On a unified integral operator for φ-convex functions

Integral operators have a very vital role in diverse fields of science and engineering. In this paper, we use φ-convex functions for unified integral operators to obtain their upper bounds and upper and lower bounds for symmetric φ-convex functions in the form of a Hadamard inequality. Also, for φ-convex functions, we obtain bounds of different known fractional and conformable fractional integrals. The results of this paper are applicable to convex functions.

For ϕ(x, y) = xy, the ϕ-convex functions reduce to the convex functions. Note that every convex function is ϕ-convex, but the converse is not true.
Integral operators play a very vital role in the study of fractional derivatives and fractional integrals. Next, we give definitions of some integral operators, which will be utilized in the results of this paper. Definition 3 ([15]) Let f ∈ L[x 0 , y 0 ], and let g be a positive increasing function on (x 0 , y 0 ] with continuous derivative on (x 0 , y 0 ). The left-and right-sided fractional integral operators of f with respect to g on [x 0 , y 0 ] of order μ, where (μ) > 0, are given by μ-1 g (t)f (t) dt, x < y 0 , (1.6) where Γ is the gamma function.
Definition 4 ([16]) Let f ∈ L[x 0 , y 0 ], and let g be positive increasing function on (x 0 , y 0 ] with continuous derivative on (x 0 , y 0 ). The left-and right-sided k-fractional integral operators of f with respect to g on [x 0 , y 0 ] of order μ, where (μ), k > 0, are given by (1.8) where Γ k is the k-gamma function.
Then for x ∈ [x 0 , y 0 ], the left and right integral operators are given by (1.14) For the particular choice of Ψ , g and the parameters involved in Mittag-Leffler functions, several conformable and fractional integrals can be obtained; see [16,Remarks 6 and 7]. In [16], some bounds of the above operators have been proved for convex functions.
Theorem 2 Along with assumptions of the Theorem 1, if f is symmetric about x 0 +y 0 2 , then we have the following inequalities: Moreover, the following result is produced by defining unified operators for the convolution f * g of functions f and g.
Theorem 3 Let f , g : [x 0 , y 0 ] − → R be a differentiable functions such that |f | is convex, 0 < x 0 < y 0 , and g is a strictly increasing function. Also, let Ψ x be an increasing function, and let η, α, ξ , γ , ζ ∈ C, p, μ, δ ≥ 0, and 0 < k ≤ δ + μ. Then we have the following modulus inequality for x ∈ (x 0 , y 0 ): In Sect. 2, we use ϕ-convex functions to obtain bounds of integral operators given in Definition 6. Moreover, we achieve Hadamard-type bounds using the additional condition of symmetry. Also, we get some particular bounds by the ϕ-convexity of |f | and defining a convenient integral operator of convolution of two functions. In Sect. 3, we give some applications of the presented results.

Main results
Throughout this section, we assume that Proof We have the following inequality for the kernel defined in (1.14) and an increasing function g: Using the ϕ-convexity of f , we have Inequalities (2.4) and (2.5) constitute the following integral inequality: Using (1.12) of Definition 6 on the left-hand side of inequality (2.6) and integrating the right-hand side, we get Now using the same technique for t ∈ (x, y 0 ] and x ∈ (x 0 , y 0 ), we can write Using the ϕ-convexity of f , we have Inequalities (2.8) and (2.9) constitute the following inequality: Using (1.13) of Definition 6 on the left-hand side and integrating by parts the right-hand side, we get We obtain inequality (2.3) by summing (2.7) and (2.10).
We will need the following lemma in proving the upcoming result.
then we have the following inequality: Using f (x 0 + y 0x) = f (x) in this inequality, we get inequality (2.12).

Results for fractional and conformable integral operators
In this section, we give bounds of some fractional and conformable fractional integral operators deduced from the results of Sect. 2. [15].

Proposition 2 Under the assumptions of Theorem 4, we have
Proof Using g = I, η = p = 0, and μ = ν in the proof of Theorem 4, bound (3.2) is satisfied.

Concluding remarks
In this paper, we study the unified integral operators (1.12) and (1.13) for the notion of ϕ-convex functions. For ϕ-convex functions, we investigated bounds of these operators in different forms, which lead to bounds of several known fractional and conformable fractional integral operators. We identified some results for fractional integral operators in Sect. 3. Also, we identified connections with the known results.