On Ostrowski Type Inequalities for Generalized Integral Operators

is


Introduction
Mathematical inequalities have been present in the development and consolidation of Science. Nowadays, inequalities are essential tools in multiple applications to di erent problems, since they are involved in the basis of the processes of approximation, estimation, interpolation, and extrenals and, in general, they appear in the models used in the study of applied problems. e formalization of mathematical inequalities begins in the 18th century, essentially, with the works of the so-called "Prince of Mathematics" Johann Carl Friedrich Gauss (1777-1855); passing through the investigations and applications of inequalities to Mathematical Analysis developed by Augustin-Louis Cauchy (1789-1857) and Pafnuti Lvóvich Chebyshev (1821-1894). It would be unfair not to mention among the formalizes of mathematical inequalities to Viktor Yakovlevich Bunyakovsky (1804-1889).
is remarkable Russian mathematician received all possible mathematical in uence from his thesis advisor Augustin-Louis Cauchy.
is remarkable scientist is credited with having proved in 1859, many years before Hermann Schwarz, the well-known Cauchy-Schwarz Inequality for the in nite-dimensional case. It is worth noting that in many texts the famous inequality is known as: Cauchy-Bunyakovsky-Schwarz. e proof of Hardy's famous inequality involved an important group of prominent mathematicians of his time such as Edmund Hermann Landau (1887-1938), George Pólya , and Issai Schur , and Marcel Riesz (1886-1969), among others. It is worth noting the coordinating role played by Godfrey Harold Hardy (1887Hardy ( -1947 in the study of inequalities; his work has been very signi cant, fundamentally, for the systematization and application of the eory of Mathematical Inequalities. Hardy was the founder of the Journal of the London Mathematical Society, a suitable publication for many articles on inequalities. In addition, along with Littlewood and Polya, Hardy was the editor of the volume Inequalities see [1], which was the rst monograph, on inequalities and immediately used as the basis for the later development of mathematical inequalities. For more information on the epistemological evolution of the eory of Mathematical Inequalities see [2]. In recent years there has been a growing interest in the study of many classical inequalities applied to integral operators associated with different types of fractional derivatives since integral inequalities and their applications play a vital role in the theory of differential equations and applied mathematics. For the most recent works, we recommend the reader [3][4][5]. Some of the inequalities studied are Gronwall, Chebyshev, Hermite-Hadamard-type, Ostrowski-type, Grüss-type, Hardy-type, Gagliardo-Nirenberg-type, reverse Minkowski and reverse Hölder inequalities (see, e.g., [6][7][8][9][10][11][12][13]). Other types of inequalities such as Hilber-type inequalities and Hadamard-type inequalities have also been recently studied in the context of integral equations and generalized mapping on fractal sets, [14,15].
In this paper, we show some new results related to Ostrowski-type inequalities via conformable and nonconformable operators.
Ostrowski-type inequalities have significant contributions to the area of numerical analysis since they provide estimates of the error of many quadrature rules, for example, the midpoint rule, Simpson's rule, the trapezoidal rule, and other generalized fractional integrals. ey also have many powerful results and a large number of applications in Probability eory and Statistics, Information eory, and Integral Operator eory. For further discussions, we refer the reader to the book by Dragomir and Rassias (see [16]).

On Generalized Ostrowski's Inequality
Alexander Ostrowski (1893Ostrowski ( -1986 was an important mathematician born in Kyiv, the former Russian empire, today the capital of Ukraine. From a mathematical point of view, he was directly influenced by great mathematicians such as Hensel, Hilbert, Klein, and Landau. Another tribute, in addition to the inequality that bears his name, is the wellknown Ostrowski Prize that is jointly sponsored by several renowned universities and the Academies of Science of the Netherlands and Denmark.
Ostrowski proved in [17] an integral inequality associated with a real differentiable function that establishes an upper bound for the difference between the function evaluated at any interior point of some interval and the average of the function over the same interval.
Since then, there are a lot of generalizations and applications of this inequality (see, e.g., [16]). In particular, Dragomir and Wang generalized this inequality to L p [a, b](p > 1) in [18] as follows: In this paper, we prove the following two weighted versions of this inequality. e main improvement is to consider general weights, but also, we prove the inequality for a larger class of functions, and we include the case p � 1. Furthermore, we prove that our inequality is sharp for every weight. (3) eorem 3 provides simple bounds, but they do not depend on the weight w. is theorem can be improved by the following bounds involving the weight.

Proofs of the Inequalities
Recall that a function f: e intermediate values theorem gives that there exists Assume first 1 < p < ∞. Hölder inequality gives e desired inequality holds since If p � 1 or p � ∞, then a similar and simpler argument gives the inequalities.

□
Proof of eorem 4. We can assume that f ′ ∈ L p [a, b], since otherwise the inequality trivially holds.
Since the integration by parts rule holds for absolutely continuous functions, we have and so, Assume first 1 < p < ∞. Hölder inequality gives us, discrete Hölder inequality and the inequality holds. Assume now that p � 1. Note that, since w ≥ 0, and the inequality also holds in this case.
If p � ∞, then a similar argument gives the inequality. Finally, let us prove (3). Fix w, 1 < p < ∞ and x ∈ [a, b], and define Since and so, f is an absolutely continuous function on [a, b] and f ′ ∈ L ∞ [a, b]. e argument in the proof of item (1) shows that it suffices to check that Note that, and so, the equality in the inequality in the first item is attained for this choice of f. Note that, if we substitute the weight w by the constant function 1 in eorem 4, then we get the classical inequality described in eorem 2.

On the Ostrowski Inequality in Conformable
and Nonconformable Context e evolution of many physical processes can be described in a more precise way by using fractional derivatives (see, e.g., [19][20][21][22][23][24]). Usually, it suffices to replace the time derivative in a given evolution equation by a fractional derivative. ere is a solid mathematical basis for proceeding this way (see, e.g., [23][24][25][26]). Recent developments on fractional calculus and its applications can be found in [27][28][29][30].
In several papers (see, e.g., [23,[31][32][33]) are defined local fractional derivatives in the following way. Given a function f(t), α ∈ (0, 1] and a kernel T(t, α), the derivative of f of order α at the point t with respect to the kernel T is defined by the following equation: Let I be an interval I⊆R, a, t ∈ I, α ∈ (0, 1] and T a positive continuous function on I × (0, 1]. In [33] the integral operator J α T,a is defined for every locally integrable function f on I as follows: Also, the integral operator J α T,a is said conformable if T(t, 1) � 1 for every t ∈ I. e following basic properties related to the operator G α T appear in [34].
Theorem 6 (see [34] eorem 2.5). Let I be an interval I⊆R , f, g: I ⟶ R and α ∈ R + . Assume that f, g are G α T -differentiable functions at t ∈ I . en the following statements hold: To review a good summary of some elementary properties associated with the integral operator J α T,a , we recommend reading the paper [33].
Theorem 8 (see [33] eorem 8). Let I be an interval I⊆R , a, b ∈ I and α ∈ R . Suppose that f, g are locally integrable functions on I , and k 1 , k 2 ∈ R . en, we have T,a f(t)(b) for every t ∈ I Proposition 1 (see [33] Proposition 6). Let I be an interval I⊆R , a ∈ I , α ∈ (0, 1] , T a positive continuous function on I × (0, 1] , and f a differentiable function on I such that f ′ is a locally integrable function on I . en, we have for all t ∈ I In [23] it is defined the integral operator J α T,a for the specific choice of the kernel T given by T(t, α) � t 1− α , and ( [23], eorem 3.1) shows for every continuous function f on I, a, t ∈ I and α ∈ (0, 1]. Hence, Proposition 2 below extends to any T this important equality (see [33]).
Proposition 2 (see [33] Proposition 7). Let I be an interval I⊆R , a ∈ I , α ∈ (0, 1] and T a positive continuous function on I × (0, 1] . en, for every continuous function f on I and a, t ∈ I .
For further information about this integral operator and its applications, we refer the readers to [25,[33][34][35].
Recall that the incomplete beta functionsB 1 and B 2 are defined, respectively, as follows:
By making the change of variables u � e − α(t− a) , we obtain 6 Journal of Mathematics By making the change of variables erefore, eorem 4 gives the inequalities. □ □ Proposition 5. Let f: [a, b] ⟶ R be an absolutely continuous function, and α < 0.

Conclusions
In this article, we continue with the study and development of an important topic in mathematics which are inequalities, particularly inequalities in fractional context. We prove two weighted versions of the generalized Ostrowsky inequality with a weight function w(t), as a consequence of these results we prove conformable and nonconformable fractional versions of this inequality, with the choice of the function w(t) � (1/(t − a) 1− α ) for the conformable case and w(t) � (1/e − α(t− a) ) for the nonconformable case, these functions have the form (1/T(t, α)), where T(t, α) represents the kernel of the fractional integral operator J α T,a . L p (φ, 〉 α )

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.