Generalized Steffensen’s Inequality by Fink’s Identity

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Grüss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of (n + 1)-convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions.


Introduction
Integral inequalities such as Hardy's inequality, Steffensen's inequality, and Ostrowski's inequality are topics of interest of many Mathematicians since their pronouncement. Several generalizations of these inequalities have been proved for different classes of functions, such as convex functions, n-convex functions, and other types of functions, for example see [1][2][3][4]. Moreover, integral inequalities have been proved for different integrals, such as Jensen-steffensen inequality for diamond integral and bounds of related identities have been obtained in [5]. Other than that, Hardy's inequality for fractional integral on general domains have been proved in [6].

Remark 1.
We can consider f to be absolute continuous instead of differentiable function and the suppositions of Theorem 1 can also be weakened. In fact for an increasing function ψ, the function Ψ(x) = x c ψ(z) dz is well defined and satisfies Ψ = ψ at all except the set of points with measure zero. One can substitute x = f (z) in (2) (see [12] (Corollary 20.5)), provided that f is absolutely continuous increasing function, therefore (ii) (5) holds reversely if t ≤ t 0 f (z) dz for every t ∈ [0, d].
Following two lemmas will be useful in our construction as well, see [14,15].
Lemma 1. For a function ψ ∈ C 2 ([c, d]), we have: Clearly, Throughout the calculations in the main results, we will use p i (ξ, u) corresponding to ∂G * ,i (ξ,u) ∂ξ for i = 1, 2, 3, and for ∂G * ,4 (ξ,u) ∂ξ , ∂G * ,5 (ξ,u) ∂ξ we use p 3 (ξ, s) and p 2 (ξ, s), respectively. We also require the classical Fink's identity given in [16]: where W [c,d] (t, u) is given by: Divided differences are fairly ascribed to Newton, and the term "divided difference" was used by Augustus de Morgan in 1842. Divided differences are found to be very helpful when we are dealing with functions having different degrees of smoothness. The following definition of divided difference is given in [8] (p. 14).
It is easy to see that (26) is equivalent to The following definition of a real valued convex function is characterized by nth-order divided difference (see [8] (p. 15)).
If this inequality is reversed, then ψ is said to be n-concave. If the inequality is strict, then ψ is said to be a strictly n-convex (n-concave) function.

Remark 2.
Note that 0-convex functions are non-negative functions, 1-convex functions are increasing functions, and 2-convex functions are simply the convex functions.
The following theorem gives an important criteria to examine the n-convexity of a function ψ (see [8] (p. 16)).
In this article, we use Fink's identity, Montgomery identities, and Green functions to prove some identities related to Steffensen's inequality. By using these identities we obtain a generalization of (4). In addition, we construct new identities which enable us to prove generalizations of inequalities (5) and (6) as one can obtain Classical Hardy-type inequalities from them, see [1]. We useČebyšev functional to construct new bounds of Grüss and Ostrowski-type inequalities. Finally, we give several applications of our work.
The first part of this section is the generalization of (4). For this, we start with the following theorem: (a) For j = 1, 2, 4, 5, we have: Proof. (a) We first prove by fixing j = 1 , other cases for j = 2, 4, 5 can be treated analogously.
Utilizing (7) and (17) for ψ and ψ respectively, we get Simplifying and employing Fubini's theorem, we get Now by replacing n with n − 2 in (24) for ψ , we have: Rest follows from simplification and Fubini's theorem. (b) Using assumption ψ (c) = 0 and employing a similar method as in (a).
From the next two theorems we get a generalization of Steffensen's inequality and its reverse by generalizing (4) and its reverse.
Theorem 6. Consider (A 2 ) and let f be as in Corollary 2 (i) then: (a) Proof. We give proof of our results by fixing j = 1, and other cases can be proved in the similar way. By using (7) and (17) for ψ and ψ respectively and applying assumption ψ(0) = 0, we get Now replacing n with n − 2 in (24) for ψ and simplifying we get the required identities.
Our next result gives a generalization of (5).
Proof. The proof is similar to that of Theorem 4 except using Theorem 6 and Corollary 2 (i).
Proof. The proof is similar to that of Theorem 5 except using Theorem 7 and Corollary 2 (i).
Next we give some generalized identities considering (6).
Theorem 9. Consider (A 2 ) and let f , λ and Λ be as in Corollary 3 (i) then: (a) For j = 1, 2, we have Proof. We give a proof of our results by fixing j = 1, and other cases can be proved in a similar way. By using (7) and (17) for ψ and ψ respectively and applying assumption ψ(0) = 0, we get: The rest follows from (24).
Proof. The proof is similar to that of Theorem 4 except using Theorem 9 and Corollary 3 (i).
Proof. The proof is similar to that of Theorem 5 except using Theorem 10 and Corollary 3 (i).

New Upper Bounds ViaČebyšev Functional
Consider theČebyšev functional for two Lebesgue integrable functions F 1 , F 2 : [c, d] → R given as: Cerone and Dragomir in [17] proposed new bounds utilizingČebyšev functional given as: The following inequality holds The constants 1 √ 2 and 1 2 are the optimal constants.
therefore the required bound in (46) follows.
Ostrowski-type inequalities associated with generalized Steffensen's inequality can be given as: The constant on the R.H.S. of (47) is sharp for 1 < s ≤ ∞ and the best possible for s = 1.

Remark 4.
We may proceed further by defining linear functionals with the inequalities proved in Theorems 8 and 11. Moreover, by proving monotonicity of new functionals we extend the inequalities in Theorems 8 and 11.

Application to Exponentially Convex Functions
We start this section by an important Remark given as: By the virtue of Theorem 4 (a), for j = 1, 2, . . . , 5, we define the positive linear functionals with respect to n-convex function ψ as follows Next we construct the non trivial examples of exponentially convex functions (see [19]) from positive linear functionals ∆ 1,j (ψ) for (j = 1, 2, . . . , 5). It is interesting to note that this is a family of n-convex functions as d n du n ψ s (u) = u s−n ≥ 0.
Since s → u s−n = e (s−n) ln u is exponentially convex function, therefore the mapping s → ∆ 1,j (ψ s ) is exponential convex and as a special case, it is also log-convex mapping. The log-convexity of this mapping enables us to construct the known Lyapunov inequality given as for r, s, t ∈ R such that r < s < t where j = 1, 2, . . . , 5.

Remark 6.
We have not given the proof of the above mentioned results (see [19] for details). The Lyapunov inequality empowered us to refine lower (upper) bound for action of the functional on the class of functions given in (53) because if exponentially convex mapping attains zero value at some point it is zero everywhere (see [19]). One can also consider some other classes of n-convex functions given in the paper [19,20] and can get similar estimations. A similar technique can also be employed by considering the results of Theorems 7 and 10.

Conclusions and Outlooks
In this article, we extended the pool of inequalities by proving generalizations of well-known Steffensen's inequalities and their reverses. The inequalities proved in the main results provide generalizations of the results from [1,4,7,11]. Moreover, Hardy's inequality is also one of the well-known inequalities. In this article, we also proved generalizations of inequalities, from [1], which are closely related to Hardy's inequality. We also developed new bounds of Grüss and Ostrowski-type inequalities. Further, the contribution of these inequalities to the theory of (n + 1)-convex functions has been presented by defining functionals from new inequalities and describing their properties. Lastly, new inequalities related to exponentially convex functions and log-convex functions, such as the Lyapunov inequality, have been developed. In the future, it can be investigated whether we can use other interpolations, such as Hermite interpolation, to prove new generalizations of Steffensen's inequality and related results.