fractional integral.

The Hermite-Hadamard inequality has been a cornerstone in the study of convex functions, providing accurate estimates for the average value of a function. In recent decades, the search for generalizations that encompass nonlocal operators and nonstandard geometries has driven the development of fractional calculus and new notions of convexity (see [1, 16, 17, 19]).

The notion of convexity (Λ, δ) presented in this work is not only a mathematical abstraction, but a unified framework that allows us to recover classic results from Godunova-Levin, Dragomir, and others, under a single formalism. This simplifies the study of inequalities by allowing us to treat various families of functions simultaneously.

Definition 1. Let D be a real interval and Λ : [0, 1] × D2 → D be a function. We say that Λ is a weighted path on D if there exist real functions Φ and Ψ of class C1 such that:

• [0, 1] ⊆ Dom(Φ) and [0, 1] ⊆ Dom(Ψ).

• Φ([0, 1]) ⊆ [0, 1] and Ψ([0, 1]) ⊆ [0, 1].

• Φ is strictly increasing on [0, 1] and Ψ is strictly decreasing on [0, 1].

• Λ(ς, ϑ, ρ) = Φ(ς)ϑ + Ψ(ς)ρ with ∂ ∂ς Λ ̸= 0.

For fixed values of ρ and ϑ, we denote Λϑ,ρ.

Definition 2. Let D ⊆ [0, +∞) be a real interval, φ : D → [0, +∞) a function, Λ(ς, ρ, ϑ) = Φ(ς)ϑ + Ψ(ς)ρ is a weighted path on D and δ : [0, 1] → Dom(Φ) ∩ Dom(Ψ) a non-negative function with δ ̸= 0. If inequality

φ (Φ(ς)ρ + mΨ(ς)ϑ) ≤ Φ(δ(ς))φ(ρ) + mΨ(δ(ς))φ(ϑ)

is fulfilled for all ρ, ϑ ∈ D and ς ∈ [0, 1] where m ∈ [0, 1]. Then the function φ is called (Λ, δ)-convex of the first type on D.

Remark 1. Considering Φ(ς) = ς, Ψ(ς) = 1 − ς and δ(ς) = hs(ς) with s ∈ (0, 1], one obtains the modified (h, m)-convexity of the first type on D (see [9], [10]). Moreover, several other types of convexity discussed in [20] can be recovered as particular cases of Definition 2:

• h-convexity (for Φ(ς) = ς and Ψ(ς) = 1 − ς);

• m-convexity (by taking Φ(ς) = δ(ς) = ς and Ψ(ς) = 1 − ς);

• (α, m)-convexity (by establishing Φ(ς) = ς, Ψ(ς) = 1 − ς and δ(ς) = ςα with α ∈ [0, 1]);

• Partially h-convexity (by selecting Φ(ς) = ς, Ψ(ς) = 1 − ς and m = 1);

• Classical convexity (if Φ(ς) = δ(ς) = ς, Ψ(ς) = 1 − ς and m = 1).

Definition 3. Let D ⊆ [0, +∞) be a real interval, φ : D → [0, +∞) a function, Λ(ς,ϑ, ρ) = Φ(ς)ϑ + Ψ(ς)ρ is a weighted path on D and δ : [0, 1] → Dom(Φ) ∩ Dom(Ψ) a nonnegative function with δ = ̸ 0. If inequality

φ (Φ(ς)ϑ + mΨ(ς)ρ) ≤ Φ(δs(ς))φ(ϑ) + mΨs(δ(ς))φ(ρ)

is fulfilled for all ρ, ϑ ∈ D and ς ∈ [0, 1] where m ∈ [0, 1] and s ∈ [−1, 1]. Then the function φ is called (Λ, δ)-convex of the second type on D.

Remark 2. Substituting Φ(ς) = ς and Ψ(ς) = 1 − ς, the (Λ, δ)-convexity of the second type reduces to the modified (h, m)-convexity of the second type on D (see [9], [10]). In addition, several other types of convexity presented in [20] can be obtained as particular cases of Definition 4:

• s-convexity in the second sense (by considering Φ(ς) = δ(ς) = ς, Ψ(ς) = 1 − ς and s ∈ (0, 1]);

• Godunova–Levin convexity (by setting Φ(ς) = δ(ς) = ς, Ψ(ς) = 1 − ς, s = −1 and m = 1);

• P-convexity (for Φ(ς) = δ(ς) = ς, Ψ(ς) = 1 − ς, s = 0 and m = 1).

By putting s = 1, the convexities mentioned in Remark 3 can be derived by choosing appropriate Φ, Ψ and δ.

The power of the (Λ, δ)-convexity framework lies in its ability to generalize beyond simple linear combinations. The conceptual key is the double nonlinearity:

(1) The argument of φ is evaluated at a nonlinear combination: Φ(ς)ϑ + Ψ(ς)ρ (2) The weights on the right-hand side of Definition 2 or Definition 4 are themselves transformed by the function δ (or its powers). Specifically, these weights become Φ(δ(ς)) and Ψ(δ(ς)) in the first kind, and Φ(δ (ς)) and (Ψ(δ(ς))) in the second kind.

This allows modeling convex behaviors with respect to unconventional interpolation paths, such as:

• Exponential path:

Φ(ς) = ee ς− −1 1 and Ψ(ς) = 1 − ee ς− −1 1

Φ(ς) = 2ς − 1 and Ψ(ς) = 2 − 2ς

• Trigonometric path:

sin( π 3 ς) sin( π 3 ς) Φ(ς) = sin π and Ψ(ς) = 1 − sin π ( 3 ) ( 3 )

cos( π 6 ς+ 32 π ) cos( π 6 ς+ 32 π ) Φ(ς) = cos 5π and Ψ(ς) = 1 − cos 5π ( 3 ) ( 3 )

Φ(ς) = tan( π 4 ς) and Ψ(ς) = 1 − tan( π 4 ς) Φ(ς) = 1−cos( 2 πς) and Ψ(ς) = 1 2 + cos( 2πς)

• Logarithmic path: Φ(ς) = ln(1+ ln 2ς) and Ψ(ς) = 1 − ln(1+ ln 2ς)

• Algebraic path: Φ(ς) = λ λς +1 r + λ+1 ς and Ψ(ς) = 1 − Φ(ς) with r ≥ 1 and λ ≥ 0

• Rational path: Φ(ς) = 1+ 2ςς and Ψ(ς) = 1 − 1+ 2ςς

It is thus possible to engineer bespoke forms of convexity, fine-tuned to the intricacies of particular problems.

Unlike traditional approaches that use fixed kernels, our use of extended heavy fractional integrals allows for unprecedented flexibility, accommodating operators with singular and non-singular kernels, such as Riemann-Liouville, Caputo-Fabrizio and Atangana-Baleanu.

Definition 4. Let f : R → R be a continuous function and let g : R → [0, ∞) be an integrable function. The convolution of f and g is defined by Z +∞ (f ∗ g)(ς) = f (ς − z)g(z) dz.

Definition 5. (see [22]) For φ ∈ L[α, β] and t ∈ (α, β), let w : D → R be a positive continuous function whose second derivative is integrable on the interior of D. The right and left-sided weighted fractional integral operators are defined respectively by

w ′′ t − ς Jα+ φ(t) = w φ(ς) dς, α β − α w ′′ ς − t Jβ− φ(t) = w φ(ς) dς.

t β − α Definition 6. (see [27]) Let φ ∈ L[α, β]. The Riemann–Liouville fractional integrals Iα η+φ and Iβ η−φ of order η > 0 with α ≥ 0 are defined by

η 1 η−1 Iα+φ(t) = (t − ς) φ(ς) dς, t > α, Γ(η) α and

η 1 η−1 Iβ−φ(t) = (ς − t) φ(ς) dς, t < β, Γ(η) t

respectively. Here, Γ(η) is the Gamma function and Iα 0+φ(t) = Iβ 0−φ(t) = φ(t).

Definition 7. (see [14]) Let φ ∈ H1(α, β), α < β and η ∈ [0, 1], then we define the Caputo-Fabrizio’s fractional integral (left and right respectively):

CF η 1 − η η α I φ (t) = φ(t) + φ(ς) dς, N(η) N(η) α

CF η 1 − η η Iβφ (t) = φ(t) + φ(ς) dς, N(η) N(η) t with H1(α, β) = { φ ∈ L2(α, β) : φ′ ∈ L2(α, β) } and N(η) a normalization function satisfying N(0) = N(1) = 1.

The Atangana-Baleanu (AB) integral operators are formulated as follows:

Definition 8. (see [8]) For β > α and η ∈ [0, 1], the left-sided fractional integral associated with the novel fractional derivative featuring a non-local kernel, for a function φ ∈ H1(α, β), is defined as:

AB η 1 − η η φ(µ) Iα+φ(t) = φ(t) + 1−η dµ.

N(η) N(η)Γ(η) α (µ − α) Definition 9. (see [7]) The right-sided fractional integral corresponding to the fractional derivative with a non-local kernel, for a function φ ∈ H1(α, β), is given by:

AB η 1 − η η φ(µ) Iβ−φ(t) = φ(t) + 1−η dµ, N(η) N(η)Γ(η) t (β − µ) in which β > α and η ∈ [0, 1].

Next we define the extended weighted fractional integral that we will use in our work.

Definition 10. Let φ ∈ L[α, β], w : D → R be a continuous and positive function with its second derivative integrable on D◦, Λ (ς, ϑ, ρ) is a weighted path on D and ρ, ϑ ∈ [α, β]. Then the extended weighted fractional integral is defined as:

Lw φ Λϑ,ρ(ς) = w′′(ς)φ (Λϑ,ρ(ς)) dς.

Remark 3. The inclusion of second derivative of the weight function w arises from the inherent nature of the problem. Alternatively, the first derivative may also be considered.

Remark 4. Consider the linear interpolation Λϑ,ρ(ς) = (1 − ς)ρ + ςϑ with ρ < ϑ. By making the substitution x = ρ + (ϑ − ρ)ς, the extended weighted fractional integral of φ coincides with ϑ− 1 ρ Jρ w + g(ϑ).

Similarly, for the linear interpolation in the reverse order Λρ,ϑ(ς) = (1−ς)ϑ+ςρ where ρ < ϑ and the substitution y = ϑ + (ρ − ϑ)ς, the extended weighted fractional integral of φ coincides with ϑ− 1 ρ Jϑ w − φ (ρ).

This article establishes novel integral inequalities within the framework of second-type (Λ, δ)-convex functions by employing extended weighted fractional integrals. Various implications for fractional integrals of the Riemann-Liouville and Atangana-Baleanu types are explored throughout the study.

We now turn to the analysis of the Hermite–Hadamard inequality within the framework of extended weighted fractional integrals. Our goal is to derive new bounds under the assumption of (Λ, δ)-convexity of the second type, which generalizes several classical convexity notions discussed in the previous section.

In what follows, we establish a Hermite-Hadamard type inequality adapted to the extended weighted integral operator introduced above.

Theorem 1. Let α, β ∈ Do with α < β and let φ : D → R be a function that satisfies the (Λ, δ)- convexity condition of the second type. Consider a weight function w : [0, 1] → R and Λ(ς, β, α) = Φ(ς)β + Ψ(ς)α a weighted path on D such that Φ( 1 2 ) = Ψ( 1 2 ) = 1 2 . Suppose that w′′(ς) ≥ 0 for all ς ∈ (0, 1), α, m β ⊂ Do, m ∈ [0, 1] and s ∈ [−1, 1]. Under these assumptions, we deduce:

α + β ′ ′ s 1 w ¯ 1 s w ¯ φ w (1) − w (0) ≤ Φ δ 2 Lφ Λβ,α(ς) + Ψ δ 2 Lφ Λα,β(ς) (1)

h i Z 1

s 1 1 s ′′ s

≤ Φ δ 2 φ(β) + Ψ δ 2 φ(α) w (ς) Φ(δ (ς)) dς

h i Z 1

s 1 α 1 s β ′′ s

+ m Φ δ 2 φ m + Ψ δ 2 φ m w (ς) (1 − Φ(δ(ς))) dς,

where Λ ¯β,α(ς) = Φ(ς)β + (1 − Φ(ς)) α.

Proof : For α < β, ς = 1 2 and m = 1 in Definition 4, we get:

α + β 1 1 s 1 1 s φ = φ Φ α + Ψ β ≤ Φ δ 2 φ (α) + Ψ δ 2 φ (β) .

2 2 2

Taking into account Λ ¯α,β(ς) = Φ(ς)α + (1 − Φ(ς)) β, we find:

α + β Λ ¯β,α(ς) + Λ ¯α,β(ς) s φ = φ ≤ Φ δs 1 φ Λ ¯β,α(ς) + Ψ δ 1 φ Λ ¯α,β(ς) .

2 2 2 2

Multiplying both members of the previous inequality by w′′(ς), integrating over the interval [0, 1] with respect to ς, we obtain:

Z 1 Z 1 Z 1 α + β ′′ s 1 ′′ ¯ 1 s ′′ ¯ φ w (ς) dς ≤ Φ δ 2 w (ς)φ Λβ,α(ς) dς + Ψ δ 2 w (ς)φ Λα,β(ς) dς 2 0 0 0 = Φ δs 1 LwΛ ¯β,α(ς) + Ψ δ 1 s LwΛ ¯α,β(ς).

2 φ 2 φ

In this way we have the first part of the desired inequality.

By means of second type (Λ, δ)- convexity property of φ, one has:

φ Λ ¯β,α(ς) = φ Φ(ς)β + m (1 − Φ(ς)) α ≤ Φ (δs(ς)) φ(β) + m (1 − Φ (δ(ς)))s φ α . (2)

In an analogous way,

φ Λ ¯α,β(ς) ≤ Φ (δs(ς)) φ(α) + m (1 − Φ (δ(ς)))s φ β . (3)

s 1 1 s

Multiplying (2) by Φ δ 2 and (3) by Ψ δ 2 and adding both results leads us to:

Φ δs 1 φ Λ ¯β,α(ς) + Ψ δ 1 s φ Λ ¯α,β(ς)

s 1 s s α

≤ Φ δ 2 Φ (δ (ς)) φ(β) + m (1 − Φ (δ(ς))) φ m

1 s s s β

+ Ψ δ 2 Φ (δ (ς)) φ(α) + m (1 − Φ (δ(ς))) φ m .

Finally, multiplying by w′′(ς), integrating over the interval [0, 1] with respect to ς and suitably rearranging the terms, we reach the right-hand side of (1). This concludes the proof. □ To illustrate the above theorem, we will perform a numerical verification. This theorem establishes an upper bound for the heavy integral operator Lw φ based on the convexity of φ.

Example 1. We select the following parameters.

• φ(µ) = µ2;

• [α, β] = [1, 2];

• w(ν) = ν2 + 1, for which w′′(ν) = 2;

• Φ(ν) = ν, Ψ(ν) = 1 − ν;

• δ(ν) = ν, m = s = 1.

To begin with, w′(1) − w′(0) = 2.

The corresponding weighted paths are given by

Λ ¯β,α(ν) = 2ν + (1 − ν) = 1 + ν,

and Λ ¯α,β(ν) = ν + 2(1 − ν) = 2 − ν.

Consequently, w ¯ 2 14 Lφ Λβ,α(ν) = 2(1 + ν) dν = ,

whereas w ¯ 2 14 Lφ Λα,β(ν) = 2(2 − ν) dν = .

Since δ = ,

the middle expression in (1) is evaluated as 1 w ¯ 1 w ¯ 14 Lφ Λβ,α(ν) + Lφ Λα,β(ν) = .

2 2 3

On the other hand, α + β ′ ′ 3 9 φ w (1) − w (0) = φ · 2 = .

2 2 2

A simple calculation yields Z 1 Z 1 w′′(ν) Φ(δs(ν)) dν = 2ν dν = 1,

and Z 1 Z 1

′′ s

w (ν) (1 − Φ(δ(ν))) dν = 2(1 − ν) dν = 1.

Accordingly, the right-hand side of (1) is 1 1 1 1 φ(2) + φ(1) + φ(1) + φ(2) = 5.

2 2 2 2

Therefore,

≤ ≤ 5 .

Corollary 1. Let the hypotheses of Theorem 15 be satisfied for Φ(ς) = ς and Ψ(ς) = 1 − ς. Under this specification, one has, for all λ ∈ N ∪ {0} :

α + β ′ ′ λ + 1 s 1 w β+λα 1 w α+λβ φ (w (1) − w (0)) ≤ δ Jα+ φ λ+1 + 1 − δ Jβ− φ λ+1 2 β − α 2 2 " # Z s 1 1 ′′ s ς ≤ δ φ(β) + 1 − δ φ(α) w (ς) δ dς (4) 2 2 0 λ + 1 " # Z

s 1 s

s 1 α 1 β ′′ ς + m δ φ + 1 − δ φ w (ς) 1 − δ dς.

2 m 2 m 0 λ + 1

Proof : Note that

α + β u + v s 1 ς λ + 1 − ς φ = φ ≤ δ φ β + α 2 2 2 λ + 1 λ + 1 1 ς λ + 1 − ς + 1 − δ φ α + β , 2 λ + 1 λ + 1 where ς λ + 1 − ς ς λ + 1 − ς u = β + α, v = α + β, λ + 1 λ + 1 λ + 1 λ + 1 with ς ∈ [0, 1].

Integrating both sides against w′′(ς) over [0, 1] gives:

Z 1 Z 1 α + β ′′ s 1 ′′ ς λ + 1 − ς φ w (ς) dς ≤ δ w (ς)φ β + α dς 2 0 2 0 λ + 1 λ + 1 s Z 1 1 ′′ ς λ + 1 − ς + 1 − δ w (ς)φ α + β dς.

2 0 λ + 1 λ + 1

After performing the change of variables induced by u and v, we obtain:

α + β ′ ′ φ (w (1) − w (0)) " Z β+λα " # λ + 1 s 1 λ+1 ′′ (u − α) ≤ h w β−α φ(u) du β − α 2 α λ+1 Z " # # 1 ′′ (β − v) + 1 − h w β−α φ(v) dv 2 α λ + +1 λβ λ+1 λ + 1 s 1 w β+λα 1 w α+λβ = δ Jα+ φ λ+1 + 1 − δ Jβ− φ λ+1 .

β − α 2 2

This verifies the first bound in (4). To proceed further, we invoke the (h, m)-convexity of φ:

ς λ + 1 − ς φ β + α (5) λ + 1 λ + 1

s ς ς α ≤ δ φ(β) + m 1 − δ φ .

λ + 1 λ + 1 m

Analogously, ς λ + 1 − ς φ α + β (6) λ + 1 λ + 1

s ς ς β ≤ δ φ(α) + m 1 − δ φ .

λ + 1 λ + 1 m

s 1 1 s

Multiplying (5) by δ 2 and (6) by 1 − δ 2 and adding both results, we find:

s 1 ς λ + 1 − ς δ φ β + α 2 λ + 1 λ + 1 1 ς λ + 1 − ς + 1 − δ φ α + β 2 λ + 1 λ + 1

s 1 s ς ς α ≤ δ δ φ(β) + m 1 − δ φ 2 λ + 1 λ + 1 m

1 s ς ς β + 1 − δ δ φ(α) + m 1 − δ φ .

2 λ + 1 λ + 1 m

Multiplying by w′′(ς), integrating over [0, 1] and collecting terms provide the second bound in (4). This completes the proof. □

Remark 5. Corollary 17 coincides with Theorem 1 of [13].

Remark 6. Setting w′′(ς) = 1, λ = 0, m = s = 1 and δ(ς) = ς in (4) yields the classical Hermite-Hadamard inequality.

Remark 7. Under the same configuration as in Remark 19, replacing w′′(ς) by exp(η −γς) with γ = 1− ηη (β − α), leads to Theorem 1 of [2].

Remark 8. Choosing w′(ς) = ςη η together with λ = 0, m = s = 1 and δ(ς) = ς transforms (4) into an expression involving Riemann-Liouville fractional integrals. In this case, one retrieves Theorem 2 in [27]: α + β Γ(η + 1) η η φ(α) + φ(β) φ ≤ η Iβ−φ(α) + Iα+φ(β) ≤ .

2 2(β − α) 2 Remark 9. For w′(ς) = ςη η and λ = m = s = 1, relation (4) reduces to Theorem 4 in [28]. Furthermore, when restricted to s-convex functions with λ = 0, it reproduces the main result of Theorem 2.1 in [30]. Additional related statements are discussed in Theorem 3 of [29] for w′(ς) = ςη. Under analogous assumptions, results for m-convex functions, including Theorem 5 of the present work, follow in a direct manner.

Remark 10. Several results formulated in terms of k-Riemann–Liouville fractional integrals, such as Theorem 5 in [31] (see also Theorem 1 in [24]), can be derived from Corollary 17 by selecting w′(ς) = ς η k , k > 0, λ = 0, m = s = 1 and δ(ς) = ς.

Remark 11. Let η ∈ [0, 1], λ = 0, m = s = 1, and δ(ς) = ς in (4), which corresponds to the convex framework. Assume further that ′′ ης η k −1 w (ς) = , kN(η)Γk(η) with k > 0. Under these assumptions, the left-hand side of (4) takes the form:

α + β 1 1 η β φ(u) φ ≤ η 1− η du 2 N(η)Γk(η) 2(β − α) k kN(η)Γk(η) α (u − α) k

η β φ(u) + 1− η du .

kN(η)Γk(η) α (β − u) k This relation can equivalently be written as:

η Z β α + β 2(β − α) k η φ(u) φ ≤ 1− η du (7) 2 N(η)Γk(η) kN(η)Γk(η) α (u − α) k η φ(u) + 1− η du.

kN(η)Γk(η) α (β − u) k

Adding the quantity N 1− (η η) (φ(α) + φ(β)) to both sides of (7) and using the estimate

α + β φ(α) + φ(β) φ ≤ ,

one arrives at:

" η # α + β 2(β − α) k (1 − η) AB η AB η φ + ≤ Iα+φ(β) + Iβ−φ(α). (8) 2 N(η)Γk(η) N(η)

A comparable argument applied to the right-hand side of (1) yields:

" Z Z # 1 η β φ(u) η β φ(u) η 1− η du + 1− η du 2(β − α) k kN(η)Γk(η) α (u − α) k kN(η)Γk(η) α (β − u) k 1 φ(α) + φ(β) N(η)Γk(η) 2

η (1−η)

After multiplying by 2(β − α) k , adding N(η) (φ(α) + φ(β)) and simplifying, we obtain:

" η # AB η AB η 2(β − α) k (1 − η) φ(α) + φ(β) Iα+φ(β) + Iβ−φ(α) ≤ + . (9) N(η)Γk(η) N(η) 2

Together (8) and (9) produce a bound closely related to Theorem 6 in [18]. In particular, the choice k = 1 leads to a formulation comparable with Proposition 2.1 in [12].

Corollary 2. By fixing m = 1, δ(ς) = ς, λ = 0, η ∈ [0, 1] and w′′(ς) = 1 in (4), one obtains:

s−1 α + β 1 φ(α) + φ(β) 2 φ ≤ φ(ς) dς ≤ .

2 β − α α s + 1

Remark 12. Corollary 25 is equivalent to Theorem 1.5 of [23].

Remark 13. By putting s = 1 in Corollary 25, one recovers Theorem 2 of [14] after straightforward algebraic manipulations.

Remark 14. For every t ∈ (α, β), the classical integral mean can be represented in terms of the left and right Caputo-Fabrizio fractional integrals (see Definition 9) as follows:

Z " Z 1 β N(η) 1 − η η t φ(ς) dς = φ(t) + φ(ς) dς β − α α η(β − α) N(η) N(η) α

1 − η η β 2(1 − η) + φ(t) + φ(ς) dς − φ(t) N(η) N(η) t N(η)

N(η) CF η CF η 2(1 − η) = α I φ (t) + Iβφ (t) − φ(t) .

η(β − α) N(η) Consequently, Theorem 2.1 of [23] is a direct outcome of combining this representation with Corollary 25.

Remark 15. Taking φ(ς) = (f ∗ g)(ς) in (4), a new formulation for convolution functions emerges:

(f ∗ g) α+ 2 β (w′(1) − w′(0))

λ + 1 s 1 w β+λα 1 s w α+λβ ≤ h 2 Jα+ (f ∗ g) λ+1 + 1 − h 2 Jβ− (f ∗ g) λ+1 β − α " Z # h i 1

s 1 1 s ′′ s ς

≤ δ 2 (f ∗ g)(β) + 1 − δ 2 (f ∗ g)(α) w (ς) δ λ+1 dς " Z # h i 1 s

s 1 α 1 s β ′′ ς

+ m δ 2 (f ∗ g) m + 1 − δ 2 (f ∗ g) m w (ς) 1 − δ λ+1 dς .

The previous remarks highlight the versatility of the main result and indicate how further estimates associated with alternative fractional operators may be obtained.

In the rest of the article, to simplify the notation of some mathematical expressions, we will adopt the following notations:

T1 := w(1) [Λβ,κ(1) + Λα,κ(1)] − w(0) [Λβ,κ(0) + Λα,κ(0)]

′ ∂Λα,κ ′ ∂Λβ,κ

−w(1) φ (Λα,κ(1)) ∂ς (1) + φ (Λβ,κ(1)) ∂ς (1)

′ ∂Λα,κ ′ ∂Λβ,κ

+w(0) φ (Λα,κ(0)) ∂ς (0) + φ (Λβ,κ(0)) ∂ς (0)

−Lw φ Λβ,κ(ς) − Lw φ Λα,κ(ς) (10)

β − κ ′ α − κ ′ T2 := w(1) φ (β) − φ (α) λ + 1 λ + 1 α − κ ′ λα + κ β − κ ′ λβ + κ +w(0) φ − φ λ + 1 λ + 1 λ + 1 λ + 1 ′ ′ λα + κ λβ + κ −w (1) φ(α) + φ(β) + w (0) φ + φ λ + 1 λ + 1 λ + 1 w λα + κ λ + 1 w λβ + κ + Jα+ φ + Jβ− φ (11) κ − α λ + 1 β − κ λ + 1

β − κ ′ α − κ ′ T3 : = w(1) (f ∗ g) (β) − (f ∗ g) (α) λ + 1 λ + 1 α − κ ′ λα + κ β − κ ′ λβ + κ +w(0) (f ∗ g) − (f ∗ g) λ + 1 λ + 1 λ + 1 λ + 1 ′ ′ λα + κ λβ + κ −w (1) (f ∗ g) (α) + φ(β) + w (0) (f ∗ g) + (f ∗ g) λ + 1 λ + 1 λ + 1 w λα + κ λ + 1 w λβ + κ + Jα+ (f ∗ g) + Jβ− (f ∗ g) (12) κ − α λ + 1 β − κ λ + 1

The subsequent identity will serve as a fundamental tool moving forward.

Lemma 1. Let be a function φ defined over a closed interval D ⊂ R, which it is twice differentiable in the interior Do. Consider a weight function w : [0, 1] → R and Λ(ς, ϑ, ρ) = Φ(ς)ϑ + Ψ(ς)ρ a weighted path with ϑ, ρ ∈ [α, β] ⊂ Do. Provided that φ′′ belongs to L[α, β], the subsequent equality can be established:

Z 1 2

′′ ∂Λα,κ

T1 = w(ς)φ (Λα,κ(ς)) ∂ς (ς) dς + E1 Z 1 2

′′ ∂Λβ,κ

+ w(ς)φ (Λβ,κ(ς)) ∂ς (ς) dς + E2, (13)

for T1 defined in (10) and κ ∈ [α, β]. Moreover,

′ ∂2Λα,κ

E1 = w(ς)φ (Λα,κ(ς)) ∂ς2 (ς) dς,

′ ∂2Λβ,κ

E2 = w(ς)φ (Λβ,κ(ς)) ∂ς2 (ς) dς.

Proof : Taking into account the Definition (12), we find:

Z 1 Z 1 Lw φ Λα,κ(ς) + Lw φ Λβ,κ(ς) = w′′(ς)φ (Λα,κ(ς)) dς + w′′(ς)φ (Λβ,κ(ς)) dς = (I1) + (I2) (14)

By applying integration by parts twice, we obtain:

′ ′ ′ ′ ∂Λα,κ

(I1) = w (1)φ (Λα,κ(1)) − w (0)φ (Λα,κ(0)) − w (ς)φ (Λα,κ(ς)) ∂ς (ς) dς

= w′(1)φ (Λα,κ(1)) − w′(0)φ (Λα,κ(0))

′ ∂Λα,κ ′ ∂Λα,κ

− w(1)φ (Λα,κ(1)) ∂ς (1) − w(0)φ (Λα,κ(0)) ∂ς (0)

1 h 2 2 i

′′ ∂Λα,κ ′ ∂ Λα,κ

− w(ς) φ (Λα,κ(ς)) ∂ς (ς) + φ (Λα,κ(ς)) ∂ς2 (ς) dς . (15)

In a manner analogous to the above,

(I2) = w′(1)φ (Λβ,κ(1)) − w′(0)φ (Λβ,κ(0))

′ ∂Λβ,κ ′ ∂Λβ,κ

− w(1)φ (Λβ,κ(1)) ∂ς (1) − w(0)φ (Λβ,κ(0)) ∂ς (0)

1 h 2 2 i

′′ ∂Λβ,κ ′ ∂ Λβ,κ

− w(ς) φ (Λβ,κ(ς)) ∂ς (ς) + φ (Λβ,κ(ς)) ∂ς2 (ς) dς . (16)

Equation (13) emerges once (14)–(16) are considered together and their terms are rearranged. □

Corollary 3. Let the assumptions of Lemma 30 be in force. We set

ς λ + 1 − ς Λα,κ(ς) = α + κ, λ + 1 λ + 1 ς λ + 1 − ς Λβ,κ(ς) = β + κ, λ + 1 λ + 1 where λ ∈ N ∪ {0}. With this specification, the identity below follows:

2 Z 1 α − κ ′′ ς λ + 1 − ς T2 = w(ς)φ α + κ dς (17) λ + 1 0 λ + 1 λ + 1 2 Z 1 β − κ ′′ ς λ + 1 − ς + w(ς)φ β + κ dς.

λ + 1 0 λ + 1 λ + 1

with T2 introduced in (11) and κ ∈ [α, β].

Proof : First we denote

2 Z 1 α − κ ′′ ς λ + 1 − ς (I1) = w(ς)φ α + κ dς.

λ + 1 0 λ + 1 λ + 1 2 Z 1 β − κ ′′ ς λ + 1 − ς (I2) = w(ς)φ β + κ dς.

λ + 1 0 λ + 1 λ + 1

We apply integration by parts twice to (I1) together with the concept of weighted fractional integrals introduced in [9], giving:

α − κ ′ ′ λα + κ (I1) = − w(1)φ (α) − w(0)φ λ + 1 λ + 1

′ ′ λα + κ − w (1)φ(α) − w (0)φ λ + 1

′′ ς λ + 1 − ς + w (ς)φ α + κ dς 0 λ + 1 λ + 1 α − κ ′ ′ λα + κ = − w(1)φ (α) − w(0)φ λ + 1 λ + 1

′ ′ λα + κ − w (1)φ(α) − w (0)φ λ + 1 Z λα+κ λα+κ !

λ + 1 ( λ+1 ) − µ + w′′ λα λ+1 +κ φ(µ) dµ κ − α α λ+1 − α α − κ ′ ′ λα + κ = − w(1)φ (α) − w(0)φ λ + 1 λ + 1

′ ′ λα + κ − w (1)φ(α) − w (0)φ λ + 1 λ + 1 w λα + κ + Jα+ φ .

κ − α λ + 1

In a manner similar to the above but for (I2), β − κ ′ ′ λβ + κ (I2) = w(1)φ (β) − w(0)φ λ + 1 λ + 1

′ ′ λβ + κ − w (1)φ(β) − w (0)φ λ + 1 λ + 1 w λβ + κ + Jβ− φ . (18) β − κ λ + 1

Adding (18) with (18) and conveniently grouping the terms, it follows (17). □

Remark 16. Corollary 31 coincides with Lemma 2.1 of [22].

Corollary 4. Assume that the hypotheses of Lemma 30 are satisfied. Consider the weight function

w(ς) = ς(1 − ς)

together with the weighted path Λα,β(ς) = ςα + (1 − ς)β.

for ς ∈ (0, 1). Then Z β 2 Z 1 φ(α) + φ(β) 1 (β − α) ′′ − φ(z) dz = ς(1 − ς)φ (ςα + (1 − ς)β) dς. (19) 2 β − α α 2 0

Proof : By applying the change of variable z = ςα + (1 − ς)β, the integral on the right-hand side of (19) becomes: Z 1 Z β ′′ 1 ′′ ς(1 − ς)φ (ςα + (1 − ς)β) dς = 3 (z − β)(α − z)φ (z) dz. (20) 0 (β − α) α

Integrating by parts in (20), yields:

Z β Z β

′′ ′ β ′

(z − β)(α − z)φ (z) dz = [(z − β)(α − z)φ (z)]α − (α + β − 2z)φ (z) dz

= − (α + β − 2z)φ′(z) dz.

A second integration by parts, produces:

Z β Z β − (α + β − 2z)φ (z) dz = − [(α + β − 2z)φ(z)]α + 2 φ(z) dz

= (β − α)(φ(α) + φ(β)) + 2 φ(z) dz. (21)

Combining (21) and (22), we obtain:

Z β Z β (z − β)(α − z)φ′′(z) dz = (β − α)(φ(α) + φ(β)) + 2 φ(z) dz. (22)

Substituting (23) into (20) gives:

Z " Z # 1 ′′ 1 β ς(1 − ς)φ (ςα + (1 − ς)β) dς = 3 (β − α)(φ(α) + φ(β)) + 2 φ(z) dz 0 (β − α) α φ(α) + φ(β) 2 = 2 + 3 φ(z) dz. (23) (β − α) (β − α) α

Finally, multiplying both sides of (23) by (β− 2α)2 leads to (19). □

Remark 17. The identity established in Corollary 33 is exactly the one appearing as Lemma 1 in [26].

In this paper, we established a broad family of Hermite–Hadamard type inequalities for twice differentiable functions within the framework of weighted paths and extended weighted fractional integral operators. The main contributions of this work can be summarized as follows.

A Fundamental Identity. The cornerstone of this study is the derivation of a new integral identity involving the second derivative of the underlying function. This identity provides a unified starting point for obtaining a variety of estimates and extends several classical identities previously used in the literature.

General Framework Based on Weighted Paths. By introducing the weighted path Λ together with the concept of (Λ, δ)-convexity, the obtained results are formulated in a highly flexible setting. This approach substantially enlarges the class of admissible convex functions and includes numerous existing notions of convexity as particular cases through suitable choices of Φ, Ψ, δ, m, and s.

Refined Hermite–Hadamard Inequalities. By combining the fundamental identity with Hölder’s, Power Mean and Young’s inequalities, several new estimates were established for twice differentiable functions whose second derivatives satisfy generalized convexity assumptions. These bounds provide quantitative refinements that extend many known Hermite–Hadamard type inequalities.

Recovery of Classical Results. An important feature of the proposed framework is its unifying character. Appropriate specifications of the weight function and the interpolation path recover several well-known inequalities involving the classical integral, weighted fractional integrals, Riemann–Liouville, Caputo–Fabrizio and Atangana– Baleanu fractional integral operators. This confirms the consistency and generality of the developed theory.

As natural extensions of this research, the study of these inequalities in the context of coordinate functions (convexity in the plane) is proposed, as well as the exploration of applications in information theory and convex optimization, where the control of integral bounds is critical for the convergence of algorithms.

The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.

L.F.A Gómez: Conceptualization, Formal analysis, Visualization, Investigation, Writing – original draft. J.E. Nápoles Valdés: Validation, Funding acquisition, Methodology, Project administration, Resources, Supervi-sion, Writing – review & editing.

All authors have read and agreed to the published version of the manuscript.

The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions. Data Availability All data supporting the reported findings in this research paper are provided within the manuscript.

The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.