Derivation of Bounds of an Integral Operator via Exponentially Convex Functions

In this paper, bounds of fractional and conformable integral operators are established in a compact form. By using exponentially convex functions, certain bounds of these operators are derived and further used to prove their boundedness and continuity. A modulus inequality is established for a diﬀerentiable function whose derivative in absolute value is exponentially convex. Upper and lower bounds of these operators are obtained in the form of a Hadamard inequality. Some particular cases of main results are also studied.


Introduction
We start with the definition of convex function.
Definition 1 (see [1]). A function f: [a, b] ⟶ R is said to be convex if holds for all x, y ∈ [a, b] and t ∈ [0, 1]. If inequality (1) is reversed, then the function f will be concave on [a, b]. Convex functions are very useful in many areas of mathematics and other subjects due to their fascinating properties and characterizations. eir geometric and analytic interpretations provide straightforward proofs of many mathematical inequalities including Hadamard, Jensen, Hölder, and Minkowski [1][2][3].
eoretically, convex functions have been generalized and extended as h-convex, m-convex, s-convex, (α, m)-convex, (h − m)-convex, (s, m)-convex, etc. Awan et al. [4] defined the function named exponentially convex function as follows: Definition 2. A function f: K ⊆ R ⟶ R, where K is an interval, is said to be an exponentially convex function if holds for all a, b ∈ K , t ∈ [0, 1] and α ∈ R. If the inequality in (2) is reversed, then f is called exponentially concave. If α � 0, then (2) gives inequality (1). For some recent citations and utilization of exponentially convex functions, one can see [5][6][7][8][9][10][11][12][13][14] and references therein. Our goal in this paper is to prove generalized integral inequalities for exponentially convex functions by using integral operators given in Definition 7. In the following, we give definitions of Riemann-Liouville fractional integrals: [a, b]. en, the left-sided and rightsided Riemann-Liouville fractional integral operators of order μ ∈ C(R(μ) > 0) are defined by Definition 4 (see [15]). Let f ∈ L 1 [a, b]. en, for k > 0, the k-fractional integral operators of f of order μ ∈ C, R(μ) > 0 are defined by A more general definition of the Riemann-Liouville fractional integral operators is given as follows: Definition 5 (see [16]). Let f: [a, b] ⟶ R be an integrable function. Also, let g be an increasing and positive function on (a, b], having a continuous derivative g ′ on (a, b). e left-sided and right-sided fractional integrals of a function f with respect to another function g on [a, b] of order μ ∈ C(R(μ) > 0) are defined by where Γ(.) is the gamma function.
Definition 6. Let f: [a, b] ⟶ R be an integrable function. Also, let g be an increasing and positive function on (a, b], having a continuous derivative g ′ on (a, b). e leftsided and right-sided k-fractional integral operators, k > 0, of a function f with respect to another function g on [a, b] of order μ, k ∈ C, R(μ) > 0 are defined by where Γ k (.) is the k-gamma function.
A compact form of integral operators defined above is given as follows: Definition 7 (see [17]). Let f, g: [a, b] ⟶ R, 0 < a < b be the functions such that f be positive and f ∈ L 1 [a, b] and g be differentiable and strictly increasing. Also, let (ϕ/x) be an increasing function on [a, ∞).
then Harmonic fractional integral operators given in [16] will be obtained and given as follows: (xiv) If we consider ϕ(t) � t μ ln t, then left-and rightsided logarithmic fractional integrals are obtained in [19] and given as follows: In the upcoming section, we will derive bounds of sum of the left-and right-sided integral operators defined in (11) and (12) for exponentially convex functions. ese bounds lead to produce results associated to several kinds of wellknown operators for exponentially convex functions, some of the results are presented in particular cases. Further in Section 3, bounds are presented in the form of a Hadamard inequality; several Hadamard type inequalities are obtained.  (11) and (12) holds

Bounds of Integral Operators and Their Consequences
Proof. For the kernel of integral operator (11), we have An exponentially convex function satisfies the following inequality: Inequalities (17) and (18) lead to the following integral inequality: while (19) gives Again, for the kernel of integral operator (12), we have An exponentially convex function satisfies the following inequality: Inequalities (21) and (22) lead to the following integral inequality: while (23) gives By adding (20) and (24), (16) can be achieved. e following remark connected the abovementioned theorem with already known results.

Theorem 2.
Let the assumptions of eorem 1 are satisfied. If f ∈ L ∞ [a, b], then integral operators defined in (11) and (12) are continuous.
Proof. From (20), we have It is given that Further g is increasing; therefore, we have g(a)).
(41) erefore, (39) gives If α > 0, then e − αx is decreasing on [a, b] and we get If α < 0, then e − αx is increasing on [a, b] and we get from (42) Hence, (F ϕ,g a + f)(x) is bounded and it is linear, and therefore, (F ϕ,g For a differentiable function f, as |f ′ | is exponentially convex, the following result holds: □ Theorem 3. Let f: I ⟶ R be a differentiable function. If |f ′ | is exponentially convex and g: I ⟶ R is a differentiable and strictly increasing function. Also, let (ϕ/x) be an increasing function on I, then for a, b ∈ I, a < b, the following inequalities for integral operators holds: where Proof. An exponentially convex function |f ′ | satisfies the following inequality: From which, we can write Inequalities (17) and (49) lead to the following integral inequality: while (50) gives From (48), we can write

Journal of Mathematics
Adopting the same method as we did for (49), the following integral inequality holds: From (51) and (53), (45) can be achieved. An exponentially convex function |f ′ | satisfies the following inequality: From which, we can write Inequalities (21) and (55) lead the following integral inequality: while (56) gives From (54), we can write Adopting the same method as we did for (55), the following inequality holds: From (57) and (59), (46) can be achieved.

Hadamard Type Inequalities for Exponentially Convex Function
In this section, we prove the Hadamard type inequality for an exponentially convex function. In order to prove this inequality result, we need the following lemma.
Proof. For the kernel of integral operator (11), we have An exponentially convex function satisfies the following inequality: Inequalities (62) and (63) lead the following integral inequality: while (64) gives On the contrary, for the kernel of integral operator (12), we have Inequalities (63) and (66) lead the following integral inequality: while the abovementioned inequality gives From (65) and (68), the following inequality can be obtained: Now, using Lemma 1 and multiplying (60) with From which, we have Again using Lemma 1 and multiplying (60) with From which, we have From (71) and (73), the following inequality can be achieved: From (69) and (74), (61) can be achieved.

Concluding Remarks
We have studied an integral operator for exponentially convex functions; this operator has direct consequences to several fractional and conformable integral operators. We have obtained bounds of the integral operator in different forms. In eorem 1, upper bounds of this operator are studied for an exponentially convex function and several Journal of Mathematics special cases have been presented in the form of propositions and corollaries. e boundedness is studied in eorem 2. In eorem 3, we have obtained results for differentiable function f such that |f ′ | is exponentially convex. A version of the Hadamard inequality is proved in eorem 4 which leads to its several variants for fractional and conformable integral operators.

Data Availability
No data were used to support this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.