On certain Ostrowski type integral inequalities for convex function via AB-fractional integral operator

: Weiinvestigate and prove a newilemma for twice di ﬀ erentiable functions with the fractional integral operator AB . Based on this newly developed lemma,iwe derive some newiresults about this identity. These new ﬁndings provide someigeneralizations of previous ﬁndings. This research buildsion a novel newiauxiliary result that allows us to create newivariants of Ostrowski typeiinequalities for twice di ﬀ erentiableiconvex mappings. Some of the newly presented results’ special cases are also discussed. As applications, several estimates involving special means of real numbers and Bessel functions are depicted.

Let g : J ⊆ → be a differentiableimapping on J o , the interior of the intervaliJ, such that g ∈ L [c, d], where c, d ∈ J with d > c.If |g (z)| ≤ K ⊆ , for all z ∈ [c, d], then the following inequality holds: holds.This result in the literature as the Ostrowski inequality.For recent result and their related some generalizations, variants and extensions concerning Ostrowski inequality (see [19][20][21][22][23][24][25][26][27]).This inequality yields an upper bound for the approximation of the integral average 1 Convexity has also played an importantirole in the advancement of inequalitiesitheory.Many well known results in inequalities theory can be obtained by exploiting the functions of convexity.Hermite Hadamard's doubleiinequality is one of the most extensively studied convexifunction results.This result provides us necessary and sufficient condition for a function to be convex.Hermite-Hadamard (H-H) inequality has been considered the most useful inequality in mathematical analysis in 1883.It is also known as classical equation of (H-H) inequality.Definition 1. Aifunction g : J ⊆ → is saidito be convex function if holdsifor all [c, d] ∈ J andiζ ∈ [0, 1].We sayithat g isiconcave if (−g) isiconvex.
TheiHermite-Hadamardiinequality assert that, ifia mapping g : J ⊂ → isiconvexiin J for c, d ∈ J and d > c, then Fractional calculus may be defined as an extension of the derivative operator idea from integer order n to arbitrary order.Fractionaliintegrals are strong tools in applied mathematics for solving a wide range of issues in science and engineering.Many mathematiciansihave merged and put effort and new ideas into fractional analysis in the present decadeito create a new dimension with various qualities in the field of mathematical analysis and appliedimathematics.Several studies have shown that fractional operators can accurately explain complex long-memory and multiscale phenomena in materials that are difficult to capture using standardimathematical methods including classical differential calculus [6,7].The significance of fractionalicalculus can be more understandable to analyzeireal world problems and several works involving fractional calculus have been done.
In recent years, some researchers have been interested in the concept of fractionaliderivative.Nonlocalifractional derivatives are classified into two types: Those with singular kernels, such as the Riemann-Liouville and Caputo derivatives, and those with nonsingular kernels, such as the Caputo-Fabrizio and Atangana-Baleanu derivatives.However, fractionaliderivative operators with non-singular kernels are very effective in resolving non-locality in real-worldiproblems.Later, we'll go through the Caputo-Fabrizioiintegral operator.

Results
In thisisection, we give Ostrowskiiinequalities for AB-fractionaliintegrals operator are obtained for twice differentiableifunctions on (c, d).For this purpose, we give a newiidentity that involve ABfractional integralsioperator whose secondiderivatives are convexifunctions.
Whichicompletes the proof.

Applications to bivariate numbers
Let's consider the followingispecial means for realinumbers c, d such that c d.

The arithmeticimean:
The logarithmic mean: The generalized logarithmic mean: Proof.Theiassertion follows from Corollaryi4 for theifunction g(u) = u k and k as specified above.

Conclusions
We defined the concept of fractional integral inequalities with convex second derivatives in this study.In addition, we investigated and demonstrated a novel lemma for the AB-fractional integral operator's second derivatives.The ideas presented in this study, we believe, will inspire scholars in functional analysis, information theory, and statistical theory.By using generalized convexities, it is possible to consider Ostrowski versions for generalized integral operators using Mittag-Leffler operators, for example.The findings reported in this article may encourage scholars to investigate comparable and more generic integral inequalities for a variety of other problems.
u) du by the value of g (u) at the point u ∈ [c, d].