Conformable Integral Inequalities of the Hermite-Hadamard Type in terms of GG-and GA-Convexities

for all κ1, κ2 ∈ I. It is well known that the convexity has been playing a key role in mathematical programming, engineering, and optimization theory. Recently, many generalizations and extensions for the classical convexity can be found in the literature [2–14]. In [15, 16], Niculescu defined the GAand GG-convex functions as follows. Definition 1 (see [15]). A function h : I 󳨀→ [0,∞) is said to be GA-convex if the inequality h (κt 1κ1−t 2 ) ≤ th (κ1) + (1 − t) h (κ2) (2)

The conformable fractional derivative of order 0 <  ≤ 1 at  > 0 for a function ℎ : [0, ∞) → R was defined in [18] as follows: ℎ is said to be -fractional differentiable if the conformable fractional derivative of ℎ of order  exists.The fractional derivative at 0 is defined as ℎ  (0) = lim →0 + ℎ  ().Theorem 8 for the conformable fractional derivative can be found in the literature [18].
where the integral is the usual Riemann improper integral and  ∈ (0, 1].
Anderson [38] provided the conformable integral version of the HH inequality as follows.
In this paper, we shall establish the Hermite-Hadamard type inequalities for GA and GG-convex functions via conformable fractional integrals and give their applications in the special bivariate means.

Main Results
In order to establish our main results, we need a lemma which we present in this section.
Proof.Using integration by parts, we have By the change of the variable  =   2  1− 1 and integration by parts, we have Now multiplying by (log  2 − log  1 ), we obtain the required result.
The desired result can be obtained by evaluating the above integral.
The desired result can be obtained by evaluating the above integral.     ℎ  ( 1 )       )) Proof.By using Lemma 12 we clearly see that Since  > 1, we can choose  > 1 such that  −1 +  −1 = 1.Applying the Hölder integral inequality and the GGconvexity of |ℎ  |  we have The desired result can be obtained by evaluating the above integral. Theorem The desired result can be obtained by evaluating the above integral. Theorem The desired result can be obtained by evaluating the above integrals.
Remark 19.By setting  = 1 in inequality ( 27), we regain inequality (5).,  (+1) 1 )) Proof.From the GA-convexity of |ℎ  |  , the power mean inequality, the property of the modulus, and Lemma 12 we clearly see that The desired result can be obtained by evaluating the above integrals.

Theorem 22.
If the function ℎ satisfies the conditions of Lemma 12 and, additionally, if |ℎ  ()| ( > 1) is GA-convex, then we have the following inequality: Proof.With the help of the GA-convexity of |ℎ  |  , the power mean inequality, the property of the modulus, and Lemma 12, we can write The desired result can be obtained by evaluating the above integrals.
Theorem 24.If ,  > 1 with  −1 + The desired result can be obtained by evaluating the above integrals.
In this section, we use the results obtained in Section 2 to present several applications to the arithmetic mean logarithmic mean and (, )-th generalized logarithmic mean Proof.Let for  > 0. Then |ℎ  ()|  is a GA-convex function on R + for  ≥ 1.Let  = 1.Then making use of function (39) in Theorem 18, we obtain the required result.

Conclusions
In the article, we derive the conformable fractional integrals' versions of the Hermite-Hadamard type inequalities for GG-and GA-convex functions.Our approach is based on an identity involving the conformable fractional integrals, the Hölder inequality, and the power mean inequality.The proven results generalized some previously obtained results.
As applications, we provide several inequalities for some special bivariate means.The present idea may stimulate further research in the theory of inequalities for other generalized integrals, for example, as presented in [35][36][37].

Theorem 14 .
If the function ℎ satisfies the conditions of Lemma 12 and, additionally, if |ℎ  ()| is GG-convex, then we

Theorem 20 .
If the function ℎ satisfies the conditions of Lemma 12 and, additionally, if |ℎ  ()|  ( > 1) is GA-convex, then we have the following inequality: −1 = 1 and the function ℎ satisfies the conditions of Lemma 12 and, additionally, if |ℎ  |  is -convex, then we have the following inequality:where (, ) represents the arithmetic mean of  and .Proof.With the help of the -convexity of |ℎ  |  , the Hölder integral inequality, the property of the modulus, and