International Journal of Analysis and Applications ISSN 2291-8639 Volume 13, Number 1 (2017), 41-53 http://www.etamaths.com SOME INTEGRAL INEQUALITIES FOR BETA-PREINVEX FUNCTIONS MUHAMMAD ASLAM NOOR∗, KHALIDA INAYAT NOOR AND SABAH IFTIKHAR Abstract. The main objective of this paper is to introduce and study a new class of preinvex functions, which is called beta-preinvex functions. Some Hermite-Hadamard type inequalities for beta-preinvex functions are established. Our results can be viewed as significant and important generalizations of several previously known results. We also establish some integral inequalities involving Euler beta functions for the class of functions whose certain powers of the absolute value are beta-preinvex function. Results proved in this paper may stimulate further research in different areas of pure and applied sciences. 1. Introduction In recent years, the convex functions and convex sets have been generalized in several directions us- ing novel and innovative techniques to study a wide class of unrelated problem in a unified and general framework. Hanson [2] introduced the concept of invex functions involving the bifunction in study of nonlinear programming. This concept stimulated much interest in applications of these invex function in different branches of pure and applied sciences. Wier and Mond [24] introduced and investigated another class of convex functions, which is called the preinvex functions. They proved that the differen- tiable preinvex functions are invex functions, but the converse may not true. It is known that the invex functions and preinvex functions are equivalent under some conditions, see [10]. Noor[5] proved that the minimum of a differentiable preinvex functions on the invex sets can be characterized by a class of variational-like inequalities. Noor [7] also proved that a function f is a preinvex function, which satisfies the Hermite-Hadmard type integral inequalities. This result can be viewed as an analogous to the con- vex functions. These results strongly influenced the recent recent research trends. For the applications, properties and other aspects of the preinvex functions, see [1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. Various refinements of the Hermite-Hadamard inequalities for the convex functions and their variant forms are being obtained in the literature by many researchers. Tunc et al.[22] introduced the concept of beta-convex functions, which include classical convex functions, s -convex functions, s -Godunova-Levin functions, tgs -convex functions and P -functions as special cases. Noor et al. [16, 16, 17] introduced the concepts of (p,q) and tgs-preinvex f unctions and obtained several integral inequalities via these preinvex functions. Motivated and inspired by the ideas and techniques of Tunc et. el.[22] and Noor et. al. [16, 17, 18], we introduce some new classes of of beta-preinvex functions and derive some estimates involving the Euler Beta function of the integral ∫a+η(b,a) a (x−a)p(a + η(b,a) −x)qf(x)dx for the class of functions whose certain powers of the absolute value are beta-preinvex function. This is the main motivation of this paper. Some special cases are discussed. Our results include the previously known results for preinvex functions and their varinat forms as special cases. It is expected that the ideas and techniques considered in this paper be staring for the future research. The interested readers are encouraged to find the novel and innovative applications of these results in other areas. Received 18th July, 2016; accepted 20th September, 2016; published 3rd January, 2017. 2010 Mathematics Subject Classification. 26D15, 26D10, 90C23. Key words and phrases. preinvex functions; beta convex function; beta functions; Hermite-Hadamard type inequality. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 41 42 NOOR, NOOR AND IFTIKHAR 2. preliminaries Let Kη be a nonempty closed set in R . Let f : Kη ⊆ R −→ R be a continuous function and η(·, ·) : Kη × Kη −→ R be a continuous bifunction. In this section, we recall the following new and known concepts. Definition 2.1. [24]. A set Kη ⊆ R is said to be an invex set with respect to the bifunction η(·, ·), if x + tη(y,x) ∈ Kη, ∀x,y ∈ Kη, t ∈ [0, 1]. If η(y,x) = y−x, then invex set Kη reduces to classical convex set. Clearly, every convex set is an invex set but the converse is not true. We new define some new concepts of beta-preinvex functions and its variant forms. Definition 2.2. A function f : Kη ⊆ R → R is said to be beta-preinvex function, where p,q > −1, if f ( x + tη(y,x) ) ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y), ∀x,y ∈ Kη, t ∈ (0, 1). (2.1) If t = 1 2 , then f ( 2x + η(y,x) 2 ) ≤ f(x) + f(y) 2p+q , ∀x,y ∈ Kη, (2.2) which is called the Jensen beta-preinvex function. We now discuss some special cases of beta-preinvex function, which appears to be new ones. I). If p = 1 and q = 0, then Definition 2.2, reduces to: Definition 2.3. [24]. A function f : Kη ⊆ R −→ R is said to be preinvex function with respect to the bifunction η(·, ·), if f ( x + tη(y,x) ) ≤ (1 − t)f(x) + tf(y), ∀x,y ∈ Kη, t ∈ [0, 1]. If t = 1 2 , then f (2x + η(y,x) 2 ) ≤ f(x) + f(y) 2 , ∀∈ Kη, which is called the Jensen preinvex function. It has been shown that a function f is a preinvex function, if and only if, it satisfies the inequality of the type f( 2a + η(b,a) 2 ) ≤ 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ f(a) + f(b) 2 , which is known as the Hermite-Hadamard-Noor integral inequality. See [5, 6, 7] for the recent results in this direction. It is worth mentioning that the minimum of of a differentiable preinvex functions on the invex sets in a normed space can be characterized by a class of variational inequalities, which is known as the variational-like inequalities. For the formulation, applications and numerical methods for solving the variational-like inequalities and related problems, see [1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and the references therein. II). If p = 0 and q = 0, then Definition 2.4. [10, 15]. A function f : Kη ⊆ R −→ R is said to be P -preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ f(x) + f(y), ∀x,y ∈ Kη. III). If p = −1 and q = 0, then Definition 2.2 reduces to: Definition 2.5. [15]. A function f : Kη ⊆ R −→ R is said to be Godunova-Levin preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ f(x) 1 − t + f(y) t , ∀x,y ∈ Kη, t ∈ (0, 1). SOME INTEGRAL INEQUALITIES FOR BETA-PREINVEX FUNCTIONS 43 IV). If p = 1 and q = 1, then Definition 2.2 reduces to: Definition 2.6. [16]. A function f : Kη ⊆ R −→ R is said to be tgs-preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ t(1 − t)[f(x) + f(y)], ∀x,y ∈ Kη, t ∈ [0, 1]. V). If p = s and q = 0, then Definition 2.2 reduces to: Definition 2.7. [17]. A function f : Kη ⊆ R −→ R is said to be s-preinvex function with respect to η(·, ·), where s ∈ [−1, 1], if f ( x + tη(y,x) ) ≤ (1 − t)sf(x) + tsf(y), ∀x,y ∈ Kη, t ∈ (0, 1). VI). If p = −s and q = 0, then Definition 2.2 reduces to: Definition 2.8. A function f : Kη ⊆ R −→ R is said to be Godunova-Levin s-preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ (1 − t)−sf(x) + t−sf(y), ∀x,y ∈ Kη, t ∈ (0, 1). VII). If p = 1 2 and q = −1 2 , then Definition 2.2, reduces to: Definition 2.9. A function f : I ⊆ R −→ R is said to be generalized MT -preinvex function with respect to η(·, ·), if f ( x + tη(y,x) ) ≤ √ 1 − t √ t f(x) + √ t √ 1 − t f(y), ∀x,y ∈ Kη, t ∈ (0, 1). Definition 2.10. A function f : Kη ⊆ R −→ R is said to be log-beta-preinvex function with respect to η(·, ·), where p,q > −1, if f ( x + tη(y,x) ) ≤ [f(x)](1−t) ptq [f(y)]t p(1−t)q, ∀x,y ∈ Kη, t ∈ (0, 1). It follows that log f ( x + tη(y,x) ) ≤ (1 − t)ptq log f(x) + tp(1 − t)q log f(y). From definition 2.10, we have f ( x + tη(y,x) ) ≤ [f(x)](1−t) ptq [f(y)]t p(1−t)q ≤ (1 − t)ptqf(x) + tp(1 − t)qf(y). This shows that, log-beta-preinvex function implies beta-preinvex function, but the converse is not true. For appropriate and suitable choices of p,q > −1, and the invex set, one can obtain several new and known classes of preinvex functions and its variant form from Definition 2.2 and Definition 2.10. This shows that the concept of beta-preinvex functions are quite general and unifying ones. We recall the following special function which is known as Beta function. β(x,y) = ∫ 1 0 tx−1(1 − t)y−1dt = Γ(x)Γ(y) Γ(x + y) , x,y > 0, where Γ(.) is the Gamma function. We also recall the well-known assumption about the bifunction η(., .), which plays an important role in the studies of the variational-like inequalities and integral inequalities. Condition C [4]: Let I ⊆ R be an invex set with respect to bifunction η(·, ·) : I × I → R. For any x,y ∈ I and any t ∈ [0, 1], we have η(y,y + tη(x,y)) = −tη(x,y) η(x,y + tη(x,y)) = (1 − t)η(x,y). 44 NOOR, NOOR AND IFTIKHAR 3. Hermite-Hadamard Inequalities In this section, we derive Hermite-Hadamard inequalities for beta-preinvex function. Without loss of generality, we denote by I = [a,a + η(b,a)] unless otherwise specified. Theorem 3.1. Let f : I = [a,a + η(b,a)] ⊆ R −→ R be beta-preinvex function with a < a + η(b,a). If f ∈ L[a,a + η(b,a)] and Condition C holds, then 2p+q−1f ( 2a + η(b,a) 2 ) ≤ 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ Γ(p + 1)Γ(q + 1) Γ(p + q + 2) [f(a) + f(b)] Proof. Let f be beta-preinvex function. Then taking x = a + tη(b,a) and y = a + (1 − t)η(b,a) in (2.2), and using condition C, we have f ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] = 1 2p+q [∫ 1 0 f ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) dt ] This implies 2p+q−1f ( 2a + η(b,a) 2 ) ≤ 1 η(b,a) ∫ a+η(b,a) a f(x)dx Now consider 1 η(b,a) ∫ a+η(b,a) a f(x)dx = ∫ 1 0 f ( a + tη(b,a) ) dt ≤ f(a) ∫ 1 0 (1 − t)ptqdt + f(b) ∫ 1 0 tp(1 − t)qdt = [f(a) + f(b)]β(p + 1,q + 1), which is the required result. � Theorem 3.2. Let f,g : I ⊂ R −→ R be beta-preinvex functions. If f,g ∈ L[a,a + η(b,a)], then 1 η(b,a) ∫ a+η(b,a) a f(x)g(2a + η(b,a) −x)dx ≤ Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) N(a,b), where M(a,b) = f(a)g(a) + f(b)g(b) (3.1) N(a,b) = f(a)g(b) + f(b)g(a) (3.2) Proof. Let f,g be beta-preinvex functions. Then f ( a + tη(b,a) ) ≤ (1 − t)ptqf(a) + tp(1 − t)qf(b) (3.3) g ( a + (1 − t)η(b,a) ) ≤ tp(1 − t)qg(a) + (1 − t)ptqg(b). (3.4) From (3.3) and (3.4), we have f ( a + tη(b,a) ) g ( a + (1 − t)η(b,a) ) ≤ [(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b)] (3.5) SOME INTEGRAL INEQUALITIES FOR BETA-PREINVEX FUNCTIONS 45 Integrating both sides of (3.5), we obtain ∫ 1 0 f ( a + tη(b,a) ) g ( a + (1 − t)η(b,a) ) dt ≤ ∫ 1 0 [(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b)]dt = [f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt + [f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt = M(a,b)β(p + q + 1,p + q + 1) + N(a,b)β(2p + 1, 2q + 1) Thus 1 η(b,a) ∫ a+η(b,a) a f(x)g(2a + η(b,a) −x)dx ≤ M(a,b)β(p + q + 1,p + q + 1) + N(a,b)β(2p + 1, 2q + 1) = Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) N(a,b), which is the required result. � If g(2a + η(b,a) −x) = g(x) in Theorem 3.2, then it reduces to the following result. Corollary 3.1. Let f,g : I ⊂ R −→ R be beta-preinvex functions. If f,g ∈ L[a,a + η(b,a)], then 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx ≤ Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) N(a,b), where M(a,b) and N(a,b) are given by (3.1) and (3.2) respectively. Theorem 3.3. Let f,g : I ⊂ R −→ R be beta-preinvex functions. If fg ∈ L[a,a+η(b,a)] and Condition C holds, then 22(p+q)−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) − 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx ≤ Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) N(a,b), where M(a,b) and N(a,b) are given by (3.1) and (3.2) respectively. Proof. Let f be beta-preinvex function. Then taking x = a + tη(b,a) and y = a + (1−t)η(b,a) in (2.2) and using condition C, we have f ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] , g ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] . 46 NOOR, NOOR AND IFTIKHAR Consider f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) ≤ 1 22p+2q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] ≤ 1 22p+2q [ f ( a + tη(b,a) ) g ( a + tη(b,a) ) +f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) +[(1 − t)ptqf(a) + tp(1 − t)qf(b)][tp(1 − t)qg(a) + (1 − t)ptqg(b) ] +[tp(1 − t)qf(a) + (1 − t)ptqf(b)][(1 − t)ptqg(a) + tp(1 − t)qg(b) ]] . Integrating over [0, 1], we have∫ 1 0 f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) dt ≤ 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2[f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt +2[f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt ] = 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2M(a,b)β(p + q + 1,p + q + 1) + 2N(a,b)β(2p + 1, 2q + 1) ] = 1 22p+2q−1 [ 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dt +M(a,b)β(p + q + 1,p + q + 1) + N(a,b)β(2p + 1, 2q + 1) ] . This completes the proof. � Theorem 3.4. Let f,g : I ⊂ R −→ R be beta-preinvex functions. If fg ∈ L[a,a + η(b,a)], then 1 η(b,a) ∫ a+η(b,a) a µ(x)[f(a)g(x) + f(b)g(x)]dx + 1 η(b,a) ∫ a+η(b,a) a µ(x)[g(a)f(x) + g(b)f(x)]dx ≤ Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) N(a,b) + 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx, where M(a,b) and N(a,b) are given by (3.1) and (3.2) respectively and µ(x) = ( ((a + η(b,a)) −x)p(x−a)q η(b,a)p+q ) . SOME INTEGRAL INEQUALITIES FOR BETA-PREINVEX FUNCTIONS 47 Proof. Let f, g be beta-preinvex functions. Then, we have f ( a + tη(b,a) ) ≤ (1 − t)ptqf(a) + tp(1 − t)qf(b), g ( a + tη(b,a) ) ≤ (1 − t)ptqg(a) + tp(1 − t)qg(b). Now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ R) and x1 < x2, x3 < x4, we have f ( a + tη(b,a) ) [(1 − t)ptqg(a) + tp(1 − t)qg(b)] +g ( a + tη(b,a) ) [(1 − t)ptqf(a) + tp(1 − t)qf(b)] ≤ [(1 − t)ptqf(a) + tp(1 − t)qf(b)][(1 − t)ptqg(a) + tp(1 − t)qg(b)] +f ( a + tη(b,a) ) g ( a + tη(b,a) ) = [f(a)g(a) + f(b)g(b)]t2p(1 − t)2q + [f(a)g(b) + f(b)g(a)]tp+q(1 − t)p+q +f ( a + tη(b,a) ) g ( a + tη(b,a) ) Integrating over [0, 1], we have g(a) ∫ 1 0 (1 − t)ptqf ( a + tη(b,a) ) dt +g(b) ∫ 1 0 tp(1 − t)qf ( a + tη(b,a) ) dt +f(a) ∫ 1 0 (1 − t)ptqg ( a + tη(b,a) ) dt +f(b) ∫ 1 0 tp(1 − t)qg ( a + tη(b,a) ) dt ≤ [f(a)g(a) + f(b)g(b)] ∫ 1 0 t2p(1 − t)2qdt +[f(a)g(b) + f(b)g(a)] ∫ 1 0 tp+q(1 − t)p+qdt + ∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt This implies 1 η(b,a) ∫ a+η(b,a) a µ(x)[f(a)g(x) + f(b)g(x)]dx + 1 η(b,a) ∫ a+η(b,a) a µ(x)[g(a)f(x) + g(b)f(x)]dx ≤ M(a,b)β(2p + 1, 2q + 1) + N(a,b)β(p + q + 1,p + q + 1) + 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx, which is the required result. � 48 NOOR, NOOR AND IFTIKHAR Theorem 3.5. Let f,g : I ⊂ R −→ R be beta-preinvex functions. If fg ∈ L[a,a + η(b,a)] and condition C holds, then f ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a g(x)dx +g ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ 1 2p+q [ 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx + Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) N(a,b) ] +2p+q−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) , where M(a,b) and N(a,b) are given by (3.1) and (3.2) respectively. Proof. Let f, g be beta-preinvex function. Then taking x = a + tη(b,a) and y = a + (1 − t)η(b,a) in (2.2) and using condition C, we have f ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] , g ( 2a + η(b,a) 2 ) ≤ 1 2p+q [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] . Now, using 〈x1 −x2,x3 −x4〉≥ 0, (x1,x2,x3,x4 ∈ R) and x1 < x2, x3 < x4, we have 1 2p+q f ( 2a + η(b,a) 2 )[ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] + 1 2p+q g ( 2a + η(b,a) 2 )[ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] ≤ 1 22p+2q [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )][ g ( a + tη(b,a) ) +g ( a + (1 − t)η(b,a) )] + f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) ≤ 1 22p+2q [ f ( a + tη(b,a) ) g ( a + tη(b,a) ) +f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) +2[f(a)g(a) + f(b)g(b)]tp+q(1 − t)p+q +2[f(a)g(b) + f(b)g(a)]t2p(1 − t)2q ] +f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) SOME INTEGRAL INEQUALITIES FOR BETA-PREINVEX FUNCTIONS 49 Integrating over [0, 1], we have 1 2p+q f ( 2a + η(b,a) 2 )∫ 1 0 [ g ( a + tη(b,a) ) + g ( a + (1 − t)η(b,a) )] dt + 1 2p+q g ( 2a + η(b,a) 2 )∫ 1 0 [ f ( a + tη(b,a) ) + f ( a + (1 − t)η(b,a) )] dt ≤ 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2[f(a)g(a) + f(b)g(b)] ∫ 1 0 tp+q(1 − t)p+qdt +2[f(a)g(b) + f(b)g(a)] ∫ 1 0 t2p(1 − t)2qdt ] + ∫ 1 0 f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) dt = 1 22p+2q [∫ 1 0 f ( a + tη(b,a) ) g ( a + tη(b,a) ) dt + ∫ 1 0 f ( a + (1 − t)η(b,a) ) g ( a + (1 − t)η(b,a) ) dt +2M(a,b)β(p + q + 1,p + q + 1) + 2N(a,b)β(2p + 1, 2q + 1) ] +f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) From the above inequality, it follows that f ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a g(x)dx +g ( 2a + η(b,a) 2 ) 1 η(b,a) ∫ a+η(b,a) a f(x)dx ≤ 1 2p+q [ 1 η(b,a) ∫ a+η(b,a) a f(x)g(x)dx + Γ(p + q + 1)Γ(p + q + 1) Γ(2p + 2q + 2) M(a,b) + Γ(2p + 1)Γ(2q + 1) Γ(2p + 2q + 2) N(a,b) ] +2p+q−1f ( 2a + η(b,a) 2 ) g ( 2a + η(b,a) 2 ) , which is the requires result. � Remark 3.1. If we take η(b,a) = b−a, in above results, we obtain the known integral inequalities for the class of beta-convex functions, see [22]. 4. Integral Inequalities We need the following Lemma in order to obtain new integral inequalities related to beta-preinvex function, which can be proved using the technique of Liu[3]. 50 NOOR, NOOR AND IFTIKHAR Lemma 4.1. If f : I = [a,a + η(b,a)] ⊆ R −→ R is a function such that f ∈ L[a,a + η(b,a)], then the following equality holds for some fixed α,β > 0.∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)βf(a + tη(b,a))dt, Theorem 4.1. Let f : I = [a,a + η(b,a)] ⊆ R −→ R be a differentiable function on the interior I◦ of I. If f ∈ L[a,a + η(b,a)] and |f| is beta-preinvex function on [a,a + η(b,a)] and α,β > 0, then∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ ηα+β+1(b,a) ( |f(a)|ϕ1(t; a,b) + |f(b)|ϕ2(t; a,b) ) , where ϕ1(t; a,b) = ∫ 1 0 tα+q(1 − t)β+pdt = β(α + q + 1,β + p + 1) (4.1) ϕ2(t; a,b) = ∫ 1 0 tα+p(1 − t)β+qdt = β(α + p + 1,β + q + 1) (4.2) Proof. Using Lemma 4.1 and beta-preinvexity of |f|, we have∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣dt ≤ ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β { (1 − t)ptq|f(a)| +tp(1 − t)q|f(b)| } dt = ηα+β+1(b,a) ( |f(a)| ∫ 1 0 tα+q(1 − t)β+pdt +|f(b)| ∫ 1 0 tα+p(1 − t)β+qdt ) = ηα+β+1(b,a) ( |f(a)|ϕ1(t; a,b) + |f(b)|ϕ2(t; a,b) ) . This completes the proof. � Theorem 4.2. Let f : I = [a,a + η(b,a)] ⊆ R −→ R be a differentiable function on the interior I◦ of I. If f ∈ L[a,a + η(b,a)] and |f|λ is beta-preinvex function on [a,a + η(b,a)] and α,β > 0, λ ≥ 1, then∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ ηα+β+1(b,a) ( ϕ3(t; a,b) )1− 1 λ( |f(a)|λϕ1(t; a,b) + |f(b)|λϕ2(t; a,b) ) 1 λ , where ϕ1(t; a,b), ϕ2(t; a,b) are given by (4.1) and (4.2) respectively, and ϕ3(t; a,b) = ∫ 1 0 tα(1 − t)βdt = β(α + 1,β + 1). SOME INTEGRAL INEQUALITIES FOR BETA-PREINVEX FUNCTIONS 51 Proof. Using Lemma 4.1, beta-preinvexity of |f|λ and power mean inequality, we have ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣dt ≤ ηα+β+1(b,a) (∫ 1 0 tα(1 − t)βdt )1− 1 λ (∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣λdt) 1λ ≤ ηα+β+1(b,a) (∫ 1 0 tα(1 − t)βdt )1− 1 λ (∫ 1 0 tα(1 − t)β { (1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ } dt ) 1 λ = ηα+β+1(b,a) (∫ 1 0 tα(1 − t)βdt )1− 1 λ ( |f(a)|λ ∫ 1 0 tα+q(1 − t)β+pdt + |f(b)|λ ∫ 1 0 tα+p(1 − t)β+qdt ) 1 λ = ηα+β+1(b,a) ( ϕ3(t; a,b) )1− 1 λ( |f(a)|λϕ1(t; a,b) + |f(b)|λϕ2(t; a,b) ) 1 λ , which the the required result. � Theorem 4.3. Let f : I = [a,a + η(b,a)] ⊆ R −→ R be a differentiable function on the interior I◦ of I. If f ∈ L[a,a + η(b,a)] and |f|λ is beta-preinvex function on [a,a + η(b,a)] and α,β > 0, then ∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx ≤ ηα+β+1(b,a) ( ϕ4(t; a,b) ) 1 µ × ( |f(a)|λ + |f(b)|λβ(p + 1,q + 1) ) 1 λ , where 1 λ + 1 µ = 1 and ϕ4(t; a,b) = ∫ 1 0 tαµ(1 − t)βµdt = β(αµ + 1,βµ + 1). 52 NOOR, NOOR AND IFTIKHAR Proof. Using Lemma 4.1, beta-preinvexity of |f|λ and the Holder’s integral inequality, we have∫ a+η(b,a) a (x−a)α(a + η(b,a) −x)βf(x)dx = ηα+β+1(b,a) ∫ 1 0 tα(1 − t)β ∣∣f(a + tη(b,a))∣∣dt ≤ ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ (∫ 1 0 ∣∣f(a + tη(b,a))∣∣λdt) 1λ ≤ ηα+β+1(b,a) (∫ 1 0 tαµ(1 − t)βµdt ) 1 µ (∫ 1 0 [(1 − t)ptq|f(a)|λ + tp(1 − t)q|f(b)|λ]dt ) 1 λ = ηα+β+1(b,a) ( ϕ4(t; a,b) ) 1 µ × ( |f(a)|λ + |f(b)|λβ(p + 1,q + 1) ) 1 λ . This completes the proof. � remarks From Definitions 2.2 and 2.3, we see that the function f satisfies the relation f( 2x + η(y,x) 2 ) = f(x) + f(y) 2 , which is called the preinvex functional equation. We remark that, if η(y,x) = y − x, then the invex set Kη becomes convex set K. In this case, the preinvex functional equations collapses to the following equation f( x + y 2 ) = f(x) + f(y) 2 , which is called the Jensen-Cauchy functional equation, see[19]. This has motivated us to consider the additive preinvex functional equation of the type f(2x + η(y,x)) = f(x) + f(y). (4.3) It is an interesting problem to study the stability criteria of the additive preinvex functional equations (4.3) using the technique of Rassias[20], Ulam[23] and Small[21]. We leave this to the interested readers to explore this aspects of the additive preinvex functional equations. Acknowledgements The authors would like to thank Dr. S. M. Junaid Zaidi, (H. I., S. I), Rector, COMSATS Institute of Information Technology, Pakistan, for providing excellent research and academic environments. References [1] I. 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Gov, Some Hermite-Hadamard inequalities for beta-convex and its fractional applications, NTMSCI 3(4)(2015), 18-33. [23] S. M. Ulam, Problems in Modern Mathmeatics, Science Editions, J. Wiley, New York, 1960. [24] T. Weir and B. Mond, Preinvex functions in multiple objective optimization. J Math. Anal. Appl., 136(1988), 29-38 Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan. ∗Corresponding author: noormaslam@gmail.com 1. Introduction 2. preliminaries 3. Hermite-Hadamard Inequalities 4. Integral Inequalities remarks Acknowledgements References