1) holds.

Theorem 4.
Let µ be a positive, finite, measure on B([a, b]) and let f and g be measurable functions such that f is non-increasing and 0 ≤ g ≤ 1. If there exists λ ∈ R + such that If f is non-decreasing, the reverse inequality in (2) holds.
Let us also recall weaker conditions for Steffensen's inequality for positive finite measures on B([a, b]) given in [10].
x)) and The generalization of the classical Steffensen inequality proved by Pečarić [17] was intensively used by many mathematicians. The following generalization in the theory of measures was proved in [11]. Theorem 6. Let µ be a positive finite measure on B([a, b]), let f , g and h be measurable functions on [a, b] such that h is positive, f /h is non-increasing and 0 ≤ g ≤ 1.
(a) If there exists λ ∈ R + such that [a,a+λ] If f /h is non-decreasing, the reverse inequalities in (5) and (6) hold.
In addition to generalization obtained by Pečarić in [17], another generalization of Steffensen's inequality which was proved by Mercer in [18] is often used. As the original Mercer generalization is wrong, there are many corrected versions of it. One corrected version was not only proved but also refined by Wu and Srivastava in [19]. Their refinement extended to the theory of measures was proved in [11]. Here, we recall the refinement of Pečarić generalization which can easily be obtained from the refinement proved in [11].
If f /h is non-decreasing, the reverse inequalities in (8) and (10) hold.
In [19], Wu and Srivastava also proved that the correction of Mercer's generalization can be sharper. Their result was extended to Borel σ-algebra in [11]. We recall this extension in the following theorem. (a) If there exists λ ∈ R + such that [a,a+λ] (12) If f /h is non-decreasing, the reverse inequalities in (11) and (12) hold.
In [16,20], the authors proved the classical Steffensen type inequalities for functions which are convex in point and also extended their result to convex functions. In this paper, we prove measure theoretic generalization of the above mentioned Steffensen type inequalities. For that purpose, we use measures on Borel σ-algebra B([a, b]).

Let us begin by proving Steffensen type inequalities for the functions from
and let λ 2 be a positive constant such that If f ∈ M c 2 [a, b] and (15) holds, the inequality in (16) is reversed.
Proof. Let A be the constant as in the Definition 1.
The function F is non-increasing on [a, c] so we can apply the inequality (1) to the function F and obtain 0 ≤ Similarly, the function F is non-decreasing on [c, b] so we can apply the reverse inequality (2) to the function F and obtain 0 ≥ Combining (17) and (18), we have we obtain the reversed inequality in (16).
and let λ 2 be a positive constant such that (21) holds, the inequality in (22) is reversed.
Proof. Let A be the constant as in the Definition 1.
The function F is non-increasing on [a, c] so we can apply the inequality (2) to the function F and obtain 0 ≤ Similarly, the function F is non-decreasing on [c, b] so we can apply the reverse inequality (1) to the function F and obtain Combining (23) and (24) we have we obtain the reversed inequality in (22) Replacing the condition 0 ≤ g ≤ 1 by the weaker one in Theorems 9 and 10 we obtain the following theorems. and Let λ 1 be a positive constant such that (13) holds and let λ 2 be a positive constant such that (14) holds (15) holds, then (16) holds. (15) holds, the inequality in (16) is reversed.
The function F is non-increasing on [a, c] and the condition (25) holds, so we can apply Theorem 5(a) and obtain (17).
Similarly, the function F is non-decreasing on [c, b] and the condition (26) holds so we can apply Theorem 5(b) to the function F and obtain (18). By similar reasoning as in the proof of Theorem 9 we obtain (16). and Let λ 1 be a positive constant such that (19) holds and let λ 2 be a positive constant such that (20) holds.
The function F is non-increasing on [a, c] and the condition (27) holds so we can apply Theorem 5(b) and obtain (23).
Similarly, the function F is non-decreasing on [c, b] and the condition (28) holds so we can apply Theorem 5(a) and obtain (24). By similar reasoning as in the proof of Theorem 10 we obtain (22).  ∈ (a, b).
We say that f /h belongs to the class M c and let λ 2 be a positive constant such that As F/h is non-decreasing on [c, b], from Theorem 6(b) we obtain and let λ 2 be a positive constant such that If f /h ∈ M c 2 [a, b] and (37) holds, the inequality in (38) is reversed.

Proof. Let A be the constant as in the Definition 2. Assume that
As

th(t)g(t)dµ(t) . (40)
The rest of the proof is the same as the proof of Theorem 13.
Proof. Taking substitutions g → g/h and f → f h in Theorem 14 we obtain the statement of this theorem.
We continue with another generalization of the Mercer type for the functions which are convex in point. It can be proved that it is equivalent to Theorems 9 and 10. For the details about this equivalence in the classical sense, the interested reader can see in [21].
and let λ 2 be a positive constant such that If f /h ∈ M c 2 [a, b] and (47) holds, the inequality in (48) is reversed.

Proof.
Taking substitutions h → kh, g → g/k and f → f k in Theorem 13 we obtain the statement of this theorem.
Theorem 18. Let µ be a positive, finite measure on B([a, b]) and let c ∈ (a, b). Let h be positive measurable function on [a, b] and let f , g and k be measurable functions on [a, b] such that 0 ≤ g ≤ k. Let λ 1 be a positive constant such that and let λ 2 be a positive constant such that (50) and (51) holds, the inequality in (52) is reversed.
Proof. Taking substitutions h → kh, g → g/k and f → f k in Theorem 14 we obtain the statement of this theorem.
In the following theorems, we prove refinements of Theorems 13 and 14. h(t)g(t)dµ(t), and let λ 2 be a positive constant such that  [a, b], and let f and g be measurable functions on [a, b] such that 0 ≤ g ≤ 1.
Let λ 1 be a positive constant such that and let λ 2 be a positive constant such that and let λ 2 be a positive constant such that (58) (60) and [x,c] k(t)g(t)dµ(t) ≥ 0, for every x ∈ [a, c] (61) and [x,b] k(t)g(t)dµ(t) ≤ [x,b] k(t)h(t)dµ(t) and  Proof. Similar to the proof of Theorem 11 using the weaker conditions proved in [11] (Theorem 3.1).
Theorem 24. Let µ be a positive, finite measure on B([a, b]) and let c ∈ (a, b). Let f , g, k, h : [a, b] → R be µ−integrable functions such that k is positive, h is non-negative and Proof. Similar to the proof of Theorem 12 using the weaker conditions proved in ( [11], Theorem 3.1).
Let λ 1 be a positive constant such that (29) holds and let λ 2 be a positive constant such that ( Proof. Similar to the proof of Theorem 11 using modification of the weaker conditions proved in [11].