Some Quantum Estimates Of Hermite-Hadamard Inequalities For ’ -Convex Functions (cid:3)

In this paper, we develop some quantum estimates of Hermite(cid:150)Hadamard type inequalities for ’ convex functions. In some special cases, these quantum estimates reduce to known results.


Introduction
In recent years, the topic of quantum calculus has attracted the attention of several scholars. Quantum calculus stands as a connection between mathematics and physics. It has large applications in many mathematical areas such as number theory, special functions, quantum mechanics and mathematical inequalities. In quantum analysis, we obtain q-analogues of mathematical objects that can be recaptured as q ! 1 . In recent years, many researchers have shown their interest in studying and investigating quantum calculus. Quantum analysis is also very helpful in numerous …elds and has large applications in various areas of pure and applied sciences. For some recent developments in quantum calculus, interested readers are referred to [18][19][20]22].
Theory of inequalities and theory of convex functions are closely related to each other, thus a rich literature on inequalities. One of the most extensively studied inequality in the literature is the Hermite-Hadamard inequality: if F : I R ! R is a convex function de…ned on the interval I of real numbers and ! 1 ; ! 2 2 I with ! 1 < ! 2 , then

Preliminaries
In this section, we recall some previously known concepts and basic results. Let I be an interval in real line R and ' : R R ! R be a bifunction.
De…nition 1 ( [21]) A function F : I ! R is called convex with respect to ' (brie ‡y '-convex), if for all ! 1 ; ! 2 2 I and t 2 [0; 1]: Remark 1 If we set '(A; B) = A B in the above de…nition, then we recover the classical de…nition of convex function.
Clearly, any convex function is '-convex function. Furthermore, there exists '-convex functions which are not convex. For example, we consider F : R ! R as Then, it is not hard to check that F is '-convex. Also, it is obvious that F is not a convex function. On the other hand, let I = [! 1 ; ! 2 ] R be an interval and 0 < q < 1 be a constant.
De…nition 2 ( [26]) Assume F : I ! R is a continuous function and let x 2 I. Then q-derivative on I of function F at x is de…ned as We say that F is q-di¤ erentiable on I provided !1 D q F(x) exists for all x 2 I. Note that if ! 1 = 0 in (2), then 0 D q F = D q F, where D q is the well-known q-derivative of the function F(x) de…ned by De…nition 3 ([26]) Let F : I ! R be a continuous function. We de…ne the second-order q-derivative on interval I, which is denoted as Similarly, we de…ne higher order q-derivative on I, !1 D n q : I ! R.
for x 2 I. Note that if ! 1 = 0, then we have the classical q-integral, which is de…ned by Also, we introduce the q-analogues of ! 1 and (x ! 1 ) n and the de…nition of q-Beta function.
De…nition 5 ( [22]) For any real number ! 1 , is called the q-analogue of ! 1 . In particular, for i 2 Z + , we denote is called the q-Beta function. Note that where [t] is the q-analogue of t.
At last, we present the following lemmas [23] that will be used in this paper.

Hermite-Hadamard Inequalities for '-Convex Functions
We need the following auxiliary result, which will be useful in proving our main results. This result was also proved by Liu and Zhuang [23].
q F continuous and integrable on F, where 0 < q < 1. Then the following identity holds: The next theorem gives some estimates for the left-hand side of the result of (5) through '-convex functions.
Theorem 2 Let F : I = [! 1 ; ! 2 ] R ! R be a twice q-di¤ erentiable function on I 0 with !1 D 2 q F continuous and integrable on I, where 0 < q < 1. If j !1 D 2 q Fj m is '-convex on [! 1 ; ! 2 ] for m 1, then the following inequality holds: where Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have Now, applying Lemma 1(b), we have It is easy to check by De…nition 4 that Thus, we get (6). Corollary 1 In Theorem 2, if q ! 1 , then we get and (6) reduces to the following inequality: Remark 3 If '(A; B) = A B and q ! 1 , then (6) reduces to the following inequality: Corollary 2 If m is a positive integer, then Theorem 2 amounts to: and (6) reduces to Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have Applying Lemmas 1(b) and 1(c), we have It is easy to check by De…nition 4 that and thus, we get (7). Corollary 3 By letting q ! 1 in Theorem 3, we get and (7) reduces to the following inequality: Remark 5 If '(A; B) = A B and q ! 1 , then (7) reduces to the following inequality: Corollary 4 In Theorem 3, if n is a positive integer , then and (7) reduces to R ! R be a twice q-di¤ erentiable function on I 0 and !1 D 2 q F be continuous and integrable on I, where 0 < q < 1.
Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have Employing Lemmas 1(a) and 1(b), we obtain It is easy to check by De…nition 4 that and thus, we get (8). Using the properties of Beta function, that is, (x; x) = 2 1 2x ( 1 2 ; x) and (x; y) = (x) (y) (xy) ; we obtain that (n + 1; n + 1) = 2 1 2(n+1) 1 2 ; n + 1 = 2 2n 1 ( 1 2 ) (n + 1) where ( 1 2 ) = p and (t) is Gamma function: Thus, inequality (8) reduces to the following inequality: Remark 7 If '(A; B) = A B and q ! 1 , then (8) reduces to the following inequality: and (8) reduces to where Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have and applying (4) in De…nition 7, we have It is easy to check by De…nition 4 that thus, we get (9). and (9) reduces to the following inequality: Remark 9 If '(A; B) = A B and q ! 1, then (9) reduces to the following inequality: and (9) reduces to R ! R be a twice q-di¤ erentiable function on I 0 with !1 D 2 q F be continuous and integrable on I where 0 < q < 1.
where M = Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have qF(! 1 ) + F(! 2 ) 1 + q and applying (4) in De…nition 7, we have It is easy to check by De…nition 4 that thus, we get (10).
R ! R be a twice q-di¤ erentiable function on I 0 with !1 D 2 q F be continuous and integrable on I where 0 < q < 1. where Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have and applying Lemma 1(a), we have It is easy to check by De…nition 4 that thus, we get (13).
Corollary 11 In Theorem 7, if q ! 1, then we have and (13) reduces to the following inequality: Remark 13 If '(A; B) = A B and q ! 1, then (13) reduces to the following inequality: and (13) reduces to Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have qF(! 1 ) + F(! 2 ) 1 + q and applying Lemma 2(a) and the fact that thus, we get (14). then (14) reduces to Remark 15 If '(A; B) = A B and q ! 1, then (14) reduces to the following inequality: R ! R be a twice q-di¤ erentiable function on I 0 with !1 D 2 q F be continuous and integrable on I where 0 < q < 1.
Proof. Using Lemma 3, Hölder inequality and the fact that j !1 D 2 q Fj m is a '-convex function, we have and applying Lemma 2(a) and (b) and the fact that (1 qt) = (1 qt) 1 q , we have qF(! 1 ) + F(! 2 ) 1 + q Hence, we get (15). and (15) reduces to R ! R be a twice q-di¤ erentiable function on I 0 with !1 D 2 q F be continuous and integrable on I where 0 < q < 1.

Conclusion
Quantum calculus has large applications in many mathematical areas such as number theory, special functions, quantum mechanics and mathematical inequalities. In this paper, develop some quantum estimates of Hermite-Hadamard type inequalities for '-convex functions. Theses results in some special cases recapture the known results. We hope that our results may be helpful for further study.