Let f : I ⊆ ℝ → ℝ be a convex function defined on the interval I of real numbers and a, b ∈ I with a < b. The inequality

is well known in the literature as Hermite-Hadamard’s inequality [5].

The most well-known inequalities related to the integral mean of a convex function f are the Hermite Hadamard inequalities or its weighted versions, the so-called Hermite-Hadamard-Fejér inequalities.

In [4], Fejér established the following Fejér inequality which is the weighted generalization of Hermite-Hadamard inequality (1.1):

Theorem 1.1

Let f : [a, b]→ ℝ be a convex function. Then the inequality

holds, where g : [a, b]→ ℝ is nonnegative, integrable and symmetric to a + b/2.

For some results which generalize, improve and extend the inequalities (1.1) and (1.2) see [1, 6, 7, 16, 18].

We recall the following inequality and special functions which are known as Beta and hypergeometric function respectively:

(see [13]).

Lemma 1.2.([15, 20])

For 0 < α ≤ 1 and 0 ≤ a < b we have |a^α − b^α| ≤ (b − a)^α.

The following definitions and mathematical preliminaries of fractional calculus theory are used further in this paper.

Definition 1.3.([13])

Let f ∈ L[a, b]. The Riemann-Liouville integrals Ja+αf and Jb-αf of order α > 0 with a ≥ 0 are defined by

and

respectively, where Γ(α) is the Gamma function defined by Γ(α)=∫0∞e-ttα-1dt and Ja+0f(x)=Jb-0f(x)=f(x).

Because of the wide application of Hermite-Hadamard type inequalities and fractional integrals, many researchers extend their studies to Hermite-Hadamard type inequalities involving fractional integrals not limited to integer integrals. Recently, more and more Hermite-Hadamard inequalities involving fractional integrals have been obtained for different classes of functions; see [3, 8, 9, 17, 19, 20].

Definition 1.4.([21])

A function f : I ⊆ (0,∞) → [0,∞) is said to be harmonically quasi-convex, if

for all x, y ∈ I and t ∈ [0, 1].

In [11], İşcan defined the so-called harmonically convex functions and established following Hermite-Hadamard type inequality for them as follows:

Definition 1.5

Let I ⊂ ℝ{0} be a real interval. A function f : I → ℝ is said to be harmonically convex, if

for all x, y ∈ I and t ∈ [0, 1]. If the inequality in (1.3) is reversed, then f is said to be harmonically concave.

Theorem 1.6.([11])

Let f : I ⊂ ℝ{0} → ℝ be a harmonically convex function and a, b ∈ I with a < b. If f ∈ L[a, b] then the following inequalities holds:

In [10], İşcan and Wu presented a Hermite-Hadamard type inequality for harmonically convex functions in fractional integral forms as follows:

Theorem 1.7

Let f : I ⊂ (0,∞) → ℝ be a function such that f ∈ L[a, b], where a, b ∈ I with a < b. If f is a harmonically convex function on [a, b], then the following inequalities for fractional integrals holds:

with α > 0 and h(x) = 1/x.

In [14] Latif et al. gave the following definition:

Definition 1.8

A function g : [a, b] ⊆ ℝ {0} → ℝ is said to be harmonically symmetric with respect to 2ab/a + b, if

holds for all x ∈ [a, b].

In [2] Chan and Wu presented a Hermite-Hadamard-Fejér inequality for harmonically convex functions as follows:

Theorem 1.9

Let f : I ⊆ ℝ{0} → ℝ be a harmonically convex function and a, b ∈ I with a < b. If f ∈ L[a, b] and g : [a, b] ⊆ ℝ{0} → ℝ is nonnegative, integrable and harmonically symmetric with respect to 2ab/a + b, then

In [12] İşcan and Kunt presented a Hermite–Hadamard-Fejér type inequality for harmonically convex functions in fractional integral forms and established following identity as follows:

Theorem 1.10

Let f : [a, b]→ ℝ be a harmonically convex function with a < b and f ∈ L[a, b]. If g : [a, b]→ ℝ is nonnegative, integrable and harmonically symmetric with respect to 2ab/a+b, then the following inequalities for fractional integrals holds:

with α > 0 and h(x) = 1/x, x∈[1b,1a].

Lemma 1.11.([12])

Let f : I ⊂ (0,∞) → ℝ be a differentiable function on I^∘such that f′ ∈ L[a, b], where a, b ∈ I and a < b. If g : [a, b]→ ℝ is integrable and harmonically symmetric with respect to 2ab/a + b, then the following equality for fractional integrals holds:

with α > 0 and h(x) = 1/x, x∈[1b,1a].

In this paper, we give some new inequalities connected with the right-hand side of Hermite-Hadamard-Fejér type integral inequality for harmonically quasi-convex function in fractional integrals.

Throughout this section, we write ‖g‖∞=supt∈[a,b]∣g(t)∣, for the continuous function g : [a, b]→ ℝ.

Theorem 2.1

Let f : I ⊂ (0,∞) → ℝ be a differentiable function on I^∘such that f′ ∈ L[a, b], where a, b ∈ I and a < b. If |f′| is harmonically quasi-convex on [a, b], g : [a, b]→ ℝ is continuous and harmonically symmetric with respect to 2ab/a + b , then the following inequality for fractional integrals holds:

where

with 0 < α ≤ 1 and h(x) = 1/x, x∈[1b,1a].

Corollary 2.2

In Theorem 2.1:

• If we take α = 1 we have the following Hermite-Hadamard-Fejér inequality for harmonically quasi-convex functions which is related to the right-hand side of (1.6):|f (a)+f (b)2∫abg (x)x2dx-∫abf (x) g (x)x2dx|≤‖g‖∞(b-a)22C1 (1) sup {∣f′ (a)∣,∣f′ (b)∣},

• If we take g (x) = 1 we have following Hermite-Hadamard type inequality for harmonically quasi-convex functions in fractional integral forms which is related to the right-hand side of (1.5):|f (a)+f (b)2-Γ (α+1)2(abb-a)α {J1/a-α(f∘h) (1/b)+J1/b+α(f∘h) (1/a)}|≤ab (b-a)2C1 (α) sup {∣f′(a)∣,∣f′ (b)∣},

• If we take α = 1 and g (x) = 1 we have the following Hermite-Hadamard type inequality for harmonically quasi-convex functions which is related to the right-hand side of (1.4):|f (a)+f (b)2-abb-a∫abf (x)x2dx|≤ab (b-a)2C1 (1) sup {∣f′ (a)∣,∣f′ (b)∣}.

Theorem 2.3

Let f : I ⊂ (0,∞) → ℝ be a differentiable function on I^∘such that f′ ∈ L[a, b], where a, b ∈ I and a < b. If |f′|^q, q ≥ 1, is harmonically quasi-convex on [a, b], g : [a, b] → ℝ is continuous and harmonically symmetric with respect to 2ab/a + b, then the following inequality for fractional integrals holds:

where

with 0 < α ≤ 1 and h(x) = 1/x, x∈[1b,1a].