A generalized Fejér-Hadamard inequality for harmonically convex functions via generalized fractional integral operator and related results

In the present research, we will develop some integral inequalities of Hermite Hadamard type for differentiable η -convex function. Moreover, our results include several new and known results as special cases.


Introduction and preliminary results
Inequalities for convex functions, for example the celebrated one is the Hadamard inequality provide a new horizon in the field of mathematical analysis.Many authors have been working on it continuously and several Hadamard like integral inequalities have been established for many kinds of functions related to convex functions.Recently a lot of integral inequalities of the Hadamard type for harmonically convex functions via fractional integrals have been published (see, [3,[6][7][8][9] and references there in).The Hadamard inequality for convex functions is stated in the following theorem.Theorem 1.Let I be an interval of real numbers and f : I → R be a convex function on I. Then for all a, b ∈ I the following inequality holds Fejér gave a weighted version of the Hadamard inequality stated as follows.It is well known as the Fejér-Hadamard inequality.In the following we give the definition of harmonically convex functions.Definition 3. [7] Let I be an interval of non-zero real numbers.Then a function f : I → R is said to be harmonically convex function if the inequality holds for a, b ∈ I and t ∈ [0, 1].If inequality in (1) is reversed, then f is said to be harmonically concave.

Definition 4. [6] A function
In the following we give the Hadamard inequality for harmonically convex functions.(2) A Fejér-Hadamard inequality for harmonically convex functions is stated as follows.
Theorem 6. [3] 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 a non negative integrable and harmonically symmetric with respect to 2ab a+b , then the following inequality holds The following definition of the Riemann-Liouville fractional integral is the asset of fractional calculus.
Then two sided Riemann-Liouville fractional integral of f of order ν > 0 is defined as

A version of the Fejér-Hadamard inequality for harmonically convex functions via
Riemann-Liouville fractional integrals is stated as follows.
Theorem 8. [9] Let f : I ⊂ (0, ∞) → R be a function such that a, b ∈ I with a < b.If f ∈ L[a, b] and f is harmonically convex function, then the following inequality for Riemann-Liouville fractional integral holds In the following we give the definition of a generalized fractional integral operator which will help us to give a generalized Fejér-Hadamard inequality for harmonically convex functions and related results.
In [14,15] properties of the generalized fractional integral operator γ,δ,k µ,ν,l,ω,a + and the generalized Mittag-Leffler function E γ,δ,k µ,ν,l (t) are studied in brief.In [14] it is proved that We use this property of generalized Mittag-Leffler function in sequal in our results.Also we use in sequal the following definitions of special functions known as beta and hypergeometric functions, (see, [10]) where 0 < b < c and |w| < 1.
In this paper we give a generalized version of the Fejér-Hadamard inequality for harmonically convex functions via generalized fractional integral operator.We also obtain bounds of the absolute differences of this generalized Fejér-Hadamard inequality for harmonically convex functions.Being generalizations, we reproduce the results proved in [8].

Main Results
To obtain our main results we need the following lemmas.
Lemma 10. [13] For 0 ≤ a < b and 0 < µ ≤ 1, we have Proof.Since f is harmonically symmetric about 2ab a+b , we have f . By definition of generalized fractional integral operator replace t by 1 a + 1 b − x in equation ( 6), we have (7) and ( 8) give the required result.
and also let g : [a, b] → R be a non-negative, integrable and harmonically symmetric function about 2ab a+b .Then the following inequalities for generalized fractional integrals hold Proof.Since f is harmonically convex function, therefore for t ∈ [0, 1], we have Multiplying both sides of (10) by t ν−1 E γ,δ,k µ,ν,l (ωt µ )g ab tb+(1−t)a and then integrating with respect to t over [0, 1], we have By This implies Using Lemma 11 in above inequality, we have To prove the second half of inequality, again from harmonically convexity of f on [a, b] and for t ∈ [0, 1] we have Multiplying both sides of (15) by t ν−1 E γ,δ,k µ,ν,l (ωt µ )g ab tb+(1−t)a , then integrating with respect to t over [0, 1], we have Setting x = tb+(1−t)a ab and by using harmonically symmetry of f with respect to 2ab a+b in (16), after simplification we have Using Lemma 11 in (17), we have By joining ( 14) and (18) we get (9).
Proof.To prove this lemma, we have 1 a This implies This implies on substracting (20) from ( 19) and using lemma 11, we get the result.
Since g is Harmonically symmetric with respect to 2ab a+b therefore g( 1 t ) = g Using ( 22) in (21), we have Setting t = ub+(1−u)a ab in (24), we have it can be written as Using ( 26) in (25), we have One can has by using Lemma 10  (ii) if we take ν = 1 along with ω = 0, then we get [8, Corollary 1(1)].

continuous and harmonically symmetric function about 2ab
a+b , then the following inequality for generalized fractional integrals holds du .
Using power means, inequality (31) becomes By using the harmonically convexity of | f | q in (32), we have . Now we evaluate the integrals of (34) by using Lemma 10 Proof.By inequality (25) of Theorem 14, we have We evaluate the integrals by using Lemma 10

Theorem 2 .
Let f : [a, b] → R be a convex function and g : [a, b] → R is non-negative, integrable and symmetric to a+b 2 .Then the following inequality

Theorem 5 . [ 7 ]x 2
Let f : I ⊂ R \ {0} → R be harmonically convex function and a, b ∈ I with a < b.If f ∈ L[a, b], then the following inequality holds dx ≤ f (a) + f (b) 2 .

Lemma 13 .
Let f : I ⊂ (0, ∞) → R be a differentiable function on the interior of I and f ∈ L[a, b] where a, b ∈ I and a < b.Also let g : I ⊂ (0, ∞) → R be an integrable and harmonically symmetric function about 2ab a+b .Then the following equality holds for generalized fractional integrals

Remark 2 .Theorem 14 .
In Lemma 13 if we take g(x) = 1 with ω = 0, then it gives [9, Lemma 3].Let f : I ⊂ (0, ∞) → R be a differentiable function on the interior of I and f ∈ L[a, b] where a, b ∈ I and a < b.If | f | is harmonically convex function on [a, b], g : [a, b] ⊂ (0, ∞) → R is a continuous and harmonically symmetric function with respect to 2ab a+b , then the following inequality for generalized fractional integrals holds f ) and (30) in (27), we get the result.Remark 3. In Theorem 14, (i) if we put ω = 0, then we get[8, Theorem 6].