Jensen–Mercer inequality for GA-convex functions and some related inequalities

In this paper, firstly, we prove a Jensen–Mercer inequality for GA-convex functions. After that, we establish weighted Hermite–Hadamard’s inequalities for GA-convex functions using the new Jensen–Mercer inequality, and we establish some new inequalities connected with Hermite–Hadamard–Mercer type inequalities for differentiable mappings whose derivatives in absolute value are GA-convex.


Introduction
Let the real function φ be defined on some nonempty interval I of the real line R. The function φ is said to be convex on I if the inequality holds for all u, v ∈ I and θ ∈ [0, 1]. The following inequality is well known in the literature as Hermite-Hadamard's inequality. Theorem 1.1 Let φ : I ⊆ R → R be a convex function defined on the interval I of real numbers and u, v ∈ I with u < v. The following double inequality holds: Since condition (3) can be written as we observe that φ : I ⊆ (0, ∞) → R is GA-convex on I if and only if φ • exp is convex on ln I := {ln x : x ∈ I}. If I = [u, v], then ln I = [ln u, ln v]. By using the useful property, we easily say that if φ : I ⊆ (0, ∞) → R is GA-convex on I and u, v ∈ I with u < v, then

Theorem 1.2 A real-valued function φ defined on an interval I is convex if and only if, for
all u 1 , u 2 , . . . , u n in I and all scalars λ i ∈ [0, 1] (i = 1, n) with n i=1 λ i = 1, we have This inequality is the well-known Jensen inequality in literature [8].
Remark 1.1 Let φ : I ⊆ (0, ∞) → R be a GA-convex function on I, then φ • exp is convex on ln I := {ln x : x ∈ I}, and by (4) we get Thus, we get Jensen's inequality for GA-convex functions as follows: In [5], Mercer proved the following variant of Jensen's inequality known as the Jensen-Mercer inequality.
We will now give definitions of the right-sided and left-sided Hadamard fractional integrals which are used throughout this paper. [u, v]. The left-sided and right-sided Hadamard fractional integrals J α [2]).
In this paper, firstly, the Jensen-Mercer inequality is proved for GA-convex functions. After that we prove weighted Hermite-Hadamard's inequalities for GA-convex functions using the new Jensen-Mercer inequality, and we establish some new fractional inequalities connected with the right sides of Hermite-Hadamard type inequalities for differentiable mappings whose derivatives in absolute value are GA-convex.

Weighted Hermite-Hadamard-Mercer inequalities for GA-convex functions
for each x ∈ [a, b].
Proof Let x ∈ [a, b] be an arbitrary point. Then there exists μ ∈ [0, 1] such that we can write x = a λ b 1-λ and ab/x = a 1-λ b λ . By using the GA-convexity of φ, we obtain Hermite-Hadamard-Fejer inequalities can be represented for GA-convex functions using a Jensen-Mercer type inequality as follows.
Proof First method: By using inequality (5) and Lemma 2.1, we can write This last inequality gives us the desired result.
Proof Since φ is a GA-convex function on [a, b], we have for all x, y ∈ [a, b] and t ∈ [0, 1].
Multiplying both sides of (10) by g(( ab x ) t ( ab y ) 1-t ), then integrating the resulting inequality with respect to t over [0, 1], we obtain and the first inequality is proved. For the proof of the second inequality in (9), by the GA-convexity of φ, we have By adding these inequalities, we have Then multiplying both sides of (11) by 1 2 g(( ab x ) t ( ab y ) 1-t ) and integrating the resulting inequality with respect to t over [0, 1], we obtain For the proof of the third inequality in (9), by inequality (7), we have The proof is completed.
If we take x = a and y = b in Theorem 2.2, then we can derive the following weighted Hermite-Hadamard inequalities for GA-convex functions.
and g : [a, b]→ R is nonnegative and integrable, then the following inequalities hold: Remark 2.1 Specially, if we choose that g is geometrically symmetric to √ ab (i.e., g(ab/u) = g(u) for all u ∈ [a, b]) in (12), then we get the following inequality: in Theorem 2.2, then we obtain the following Hermite-Hadamard-Mercer inequalities for GA-convex functions via Hadamard fractional integrals.
, then the following inequalities for Hadamard fractional integrals hold: for all x, y ∈ [a, b] and α > 0. Specially, we take α = 1 in the above inequalities, then we get Remark 2.2 Specially, if we choose x = a and y = b in (13), then we get the following inequalities: which coincide with the inequality in [1, Theorem 2.1].
Let w : [a, b]→ R be a nonnegative and integrable function.
in Theorem 2.2, then we obtain the following weighted Hermite-Hadamard-Mercer inequalities for GA-convex functions via Hadamard fractional integrals.
for all x, y ∈ [a, b] with x < y. Also, we obtain the following inequalities from both inequalities above: for all x, y ∈ [a, b] with x < y.  (14), then we get the following inequalities: which coincide with the inequality in [3, Theorem 2.1].
If we choose g(u) = 1 in Theorem 2.2, then we obtain the following Hermite-Hadamard-Mercer inequalities for GA-convex functions. φ is a GA-convex function on [a, b], then the following inequalities hold:

Corollary 2.4 Let
for all x, y ∈ [a, b] with x < y.
and similarly for all x, y ∈ [a, b] and t ∈ [0, 1]. By adding these inequalities, we have Multiplying both sides of (17) by 1 2 g( ab x t y 1-t ), then integrating the resulting inequality with respect to t over [0, 1], we obtain for all x, y ∈ [a, b].Thus the second inequality is proved. For the proof of the last inequality in (16), by the GA-convexity of φ, we have for all x, y ∈ [a, b]. Multiplying both sides of (18) by g( ab x t y 1-t ), then integrating the resulting inequality with respect to t over [0, 1], we obtain If we take x = a and y = b in Theorem 2.3, then we can derive the following weighted Hermite-Hadamard inequalities for GA-convex functions.

Corollary 2.6 Let
for all x, y ∈ [a, b] with x < y and α > 0. Specially, we take α = 1 in the above inequalities, then we get for all x, y ∈ [a, b] with x < y. Let w : [a, b]→ R be a nonnegative and integrable function.
for all x, y ∈ [a, b] with x < y. Also, we obtain the following inequalities from both inequalities above: for all x, y ∈ [a, b] with x < y.
Specially, if we choose x = a and y = b in (21), then we get the following inequalities. φ is a GA-convex function on [a, b] and w : [a, b]→ R is nonnegative and integrable, then the following inequalities for Hadamard fractional integrals hold: If we choose g(u) = 1 in Theorem 2.3, then we obtain the following Hermite-Hadamard-Mercer inequalities for GA-convex functions. φ is a GA-convex function on [a, b], then the following inequalities hold: for all x, y ∈ [a, b] with x < y.

Some Hermite-Hadamard type inequalities via Jensen-Mercer inequality for GA-convex functions
We will use the following notations throughout this section: In order to prove our main results, we need the following identity which is related to the second inequality in (15).
Proof Integrating by parts and changing variables of integration yields ab ln y/x 2y This completes the proof.
for all x, y ∈ [a, b],with x < y.
Proof Since |φ | q is GA-convex on [a, b], from Lemma 3.1 and the power mean inequality, If we take q = 1 in Theorem 3.1, we can derive the following corollary.
for all x, y ∈ [a, b] with x < y.
If we take x = a and y = b in Theorem 3.1, we can derive the following corollary.
Proof Since |φ | q is GA-convex on [a, b], from Lemma 3.1 and Hölder's inequality, we have If we take x = a and y = b in Theorem 3.2, we can derive the following corollary.
If we take x = a and y = b in Theorem 3.3, we can derive the following corollary.

Conclusion
This article aims to investigate certain weighted Hermite-Hadamard-Mercer type inequalities for a GA-convex function, which are related to the Hermite-Hadamard-Fejér inequality and fractional Hermite-Hadamard type inequalities. It is worth mentioning that certain results proved in this article generalize parts of the results provided by İşcan [1], Kunt and İşcan [3], and Latif et al. [4]. Certain estimates related to the second Hermite-Hadamard-Mercer inequality for GA-convex functions given in (15) are obtained. For this purpose, an identity for differentiable mappings is established.