International Journal of Analysis and Applications ISSN 2291-8639 Volume 15, Number 1 (2017), 46-56 http://www.etamaths.com SOME CHARACTERIZATIONS OF GENERAL PREINVEX FUNCTIONS MUHAMMAD UZAIR AWAN1, MUHAMMAD ASLAM NOOR2,3, VISHNU NARAYAN MISHRA4,5,∗ AND KHALIDA INAYAT NOOR3 Abstract. In this paper, we consider a new class of general preinvex functions involving an arbitrary function. We show that the optimality condition for general preinvex functions on general invex set can be characterized by a class of variational-like inequality. We also derive some integral inequalities of Hermite-Hadamard type via general preinvex functions. Some special cases are also discussed. Our results represent a significant refinement of the previously known results. These results may stimulate further research in this area. 1. Introduction In recent years, several extensions and generalizations have been introduced and considered for classical convexity using novel and innovative techniques, see [1, 13]. A significant generalization of convex functions was that of invex functions which was introduced by Hanson [7]. Ben-Israel and Mond [8] introduced another class of convex functions, which is called preinvex functions. We remark that the differentiable preinvex functions are invex functions, but the converse may be true. It is well-known that the preinvex functions and invex sets may not be convex functions and convex sets. Many researchers have investigated different properties of the preinvex functions and their role in different fields of sciences such as optimization, variational inequalities, equilibrium problems and integral inequalities, see [14, 15, 17, 18, 22–24]. Another significant generalization of classical convex sets and functions was the introduction of general nonconvex (ϕ-convex) sets and general nonconvex (ϕ-convex) functions with respect to an arbitrary function, respectively by Youness [25]. These general convex set may not be a classical convex set, see [5]. Noor [16] has investigated the applications of general nonconvex functions in variational inequalities and optimization theory. It is obvious that preinvex functions and general convex functions are distinctly two different classes of convex functions. This motivated Fulga et al. [6], to consider another class of convex functions by combining these two classes. This new class of convex function is called the general preinvex function. In this paper, we discuss some properties of the general preinvex functions. We show that the optimization of the differentiable general preinvex functions can be characterized by a class of variational-like inequality, which is called general variational-like inequality. In the last section, we derive some Hermite-Hadamard type inequalities via general preinvex functions. This may be starting point for some new research in this field. 2. Preliminaries In this section, we define some basic results and also discuss several special cases. Before proceeding further, we suppose Kηϕ be a nonempty closed set in a Hilbert space H. We denote 〈., .〉 by norm and ‖.‖ by inner product, respectively. Also suppose that η(., .) : Kηϕ × Kηϕ → H and ϕ : H → H be arbitrary functions. Definition 2.1 ( [25]). A set Kϕ ⊆ Rn is said to be a general convex (ϕ-convex) set, if and only if, there exists an arbitrary function ϕ such that, (1 − t)ϕ(u) + tϕ(v) ∈ Kϕ, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kϕ, t ∈ [0, 1]. Received 8th May, 2017; accepted 14th July, 2017; published 1st September, 2017. 2010 Mathematics Subject Classification. 26A51, 26D15, 49J40, 47H10, 90C33. Key words and phrases. convex functions; general preinvex functions; differentiability; Hermite-Hadamard 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. 46 GENERAL PREINVEX FUNCTIONS 47 Definition 2.2 ( [6]). A set Kηϕ is said to be general invex set with respect to η(., .) and ϕ, if ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ Kηϕ, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. (2.1) It is known that general invex set may not be a general convex set see [6]. Note that, if η(ϕ(v),ϕ(u)) = ϕ(v)−ϕ(u), then our definition reduces to the definition of general convex set, which is mainly due to Youness [25]. If along with η(ϕ(v),ϕ(u)) = ϕ(v) −ϕ(u), we have ϕ = I, where I is identity function, then we have the definition of classical convex set. Definition 2.3 ( [6]). A function F on Kηϕ is said to be general preinvex with respect to arbitrary functions η and ϕ, if F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)F(ϕ(u)) + tF(ϕ(v)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. (2.2) If η(ϕ(v),ϕ(u)) = ϕ(v)−ϕ(u), then our definition reduces to the definition of general convex function [25]. If ϕ = I, where I is identity function, then we have the definition of preinvex functions [24]. Definition 2.4. A function F is said to be general mid-preinvex with respect to arbitrary functions η and ϕ, if F ( 2ϕ(u) + η(ϕ(v),ϕ(u)) 2 ) ≤ F(ϕ(u)) + F(ϕ(v)) 2 , ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ. Definition 2.5. A function F on Kηϕ is said to be general semistrictly preinvex with respect to arbitrary functions η and ϕ, if and only if ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, F(ϕ(u)) 6= F(ϕ(v)), t ∈ (0, 1), we have F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)F(ϕ(u)) + tF(ϕ(v)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. (2.3) Definition 2.6. A function F on Kηϕ is said to be general strictly preinvex with respect to arbitrary functions η and ϕ, if and only if ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, ϕ(u) 6= ϕ(v), t ∈ (0, 1), we have F(ϕ(u) + tη(ϕ(v),ϕ(u))) < (1 − t)F(ϕ(u)) + tF(ϕ(v)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. (2.4) Definition 2.7 ( [6]). A function F on general invex set Kηϕ is said to be quasi general preinvex function, if F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ max{F(ϕ(u)),F(ϕ(v))}, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. Definition 2.8. A function F on general invex set Kηϕ is said to be general logarithmic preinvex function, if F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (F(ϕ(u)))1−t(F(ϕ(v)))t, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1], where F(.) > 0. From above definition, we have F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (F(ϕ(u)))1−t(F(ϕ(v)))t ≤ (1 − t)F(ϕ(u)) + tf(ϕ(v)) ≤ max{F(ϕ(u)),F(ϕ(v))}. Following conditions are useful in studying various properties of our proposed results. Condition C. Let η(., .) : Kηϕ ×Kηϕ → H satisfies the following assumptions η(ϕ(u),ϕ(u) + tη(ϕ(v),ϕ(u))) = −tη(ϕ(v),ϕ(u)), η(ϕ(v),ϕ(u) + tη(ϕ(v),ϕ(u))) = (1 − t)η(ϕ(v),ϕ(u)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. 48 AWAN, NOOR, MISHRA AND NOOR For more information, see [9]. For t = 1 in Definition 2.3, we have following condition. Condition A. Let F be general preinvex function, then F(ϕ(u) + η(ϕ(v),ϕ(u))) ≤ F(ϕ(v)). Let Kηϕ = Iηϕ = [ϕ(a),ϕ(a) +η(ϕ(b),ϕ(a))] be the interval. We now define general preinvex functions on I. Definition 2.9. Let Iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))]. Then F is a general preinvex function, if and only if,∣∣∣∣∣∣ 1 1 1 ϕ(a) ϕ(x) ϕ(a) + η(ϕ(b),ϕ(a)) F(ϕ(a)) F(ϕ(x)) F(ϕ(a) + η(ϕ(b),ϕ(a))) ∣∣∣∣∣∣ ≥ 0; ϕ(a) ≤ ϕ(x) ≤ ϕ(a) + η(ϕ(b),ϕ(a)). One can easily show that the following are equivalent: (1) F is general preinvex function on general invex set. (2) F(ϕ(x)) ≤ F(ϕ(a)) + F(ϕ(b))−F(ϕ(a)) η(ϕ(b),ϕ(a)) (ϕ(x) −ϕ(a)). (3) F(ϕ(x))−F(ϕ(a)) ϕ(x)−ϕ(a) ≤ F(ϕ(b))−F(ϕ(a)) η(ϕ(b),ϕ(a)) ≤ F(ϕ(b))−F(ϕ(x)) ϕ(a)+η(ϕ(b),ϕ(a))−ϕ(x) . (4) [ϕ(x) − (ϕ(a) + η(ϕ(b),ϕ(a)))]F(ϕ(a)) + η(ϕ(b),ϕ(a))F(ϕ(x)) + [ϕ(a) −ϕ(x)]F(ϕ(b)) ≥ 0. (5) F(ϕ(a)) η(ϕ(b),ϕ(a))(ϕ(a)−ϕ(x)) + F(ϕ(x)) [ϕ(x)−(ϕ(a)+η(ϕ(b),ϕ(a)))][ϕ(a)−ϕ(x)] + F(ϕ(b)) η(ϕ(b),ϕ(a))[ϕ(x)−(ϕ(a)+η(ϕ(b),ϕ(a)))] ≥ 0, where ϕ(x) = ϕ(a) + tη(ϕ(b),ϕ(a)) and t ∈ [0, 1]. Remark 2.1. Note that for ϕ = I, where I is the identity function, the above definition reduces to the definition for preinvex functions on an interval. Definition 2.10. Let Iη = [a,a + η(b,a)]. Then F is called preinvex function, if and only if,∣∣∣∣∣∣ 1 1 1 a x a + η(b,a) F(a) F(x) F(a + η(b,a)) ∣∣∣∣∣∣ ≥ 0; a ≤ x ≤ a + η(b,a). One can easily show that the following are equivalent: (1) F is preinvex function on invex set. (2) F(x) ≤ F(a) + F(b)−F(a) η(b,a) (x−a). (3) F(x)−F(a) x−a ≤ F(b)−F(a) η(b,a) ≤ F(b)−F(x) a+η(b,a)−x. (4) [x− (a + η(b,a))]F(a) + η(b,a)F(x) + (a−x)F(b) ≥ 0. (5) F(a) η(b,a)(a−x) + F(x) [x−(a+η(b,a))][a−x] + F(b) η(b,a)[x−(a+η(b,a))] ≥ 0, where x = a + tη(b,a) and t ∈ [0, 1]. Remark 2.2. If in Definition 2.8, η(ϕ(b),ϕ(a)) = ϕ(b)−ϕ(a). Then, we have the definition of general convex functions on interval. Definition 2.11. Let Iϕ = [ϕ(a),ϕ(b)]. Then F is called general convex function, if and only if,∣∣∣∣∣∣ 1 1 1 ϕ(a) ϕ(x) ϕ(b) F(ϕ(a)) F(ϕ(x)) F(ϕ(b)) ∣∣∣∣∣∣ ≥ 0; ϕ(a) ≤ ϕ(x) ≤ ϕ(b). One can easily show that the following are equivalent: (1) F is general convex function on general convex set. (2) F(ϕ(x)) ≤ F(ϕ(a)) + F(ϕ(b))−F(ϕ(a)) ϕ(b)−ϕ(a) (ϕ(x) −ϕ(a)). (3) F(ϕ(x))−F(ϕ(a)) ϕ(x)−ϕ(a) ≤ F(ϕ(b))−F(ϕ(a)) ϕ(b)−ϕ(a) ≤ F(ϕ(b))−F(ϕ(x)) ϕ(b)−ϕ(x) . (4) (ϕ(x) −ϕ(b))F(ϕ(a)) + (ϕ(b) −ϕ(a))F(ϕ(x)) + (ϕ(a) −ϕ(x))F(ϕ(b)) ≥ 0. (5) F(ϕ(a)) (ϕ(b)−ϕ(a))(ϕ(a)−ϕ(x)) + F(ϕ(x)) (ϕ(x)−ϕ(b))(ϕ(a)−ϕ(x)) + F(ϕ(b)) (ϕ(b)−ϕ(a))(ϕ(x)−ϕ(b)) ≥ 0, where ϕ(x) = ϕ(a) + t(ϕ(b) −ϕ(a)) and t ∈ [0, 1]. GENERAL PREINVEX FUNCTIONS 49 Definition 2.12. A differentiable function F on general invex set Kηϕ is said to be general invex function, if there exists arbitrary functions η and ϕ, such that F(ϕ(v)) −F(ϕ(u)) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, where F ′ is the differential of F . If ϕ = I, our definition of general invexity reduces to the definition of invexity which is mainly due to Hanson [5]. Definition 2.13. A function F is said to be pseudo general preinvex with respect to η, if there exists a strictly positive bifunction b such that F(ϕ(v)) < F(ϕ(u)) ⇒ F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ F(ϕ(u)) + t(t− 1)b(ϕ(v),ϕ(u)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ (0, 1). Definition 2.14. A function T is said to be η-monotone, if and only if 〈T(ϕ(u)),η(ϕ(v),ϕ(u))〉 + 〈T(ϕ(v)),η(ϕ(v),ϕ(u))〉≤ 0, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ. 3. Results and Discussions 3.1. Variational-Like Inequalities. In this section, we derive some general variational-like inequal- ities. Theorem 3.1. If F is a general preinvex function on Kηϕ, then, the lower level set Lα = {u ∈ H : ϕ(u) ∈ Kηϕ : F(ϕ(u)) ≤ α,α ∈ R} is a general invex set. Theorem 3.2. A function F is general preinvex function on Kηϕ if and only if, epi(F) = {(ϕ(u),α) : ϕ(u) ∈ Kηϕ,α ∈ R,F(ϕ(u)) ≤ α} is general invex set. Theorem 3.3. Let F be a general preinvex function. Suppose µ = inf ϕ(u)∈Kηϕ F(ϕ(u)). Then the set Aη = {u ∈ H : ϕ(u) ∈ Kηϕ : F(ϕ(u)) = µ} is general invex set of Kηϕ. If F is general strictly preinvex, then Aη is a singelton. Proof. Let ϕ(u),ϕ(v) ∈ Aη. Then for 0 < t < 1, we suppose ϕ(z) = ϕ(u) + tη(ϕ(v),ϕ(u)). Since F is preinvex function. Then F(ϕ(z)) = F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)F(ϕ(u)) + tf(ϕ(v)) = µ, this implies that ϕ(z) ∈ Aη thus Aη is general invex set. Now for other part of the theorem, we assume contrary that F(ϕ(u)) = F(ϕ(v)) = µ. Since Kηϕ is general invex set, then for 0 < t < 1, ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ Kηϕ. Also, since F is strictly general preinvex function. F(ϕ(u) + tη(ϕ(v),ϕ(u))) < (1 − t)F(ϕ(u)) + tf(ϕ(v)) = µ, which is contradiction that µ = inf ϕ(u)∈Kηϕ F(ϕ(u)), this completes the proof. � Theorem 3.4. Let F be a general preinvex function on Kη. If φ is a nondecreasing convex function, then φ◦F is a general preinvex function. Proof. Since F is a general preinvex function and φ is nondecreasing, then φ◦F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ φ[F(ϕ(u) + tη(ϕ(v),ϕ(u)))] ≤ φ[(1 − t)F(ϕ(u)) + tf(ϕ(v))] ≤ (1 − t)φ◦F(ϕ(u)) + tφ◦F(ϕ(v)). This completes the proof. � Theorem 3.5. Let F be a semistrictly general preinvex function on Kη. If φ is a nondecreasing convex function, then φ◦F is a semistrictly general preinvex function. Lemma 3.1. Let Kηϕ be general convex set and let F be be general invex function on K. The 50 AWAN, NOOR, MISHRA AND NOOR (1) If 〈F ′(ϕ(u)),ϕ(v) −ϕ(u)〉≤ 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, such that F(ϕ(v)) ≤ F(ϕ(u)), then F is a general pseudo-convex function. (2) If 〈F ′(ϕ(u)),ϕ(v) −ϕ(u)〉≤ 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, such that F(ϕ(v)) < F(ϕ(u)), then F is strictly general pseudo-convex function. Proof. Let u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ and F(ϕ(v)) ≤ F(ϕ(u)). Then 〈F ′(ϕ(u)),ϕ(v) −ϕ(u)〉 = 〈F ′(ϕ(u)),ϕ(v) −ϕ(u) −η(ϕ(v),ϕ(u))〉 + 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉 ≤ 〈F ′(ϕ(u)),ϕ(v) −ϕ(u) −η(ϕ(v),ϕ(u))〉 + F(ϕ(v)) −F(ϕ(u)) ≤〈F ′(ϕ(u)),ϕ(v) −ϕ(u) −η(ϕ(v),ϕ(u))〉≤ 0. This completes the proof. The proof of second part is on similar lines. � Theorem 3.6. If F be a general preinvex function. Then any local minimum of F is a global minimum. Proof. Let F has a local minimum at ϕ(u) ∈ Kηϕ. Assume contrary, that F(ϕ(v)) < F(ϕ(u)) for some ϕ(v) ∈ K. Now since F is general preinvex function. Then F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)F(ϕ(u)) + tf(ϕ(v)). This implies that F(ϕ(u) + tη(ϕ(v),ϕ(u))) −F(ϕ(u)) ≤ t(F(ϕ(v)) −F(ϕ(u))) < 0. Thus, we have F(ϕ(u) + tη(ϕ(v),ϕ(u))) < F(ϕ(u)), a contradiction. This completes the proof. � Theorem 3.7. If F be a semistrictly general preinvex function. Then any local minimum of F is a global minimum. Proof. The proof is similar to previous. � Theorem 3.8. Let F be general preinvex function with respect to η, i = 1, 2, . . . ,n. Then n∑ i=0 µiFi(ϕ(u)) is general preinvex with respect to η, where µi ≥ 0. Proof. Let Fi be general preinvex functions. Then Fi(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)F(ϕ(u)) + tf(ϕ(v)). Now (1 − t) n∑ i=0 µiFi(ϕ(u)) + t n∑ i=0 µiFi(ϕ(v)) = n∑ i=0 µi[(1 − t)Fi(ϕ(u)) + tfi(ϕ(v))] ≥ n∑ i=0 µiFi(ϕ(u) + tη(ϕ(v),ϕ(u))). This implies that n∑ i=0 µiFi(ϕ(u)) is general preinvex function. � Theorem 3.9. If F is general preinvex function with respect to η such that F(ϕ(v)) < F(ϕ(u)), then F is general pseudo preinvex function with respect to same η. GENERAL PREINVEX FUNCTIONS 51 Proof. Since F(ϕ(v)) < F(ϕ(u)) and F is general preinvex function with respect to η, then for all u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ and t ∈ (0, 1), we have F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ F(ϕ(u)) + t(F(ϕ(v)) −F(ϕ(u))) < F(ϕ(u)) + t(1 − t)(F(ϕ(v)) −F(ϕ(u))) = F(ϕ(u)) + t(t− 1)(F(ϕ(u)) −F(ϕ(v))) = F(ϕ(u)) + t(t− 1)b(ϕ(u),ϕ(v)), where b(ϕ(u),ϕ(v)) = F(ϕ(u)) −F(ϕ(v)) > 0. This completes the proof. � Theorem 3.10. Let F be a differentiable general preinvex function on Kηϕ. Then ϕ(u) ∈ Kηϕ is the minimum of F on Kηϕ if and only if ϕ(u) ∈ Keta satisfies the inequality 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, (3.1) where F ′ is the differential of F at ϕ(u) ∈ Kηϕ. The inequality (3.1) is called the general variational-like inequality. Proof. Let ϕ(u) ∈ Kηϕ be a minimum of general preinvex function F on Kηϕ. Then by definition of minimum, we have, F(ϕ(u)) ≤ F(ϕ(v)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ. (3.2) Since Kηϕ is a general invex set, so ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1], we have ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ Kηϕ. (3.3) Replacing ϕ(v) by ϕ(vt) in (3.2) we get F(ϕ(u)) ≤ F(ϕ(vt)) = F(ϕ(u) + tη(ϕ(v),ϕ(u))), which implies that F(ϕ(u) + tη(ϕ(v),ϕ(u))) −F(ϕ(u)) ≥ 0. Since F is differentiable, so dividing both sides of the above inequality by t and then taking the limit as t → 0, we have 0 ≤ lim t→0 ( F(ϕ(u) + tη(ϕ(v),ϕ(u))) −F(ϕ(u)) t ) = 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, that is ϕ(u) ∈ Kηϕ satisfies the inequality 〈F ′(u),η(ϕ(v),ϕ(u))〉≥ 0, ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ. Conversely, let inequality (3.1) holds. We have to show that ϕ(u) ∈ Kηϕ, is the minimum of F on the general invex set Kηϕ. Since F is general preinvex function, then F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ F(ϕ(u)) + t(F(ϕ(v)) −F(ϕ(v))). Now taking limit as t → 0, we have F(ϕ(v)) −F(ϕ(u)) ≥ lim t→0 F(ϕ(u) + tη(ϕ(v),ϕ(u))) −F(ϕ(u)) t = 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉 ≥ 0, thus, it follows that F(ϕ(u)) ≤ F(ϕ(v)), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, which completes the proof. � Theorem 3.11. Let F be a differentiable function on general invex set Kηϕ and suppose condition C holds. Then F is general preinvex function if and only if F is a general invex function. 52 AWAN, NOOR, MISHRA AND NOOR Proof. Let F be a differentiable general preinvex function, then F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ F(ϕ(u)) + t(F(ϕ(v))−F(ϕ(u))), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1], since F is differentiable taking limit as t → 0, we have F(ϕ(v)) −F(ϕ(u)) ≥ lim t→0 F(ϕ(u) + tη(ϕ(v),ϕ(u))) −F(ϕ(u)) t = 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. This implies that F is general invex function. Conversely, suppose that F is general invex function, that is F(ϕ(v)) −F(ϕ(u)) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.4) Since Kηϕ is a general invex set. Then ∀ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1], we have ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ Kηϕ. Replacing ϕ(u) by ϕ(vt) in (3.4) and using condition C, we have F(ϕ(v)) −F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≥ (1 − t)〈F ′(ϕ(u) + tη(ϕ(v),ϕ(u))),η(ϕ(v),ϕ(u))〉. (3.5) Similarly, we have F(ϕ(u)) −F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≥−t〈F ′(ϕ(u) + tη(ϕ(v),ϕ(u))),η(ϕ(v),ϕ(u))〉. (3.6) Multiplying (3.5) by t and (3.6) by (1 − t), and then adding the resultant, we have F(ϕ(u) + tη(ϕ(v),ϕ(u))) ≤ (1 − t)F(ϕ(u)) + tf(ϕ(v)). (3.7) This completes the proof. � Theorem 3.12. Let F be a differentiable function on general invex set Kηϕ and suppose Condition A holds. Then the differential F ′ of F is η-monotone if and only if F is a general invex function. Proof. Let F be general invex function. Then F(ϕ(v)) −F(ϕ(u)) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.8) Interchanging ϕ(u) and ϕ(v) in above inequality, we have F(ϕ(u)) −F(ϕ(v)) ≥〈F ′(ϕ(v)),η(ϕ(u),ϕ(v))〉. (3.9) Adding (3.8) and (3.9), we have 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉 + 〈F ′(ϕ(v)),η(ϕ(u),ϕ(v))〉≤ 0. (3.10) This implies that F ′ is η-monotone. Conversely, suppose that F ′ is η-monotone, that is F ′ satisfies inequality (3.10). Then, from (3.10), we have 〈F ′(ϕ(v)),η(ϕ(u),ϕ(v))〉≤−〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.11) Now, since Kηϕ is general invex set, then, ∀ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1], we have ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ Kηϕ. Taking ϕ(v) as ϕ(vt) in (3.11), and applying condition C, we get 〈F ′(ϕ(u) + tη(ϕ(v),ϕ(u))),η(ϕ(v),ϕ(u))〉≥ 〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.12) Consider an auxiliary function ϕ(t) = F(ϕ(vt)) ≡ F(ϕ(u) + tη(ϕ(v),ϕ(u))), ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1]. (3.13) Now using the fact that F is differentiable, we have ϕ′(t) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. Integrating above inequality with respect to t on [0, 1], we have ϕ(1) −ϕ(0) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. (3.14) Using (3.13) and (3.14), we have F(ϕ(u) + η(ϕ(v),ϕ(u))) −F(u) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉. Now using Condition A, we have F(ϕ(v)) −F(ϕ(u)) ≥〈F ′(ϕ(u)),η(ϕ(v),ϕ(u))〉, which shows that F is general invex function. This completes the proof. � GENERAL PREINVEX FUNCTIONS 53 Theorem 3.13. Let Kηϕ be a general invex set in H. Suppose function F be η-pseudomonotone and η-hemicontinuous. If condition C holds, then ϕ(u) ∈ Kηϕ satisfies 〈F(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ H : ϕ(v) ∈ Kηϕ, (3.15) if and only if ϕ(u) ∈ Kηϕ satisfies 〈F(ϕ(v)),η(ϕ(u),ϕ(v))〉≤ 0, ∀v ∈ H : ϕ(v) ∈ Kηϕ. (3.16) Proof. Let u ∈ H : ϕ(u) ∈ Kηϕ satisfies the following inequality 〈F(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ H : ϕ(v) ∈ Kηϕ, which implies that 〈F(ϕ(v)),η(ϕ(u),ϕ(v))〉≤ 0, ∀v ∈ H : ϕ(v) ∈ Kηϕ, where F is η-pseudomonotone. Conversely, let (3.16) holds. Since Kηϕ is general invex set, then ∀u,v ∈ H : ϕ(u),ϕ(v) ∈ Kηϕ, t ∈ [0, 1], ϕ(vt) ≡ ϕ(u) + tη(ϕ(v),ϕ(u)) ∈ Kηϕ. Taking ϕ(v) = ϕ(vt) in (3.16) and using condition C, we have 0 ≥〈F(ϕ(vt)),η(ϕ(u),ϕ(u) + tη(ϕ(v),ϕ(u)))〉 = −t〈F(ϕ(vt)),η(ϕ(v),ϕ(u))〉, from which we have 〈F(ϕ(vt)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ H : ϕ(v) ∈ Kηϕ. Taking limit as t → 0 on both sides of above inequality, we have 〈F(ϕ(u)),η(ϕ(v),ϕ(u))〉≥ 0, ∀v ∈ H : ϕ(v) ∈ Kηϕ, (3.17) where we have used the fact that F is η-hemicontinuous. This completes the proof. � 3.2. Hermite-Hadamard type Inequalities. Hermite-Hadamard type inequalities provides us nec- essary and sufficient condition for a function to be convex. In recent years many new generalizations of these inequalities have been obtained via different classes of convex functions. For more information, see [2, 4, 19–21, 23]. In this section, we derive some Hermite-Hadamard type inequalities via general preinvex functions. Theorem 3.14. Let F : Iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))] → R be a general preinvex function with η(ϕ(b),ϕ(a)) > 0. If η(., .) satisfies the condition C, then we have F ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) ≤ 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) F(ϕ(x))dϕ(x) ≤ F(ϕ(a)) + F(ϕ(b)) 2 . Proof. Since F is general preinvex function and η(., .) satisfies the condition C, we have F ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) ≤ 1 2 [F (ϕ(a) + tη(ϕ(b),ϕ(a))) + F(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] . Integrating above inequality with respect to t on [0, 1], we have F ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) ≤ 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) F(ϕ(x))dϕ(x). (3.18) Also F(ϕ(a) + tη(ϕ(b),ϕ(a))) ≤ (1 − t)F(ϕ(a)) + tF(ϕ(b)). Integrating above inequality with respect to t on [0, 1], we have 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) F(ϕ(x))dϕ(x) ≤ F(ϕ(a)) + F(ϕ(b)) 2 . (3.19) Combining (3.18) and (3.19) completes the proof. � 54 AWAN, NOOR, MISHRA AND NOOR Theorem 3.15. Let F,W : Iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))] → R be general preinvex functions respectively with η(ϕ(b),ϕ(a)) > 0. Suppose η(., .) satisfies Condition C, then we have F ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) W ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) − 1 2η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) F(ϕ(x))W(ϕ(x))dϕ(x) ≤ 1 6 M(ϕ(a),ϕ(b)) + 1 3 N(ϕ(a),ϕ(b)), where M(ϕ(a),ϕ(b)) = F(ϕ(a))W(ϕ(a)) + F(ϕ(b))W(ϕ(b)) (3.20) and N(ϕ(a),ϕ(b)) = F(ϕ(a))W(ϕ(b)) + F(ϕ(b))W(ϕ(a)). (3.21) Proof. Since F and W are general preinvex functions respectively and η(., .) satisfies Condition C, we have F ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) W ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) = F(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)) + 1 2 η(ϕ(a) + tη(ϕ(b),ϕ(a)),ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))) ×W(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)) + 1 2 η(ϕ(a) + tη(ϕ(b),ϕ(a)),ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))) ≤ 1 4 [F(ϕ(a) + tη(ϕ(b),ϕ(a))) + F(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] × [W(ϕ(a) + tη(ϕ(b),ϕ(a))) + W(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] ≤ 1 4 [F(ϕ(a) + tη(ϕ(b),ϕ(a)))W(ϕ(a) + tη(ϕ(b),ϕ(a))) + F(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))W(ϕ(a) + (1 − t)η(ϕ(b),ϕ(a)))] + 1 2 { 2(t− t2)M(a,b) + [t2 + (1 − t)2]N(a,b) } . Integrating above inequality with respect to t on [0,1], we have F ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) W ( 2ϕ(a) + η(ϕ(b),ϕ(a)) 2 ) − 1 2η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) F(ϕ(x))W(ϕ(x))dϕ(x) ≤ 1 6 M(ϕ(a),ϕ(b)) + 1 3 N(ϕ(a),ϕ(b)). The proof is complete. � Theorem 3.16. Let F,W : Iηϕ = [ϕ(a),ϕ(a) + η(ϕ(b),ϕ(a))] → R be general preinvex function with η(b,a) > 0, then we have 1 η(ϕ(b),ϕ(a)) ϕ(a)+η(ϕ(b),ϕ(a))∫ ϕ(a) F(ϕ(x))W(ϕ(x))dϕ(x) ≤ 1 3 M(ϕ(a),ϕ(b)) + 1 6 N(ϕ(a),ϕ(b)). where M(ϕ(a),ϕ(b)) and N(ϕ(a),ϕ(b)) are given by (3.20) and (3.21) respectively. GENERAL PREINVEX FUNCTIONS 55 Proof. Let F,W be general preinvex functions, then for all t ∈ [0, 1], we have F(ϕ(a) + tη(ϕ(b),ϕ(a)))W(ϕ(a) + tη(ϕ(b),ϕ(a))) ≤ [(1 − t)F(ϕ(a)) + tF(ϕ(b))][(1 − t)W(ϕ(a)) + tW(ϕ(b))] = (1 − t)2F(ϕ(a))W(ϕ(a)) + t(1 − t)F(ϕ(b))W(ϕ(a)) + t(1 − t)F(ϕ(a))W(ϕ(b)) + t2F(ϕ(b))W(ϕ(b)). 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Theory Appl., 102 (1999), 439–450. 1Department of Mathematics, Government College University,, [0pt] Faisalabad, Pakistan. 2Mathematics Department, King Saud University, Riyadh, Saudi Arabia. 3Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan. 4Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India 5L. 1627 Awadh Puri Colony Beniganj, Phase - III, Opposite - Industrial Training Institute (I.T.I.), Faiz- abad 224 001, Uttar Pradesh, India ∗Corresponding author: vishnunarayanmishra@gmail.com 1. Introduction 2. Preliminaries 3. Results and Discussions 3.1. Variational-Like Inequalities 3.2. Hermite-Hadamard type Inequalities 4. Conclusion References