Weighted fractional inequalities for new conditions on h -convex functions

We use a new function class called B -function to establish a novel version of Hermite–Hadamard inequality for weighted ψ -Hilfer operators. Additionally, we prove two new identities involving weighted ψ -Hilfer operators for diﬀerentiable functions. Moreover, by employing these equalities and the properties of the B -function, we derive several trapezoid-and midpoint-type inequalities for h -convex functions. Furthermore, the obtained results are reduced to several well-known and some new inequalities by making speciﬁc choices of the function h


Introduction & preliminaries
In recent decades, various publications have focused on generalizing the Hermite-Hadamard inequality and developing trapezoid-and midpoint-type inequalities that provide bounds for the right-and left-hand sides of the aforementioned inequality.The authors [11] demonstrated various similar trapezoid-type inequalities and developed the Hermite-Hadamard inequality for Riemann-Liouville fractional integrals.Kara et al. [8] identified the following Hermite-Hadamard inequalities: Let ψ : [a, b] → R be a monotone increasing function such that the derivative ψ > 0 is continuous on (a, b).If g is a convex function on [a, b], then where the ψ-Hilfer operators are defined as follows: and See [3,7,9,12] for further information on comparable results.
In [13], the author introduces a novel class of functions, called h-convex functions.
Definition 1 Let h : J ⊆ R → R, where (0, 1) ⊆ J, be a nonnegative function, h = 0. We say that f : I ⊆ R → R is an h-convex function if f is nonnegative and for all x, y ∈ I, α ∈ (0, 1) we have If the inequality in (1.2) is reversed, then f is said to be h-concave.

By setting
• h(λ) = λ, Definition 1 reduces to that of the classical convex function.
The weighted fractional integrals are defined as follows: J where w is a weighted function and the gamma function defined by For these operators, consider the following space: For special choices of ψ, w, and β, we get already known results.
(3) For ψ(t) = t and w(t) = 1, the operators are simplified to Riemann-Liouville integral operators.(4) Taking ψ(t) = t, w(t) = 1, and β = 1, the operators reduce to classical Riemann integrals.(5) Setting ψ(t) = ln(t) and a > 1, we get the weighted Hadamard operators of order β > 0. (6) Setting ψ(t) = ln(t), w(t) = 1, and a > 1, the operators are simplified to Hadamard operators of order β > 0. The purpose of this study is to generalize the Hermite-Hadamard inequality given in [8] for the h-convex function and weighted ψ-Hilfer operator with conditions.For this aim, we assume h is a B-function.

Hermite-Hadamard inequality
This section establishes Hermite-Hadamard-type inequalities for h-convex functions using ψ-Hilfer operators.Throughout this paper, we consider that 0 ≤ a < b < ∞, β > 0, and ψ is a positive differentiable increasing function on (a, b).
The following results are dependent on the function h presented in Theorem 2.1.First, assuming h(α) = α, we get the following result using the weighted ψ-Hilfer operators for convex functions.

Corollary 2 Let f ∈ X[a, b] be a convex function. Then the following inequalities hold:
where F(t) and (ψ, β) are defined by (2.2) and (2.3), respectively.
By setting h(α) = 1, we get the following result using the weighted ψ-Hilfer operators with an f being a P-function.
Using h(α) = α s , we obtain the following result through the weighted ψ-Hilfer operators and s-convex functions.
, we deduce the following result through the weighted ψ-Hilfer operators and n-fractional polynomial convex functions.

Weighted trapezoid-type inequalities
This section presents weighted trapezoid inequalities and their particular results utilizing weighted ψ-Hilfer operators with w being symmetric with respect to a+b 2 (i.e., w(t) = w(b + at)).To accomplish this, we must first establish an equality in the following lemma.

Lemma 3.1 Assume w is a differentiable and symmetric with respect to a+b
2 function, and suppose h is a B-function.Let f : [a, b] → R be a function where (wf ) is a differentiable mapping on (a, b).Then the following identity holds: where Integrating by parts (3.4) and using (2.2), we get Similarly, let Integrating by parts (3.6), we obtain On the other hand, since From (3.4), we get By changing the variable τ = 1+s 2 a + 1-s 2 b, we obtain Similarly, from (3.6) we deduce Consequently, Finally, we acquire the needed equality (3.1) by substituting (3.9) into (3.8).
A ψ,β (s) ds. (3.10) Proof Taking the absolute value of the identity (3.1) and using the h-convexity of the function |(wf ) |, we get Given that h is a B-function, setting α = 1-s 2 and 1α = 1+s 2 yields The following results are obtained via the weighted ψ-Hilfer operators and depend on the function h given in Theorem 3.1.
Notice that the inequality A p + B p ≤ (A + B) p yields the second inequality in (3.11).
Setting w = 1 and h(s) = s in Theorem 3.2, we get the following corollary.
which is a better estimate compared with [8, Theorem 3.5].

Weighted midpoint-type inequalities
This section establishes some weighted midpoint inequalities for weighted ψ-Hilfer operators using the identity in the following lemma.
Similarly, let (4.6) In addition, according to (4.2), Similarly, from (4.4) we get As a result, Proof Taking the absolute value of the identity (4.1) and using the h-convexity of |(wf ) | and inequality (1.4), we deduce This ends the proof.
The following results are obtained using the weighted ψ-Hilfer operators and depend on the function h given in Theorem 4.1.
(  (4.9) Proof Taking the absolute value of (4.1) and using the well-known Hölder's inequality, we obtain

Conclusions
In this study, we recalled a new function class, namely that of B-functions, and utilized it to derive a novel version of the Hermite-Hadamard inequality for weighted ψ-Hilfer operators.We also established two new identities involving weighted ψ-Hilfer operators for differentiable functions.By combining these identities and the properties of the B-function, we obtained several trapezoid-and midpoint-type inequalities for h-convex functions.
Our results not only extend the existing literature on inequalities involving fractional operators but also provide new insights into the behavior of h-convex functions under these operators.Additionally, our methods can be applied to other fractional integral operators by using B-functions.

Definition 2
presented a new class of function, called B-function.Let a < b and g : (a, b) ⊂ R → R be a nonnegative function.The function g is a B-function, or g belongs to the class B(a, b), if for all x ∈ (a, b), we have g(xa) + g(bx) ≤ 2g a + b 2 .(1.3)If the inequality (1.3) is reversed, g is called an A-function, or we say that g belongs to the class A(a, b).If we have the equality in (1.3), then g is called an AB-function, or we say that g belongs to the class AB(a, b).

Definition 3 (
[6]) Let [a, b] ⊆ [0, +∞).Let β > 0 and ψ be a positive, increasing differentiable function such that ψ (s) = 0 for all s ∈ [a, b].The left-and right-sided weighted fractional integrals of a function f with respect to the function ψ on [a, b] are respectively defined as follows:

Corollary 3
Let β > 0 and f ∈ X[a, b] be a P-function.Then the following inequalities hold:

Remark 2 Theorem 3 . 1
Putting w = 1 in Lemma 3.1, we get [8, Lemma 3.1].Under the hypotheses of Lemma 3.1, if |(wf ) | is an h-convex mapping on [a, b] and h is a B-function, then the trapezoid-type inequality holds, namely

1 p(Corollary 9 1 p = 1 .
and |(wf ) | p is an h-convex function, we conclude first inequality in (4.9).The second inequality in (4.9) is clear from the inequality A p + B p ≤ (A + B) p .Setting w = 1 and h(s) = s in Theorem 4.2, we get the following corollary.Let p > 1 and 1 p + If |f | p is a convex mapping on [a, b], then which is a better estimate compared with [8, Theorem 4.5].