Generalizations of Ste ensen ’ s inequality via the extension of Montgomery identity

Abstract: In this paper, we obtained new generalizations of Ste ensen’s inequality for n-convex functions by using extension of Montgomery identity via Taylor’s formula. Since 1-convex functions are nondecreasing functions, new inequalities generalize Stefensen’s inequality. Related Ostrowski type inequalities are also provided. Bounds for the reminders in new identities are given by using the Chebyshev and Grüss type inequalities.


Introduction
In [1], the authors obtain the following extension of Montgomery identity using Taylor's formula: Remark 1.2. The last identity holds also for n = . In this special case, we assume that ∑ n− i= ⋯ is an empty sum. Thus (1) reduces to well-known Montgomery identity (e.g. [2]) where the Peano kernel is The aim of this paper is to obtain some new generalizations of Ste ensen's inequality using above extension of Montgomery identity. The Ste ensen's inequality was rst given and proved by Ste ensen in 1918 ( [3]): The inequalities are reversed for f nondecreasing.
In [4] Jakšetić and Pečarić generalized Ste ensen's inequality for positive measures. Mitrinović stated in [5] that the inequalities in (2) follow from the identities which will be the starting point for our generalizations of Ste ensen's inequality.
First, let us recall the de nition of n-convex functions. Let f be a real-valued function de ned on the segment [a, b]. The divided di erence of order n of the function f at distinct points x , ..., x n ∈ [a, b], is de ned recursively (see [6,7]) by The value f [x , . . . , x n ] is independent of the order of the points x , . . . , x n . The de nition may be extended to include the case when some (or all) of the points coincide. Assuming that f (j− ) (x) exists, we de ne The notion of n-convexity goes back to Popoviciu ([8] Note that, −convex functions are nondecreasing functions. If f (n) exists, then f is n−convex i f (n) ≥ . The paper is organized as follows. After this Introduction, in Section 2 we obtain new identities related to Ste ensen's inequality. Using these new identities we generalize Ste ensen's inequality for n−convex functions. Further, in Section 3 we give Ostrowski-type inequalities related to our new generalizations. We conclude this paper with some new bounds for our identities, using the Chebyshev and Grüss type inequalities.
Throughout the paper, it is assumed that all integrals under consideration exist and that they are nite.

Generalizations of Ste ensen's inequality via the extension of Montgomery identity
In this section we obtain generalizations of Ste ensen's inequality for n-convex functons using identity (1).

t)p(t)dt and let the function G be de ned by
Then Proof. Using identity and integration by parts we have Applying identity (1) to f ′ and replacing n with n − we have After applying Fubini's theorem on the last term in (6) we obtain (5).

t)p(t)dt and let the function G be de ned by
Then Proof. Similarly as in the proof of Theorem 2.1, we use the identity Now, using the above obtained identites we give generalization of Ste ensen's inequality for n-convex functions. Proof.
If the function f is n-convex, without loss of generality we can assume that f is n−times di erentiable and f (n) ≥ see [7, p. 16 and p. 293]. Now we can apply Theorem 2.1 to obtain (10).

t)p(t)dt and let the function G be de ned by (7). If f is n−convex and
Proof. Similar to the proof of Theorem 2.3. (9) and (11) are nonpositive, then the reverse inequalities in (10) and (12) hold. Note that in this case for some odd n ≥ , functions G i , i = , are nonnegative so integrals in (9) and (11) are nonpositive. Hence, inequalities (10) and (12) are reversed.

Ostrowski-type inequalities
In this section we give the Ostrowski-type inequalities related to generalizations obtained in the previous section.
Here, the symbol L p [a, b] ( ≤ p < ∞) denotes the space of p-power integrable functions on the interval [a, b] equipped with the norm The constant on the right-hand side of (13) is sharp for < p ≤ ∞ and the best possible for p = .
Proof. Let's denote By taking the modulus of (5) and applying Hölder's inequality we obtain For the proof of the sharpness of the constant C q let us nd a function f for which the equality in (13) is obtained. For < p < ∞ take f to be such that is the best possible inequality. C (⋅) is a continuous function on [a, b] and so is C(⋅) . Suppose that C(⋅) attains its maximum at s ∈ [a, b]. First we assume that C(s ) > . For ε > small enough we de ne f ε (s) by and the rest of the proof is the same as above.
The constant on the right-hand side of (15) is sharp for < p ≤ ∞ and the best possible for p = .
Proof. Similar to the proof of Theorem 3.1.

Generalizations related to the bounds for the Chebyshev functional
Let f , h ∶ [a, b] → R be Lebesgue integrable functions. We de ne the Chebyshev functional T(f , h) by In 1882 Chebyshev proved that . In 1934, Grüss in his paper [10] proved that provided that there exist the real numbers m, M, n, N such that is the best possible. In [11] Cerone and Dragomir proved the following theorems: The constant √ in (16) is the best possible.
The constant in (17) is the best possible.
In the sequel we use the above theorems to obtain some new bounds for integrals on the left hand side in the perturbed version of identities (5) and (8).
Firstly, let us denote