International Journal of Analysis and Applications ISSN 2291-8639 Volume 5, Number 2 (2014), 115-122 http://www.etamaths.com NEW INTEGRAL INEQUALITIES THROUGH INVEXITY WITH APPLICATIONS SHAHID QAISAR1,∗, CHUANJIANG HE1 AND SABIR HUSSAIN2 Abstract. In this paper, we obtain some inequalities of Simpson’s inequality type for functions whose derivatives absolute values are quasi-preinvex func- tion. Applications to some special means are considered. 1. Introduction The Simpson’s inequality is very important and the well-known in the literature: This inequality is stated that: If f : [a,b] → R be a four times continuously differentiable mapping on (a,b) and ∥∥f(4)∥∥∞ = sup x∈(a,b) ∣∣f(4) (x)∣∣ < ∞.then the following inequality holds:∣∣∣∣∣∣13 [ f(a) + f(b) 2 + 2f ( a + b 2 )] − 1 b−a b∫ a f(x)dx ∣∣∣∣∣∣ ≤ 12880 ∥∥∥f(4)∥∥∥ ∞ (b−a)4 Recently, many others [5-7], [1] developed and discussed error estimates of the Simpson’s type inequality interms of refinement, counterparts, generalizations and new Simpson’s type inequalities. In [1],Dragomir et.al. proved the following recent developments on Simpson’s inequality for which the reminder is expressed in terms of lower derivatives than the fourth. Theorem 1. Suppose f : [a,b] → R is a differentiable mapping whose derivative is continuous on (a,b) and f′ ∈ L [a,b] .then the following inequality (1.1) ∣∣∣∣∣13 [ f(a)+f(b) 2 + 2f ( a+b 2 )] − 1 b−a b∫ a f(x)dx ∣∣∣∣∣ ≤ b−a3 ‖f′‖1 holds, where ‖f′‖1 = b∫ a |f′ (x)|dx. The bound of (1.1) for L-Lipschitian mapping was given in [1] by 5 36 L (b−a). Theorem 2. Suppose f : [a,b] → R is an absolutely continuous mapping on [a,b] whose derivative belongs to Lp [a,b] .then the following inequality holds, (1.2) ∣∣∣∣∣13 [ f(a)+f(b) 2 + 2f ( a+b 2 )] − 1 b−a b∫ a f(x)dx ∣∣∣∣∣ ≤ 1 6 [ 2q+1+1 3(q+1) ]1 q (b−a) 1 q ‖f′‖p 2010 Mathematics Subject Classification. 26D07; 26D10; 26D15. Key words and phrases. Simpson’s inequality; quasi-convex function; power-mean inequality; Holder’s integral inequality. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 115 116 QAISAR, HE AND HUSSAIN Where 1 p + 1 q = 1. In [2], Kirmaci established the following Hermite-Hadamard inequalities for dif- ferent convex functions as: Theorem 3. Let f : I ⊂ R → R be a differentiable function on I0 interior of I0a,b ∈ I with a < b.if the mapping |f′| is convex on [a,b] , and the following inequality holds (1.3) ∣∣∣ 1b−a ∫ ba f (x) dx−f (a+b2 )∣∣∣ ≤ b−a4 [|f′ (a)| + |f′ (b)|] . Let K be a closed set Rnand let f : K → Rand η : K × K → Rbe continuous functions. Let x ∈ K,then the set K is said to be invex at xwith respect to η (., .) , If x + tη (y,x) ∈ K,∀x,y ∈ K,t ∈ [0, 1] . Kis said to be invex set with respect to η if Kis invex at each x ∈ K.The invex set Kis also called a η-connected set. Definition 5[3]. The function f on the invex set Kis said to be preinvex with respect to η,if f (u + tη (v,u)) ≤ + (1 − t) f (u) + tf (v) ,∀u,v ∈ K,t ∈ [0, 1] . The function f is said to be preconcave if and only if −fis preinvex. It is to be noted that every convex function is preinvex with respect to the map η (x,y) = x−y but the converse is not true. Definition 6[4]. The function f on the invex set Kis said to be preinvex with respect to η,if f (u + tη (v,u)) ≤ max{f (u) ,f (v)} ,∀u,v ∈ K,t ∈ [0, 1] . Also every quasi-convex function is a prequasiinvex with respect to the map η (u,v) but the converse does not hold, see for example [8]. The main aim of this paper is to establish new Simpson’s type inequalities for the class of functions whose derivatives in absolute values are quasi-preinvex . . 2. Main results Before proceeding towards our main theorem regarding generalization of the Simpson’s type inequality using prequasiinvex . We begin with the following Lem- ma. Lemma 2.1. Let K ⊆ R be an open invex subset with respect to η : K ×K → Rand a,b ∈ Kwith a < a + η (b,a)suppose f : K → Ris a differentiable mapping on Ksuch that f′ ∈ L ([a,a + η (b,a)]) .Then for every a,b ∈ K with η (b,a) 6= 0the following inequality holds: ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ = η(b,a) 2 [ 1∫ 0 ( λ 2 − 1 3 ) f′ ( a + ( 1+λ 2 ) η (b,a) ) dλ + 1∫ 0 ( 1 3 − λ 2 ) f′ ( a + ( 1−λ 2 ) η (b,a) ) dλ ] . NEW INTEGRAL INEQUALITIES THROUGH INVEXITY WITH APPLICATIONS 117 Proof. Integrating by parts, we have I1 = 1∫ 0 ( λ 2 − 1 3 ) f′ ( a + ( 1+λ 2 ) η (b,a) ) dλ = 2( λ2 − 1 3 )f(a+( 1+λ 2 )η(b,a)) η(b,a) ∣∣∣∣1 0 − 1 η(b,a) 1∫ 0 f ( a + ( 1+λ 2 ) η (b,a) ) = 2 6η(b,a) f (a + η (b,a)) + 2 3η(b,a) f ( 2a+η(b,a) 2 ) − 1 η(b,a) 1∫ 0 f ( a + ( 1+λ 2 ) η (b,a) ) dλ Setting x = a+ ( 1+λ 2 ) η (b,a) and dx = η(b,a) 2 dλ which gives I1 = 2 6η(b,a) f (a + η (b,a))+ 2 3η(b,a) f ( 2a+η(b,a) 2 ) − 2 (η(b,a))2 a+η(b,a)∫ a+ 1 2 η(b,a) f (x) dx Similarly we can show that I2 = 1∫ 0 ( 1 3 − λ 2 ) f′ ( a + ( 1−λ 2 ) η (b,a) ) dλ = 2 6η(b,a) f (a) + 2 3η(b,a) f ( 2a+η(b,a) 2 ) − 2 (η(b,a))2 a+ 1 2 η(b,a)∫ a f (x) dx Therefore η(b,a) 2 [I1 + I2] = [ f(a)+f(a+η(b,a)) 6 + 2 3 f ( 2a+η(b,a) 2 ) − 1 η(b,a) a+η(b,a)∫ a f(x)dx ] Which completes the proof. In the following theorem, we shall propose some new upper bound for the right- hand side of Simpson’s type inequality for functions whose derivatives absolute values are prequasiinvex. Theorem 2.2. Let K ⊆ [0,∞) be an open invex subset with respect to η : K×K → R and a,b ∈ Kwith a < a + η (b,a) suppose f : K → R is a differentiable mapping on K such that f′ ∈ L ([a,a + η (b,a)]) . If |f′| is preinvex on K, then for every a,b ∈ K with η (b,a) 6= 0 the following inequality holds: (2.1) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 [ sup { |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣} + sup {∣∣f′(a + 1 2 η (b,a) )∣∣ , |f′ (a + η (b,a))|} ] . 118 QAISAR, HE AND HUSSAIN Proof. From Lemma 2.1, and since |f′| is prequasiinvex, then we have∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 2 [ 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣dλ + 1∫ 0 ∣∣1 3 − λ 2 ∣∣ ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣dλ] ≤ η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ sup {|f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣}dλ + η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ sup {∣∣f′(a + 1 2 η (b,a) )∣∣ , |f′ (a + η (b,a))|}dλ ≤ η(b,a) 2 sup { |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣} 1∫ 0 ∣∣λ 2 − 1 3 ∣∣dλ + η(b,a) 2 sup {∣∣f′(a + 1 2 η (b,a) )∣∣ , |f′ (a + η (b,a))|} 1∫ 0 ∣∣λ 2 − 1 3 ∣∣dλ = 5η(b,a) 72 sup { |f′′ (a)| , ∣∣f′′(a + 1 2 η (b,a) )∣∣} + 5η(b,a) 72 sup {∣∣f′′(a + 1 2 η (b,a) )∣∣ , |f′′ (a + η (b,a))|} . Which completes the proof. The upper bound for the midpoint inequality for the first derivative is presented as Corollary 2.3. Let f as in Theorem 2.2, if in addition (1) |f′| is increasing, then we have (2.2) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 [ |f′ (a + η (b,a))| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . (2) |f′| is decreasing, then we have (2.3) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 [ |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . Proof. It follows directly by Theorem 2.2. Remark 2.4. We note that the inequalities (2.2) and (2.3), are two new refine- ments of the trapezoid inequality for prequasiinvex functions, and thus for convex functions. The corresponding version for powers of the absolute value of the first derivative is incorporated in the following result: Theorem 2.5. Let K ⊆ [0,∞) be an open invex subset with respect to η : K×K → R and a,b ∈ K with a < a+η (b,a) suppose f : K → R is a differentiable mapping on K such that f′ ∈ L ([a,a + η (b,a)]) . If |f′|p is preinvex on K,from some p > 1, then for every a,b ∈ K with η (b,a) 6= 0 the following inequality holds: (2.4) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 12 ( 1+2p+1 3(p+1) )1/p ( sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 })p−1p + η(b,a) 12 ( 1+2p+1 3(p+1) )1/p ( sup {∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 , |f′ (a + η (b,a))| pp−1 })p−1p NEW INTEGRAL INEQUALITIES THROUGH INVEXITY WITH APPLICATIONS 119 Where q = p/(p− 1). Proof . From Lemma 2.1, and using the well known Holder integral inequality, we have∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣dλ + η(b,a) 2 1∫ 0 ∣∣1 3 − λ 2 ∣∣ ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣dλ ≤ η(b,a) 2 ( 1∫ 0 ( λ 2 − 1 3 )p)1p ( 1∫ 0 ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣ pp−1 dλ)p−1p + η(b,a) 2 ( 1∫ 0 ( 1 3 − λ 2 )p)1p ( 1∫ 0 ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣ pp−1 dλ)p−1p ≤ η(b,a) 2 ( 1∫ 0 ( λ 2 − 1 3 )p)1p ( 1∫ 0 sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 }dλ)p−1p + η(b,a) 2 ( 1∫ 0 ( 1 3 − λ 2 )p)1p ( 1∫ 0 sup {∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 , |f′ (a + η (b,a))| pp−1 }dλ)p−1p = η(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( 1∫ 0 sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 }dλ)p−1p + η(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( 1∫ 0 sup {∣∣f′(a + 1 2 η (b,a) )∣∣ pp−1 , |f′ (a + η (b,a))| pp−1 }dλ)p−1p Which completes the proof. Corollary 2.6. Let f as in Theorem 2.5, if in addition (1) |f′|p/(p−1) is increasing, then we have (2.5) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 12 ( 1+2p+1 3(p+1) )1/p [ |f′ (a + η (b,a))| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . (2) |f′|p/(p−1) is decreasing, then we have (2.6) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 12 ( 1+2p+1 3(p+1) )1/p [ |f′ (a)| , ∣∣f′(a + 1 2 η (b,a) )∣∣] . An improvement of constants in Theorem 2.5 and a consolidation of this result with Theorem 2.2. are given in the following theorem. Theorem 2.7. Let K ⊆ [0,∞) be an open invex subset with respect to η : K×K → R and a,b ∈ K with a < a+η (b,a) suppose f : K → R is a differentiable mapping on K such that f′ ∈ L ([a,a + η (b,a)]) . If |f′|q is preinvex on K, q ≥ 1, 120 QAISAR, HE AND HUSSAIN then for every a,b ∈ K with η (b,a) 6= 0 the following inequality holds: (2.7) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 ( sup { |f′ (a)|q , ∣∣f′(a + 1 2 η (b,a) )∣∣q})1q + 5η(b,a) 72 ( sup {∣∣f′(a + 1 2 η (b,a) )∣∣q , |f′ (a + η (b,a))|q})1q . Proof . Suppose thatq ≥ 1. From Lemma 2.1 and using the well known power mean inequality, we have∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ η(b,a) 2 1∫ 0 ∣∣λ 2 − 1 3 ∣∣ ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣dλ + η(b,a) 2 1∫ 0 ∣∣1 3 − λ 2 ∣∣ ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣dλ ≤ η(b,a) 2 ( 1∫ 0 ( λ 2 − 1 3 ) dλ )1−1 q ( 1∫ 0 ∣∣f′(a + (1+λ 2 ) η (b,a) )∣∣q dλ)1q + η(b,a) 2 ( 1∫ 0 ( 1 3 − λ 2 ) dλ )1−1 q ( 1∫ 0 ∣∣f′(a + (1−λ 2 ) η (b,a) )∣∣q dλ)1q Since |f′|q is quasi-preinvexity , we have∣∣∣∣f′ ( a + ( 1 + λ 2 ) η (b,a) )∣∣∣∣q ≤ sup ( |f′ (a)|q , ∣∣∣∣f′ ( a + 1 2 η (b,a) )∣∣∣∣q ) And∣∣∣∣f′ ( a + ( 1 −λ 2 ) η (b,a) )∣∣∣∣q ≤ sup (∣∣∣∣f′ ( a + 1 2 η (b,a) )∣∣∣∣q , |f′ (a + η (b,a))|q ) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+η(b,a) 2 ) + f (a + η (b,a)) ] − 1 η(b,a) a+η(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5η(b,a) 72 ( sup { |f′ (a)|q , ∣∣f′(a + 1 2 η (b,a) )∣∣q})1q + 5η(b,a) 72 ( sup {∣∣f′(a + 1 2 η (b,a) )∣∣q , |f′ (a + η (b,a))|q})1q . Which completes the proof. 3. Application to some Special Means In what follows we give certain generalization of some notions for a positive valued function of a positive variable. Definition 3[9]. A function M : R → R,is called a mean function if it has the following properties: (1) Homogeneity: M (ax,ay) = aM (x,y) ,for all a > 0, (2) Symmetry: M (x,y) = M (x,y) , (3) Reflexivity: M (x,x) = x, (4) Monotonicity: If x ≤ x′ and y ≤ y′, then M (x,y) = M (x′,y′) , (5) Internality: min{x,y}≤ M (x,y) ≤ max{x,y} . NEW INTEGRAL INEQUALITIES THROUGH INVEXITY WITH APPLICATIONS 121 We consider some means for arbitrary positive real numbers a,b (see for instance [9]). We now consider the applications of our theorem to the special means. The arithmetic mean; A := A (a,b) = a + b 2 The geometic mean; G := G(a,b) = √ ab The power mean; Pr := Pr(a,b) = ( ar + br 2 )1 r , r ≥ 1, The indentric mean: I = I (a,b) = { 1 e ( bb aa ) , ifa 6= b a , ifa = b The Harmonic mean: H := H (a,b) = 2ab a + b , The logarithmic mean: L = L (a,b) = a− b ln |a|− ln |b| , |a| 6= |b| The p- logarithmic mean: Lp ≡ Lp (a,b) = [ bp+1 −ap+1 (p + 1) (b−a) ] , a 6= b p ∈ < \ {– 1, 0}: a, b > 0. It is well known that LP is monotonic nondecreasing over p ∈ R with L−1 := Land L0 := I. In particular, we have the following inequalities H ≤ G ≤ L ≤ I ≤ A. Now let a and bbe positive real numbers such that a < b.consider the function a < b. M : M (b,a) : [a,a + η (b,a)] × [a,a + η (b,a)] → R, which is one of the above mentioned means, therefore one can obtain variant inequalities for these means as follows: η (b,a) = M (b,a)in (2.1), (2.4) and (2.7), one can obtain the following interesting inequalities involving means: ∣∣∣∣∣16 [ f (a) + 4f ( 2a+M(b,a) 2 ) + f (a + M (b,a)) ] ? − 1 M(b,a) a+M(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5M(b,a) 72 [ sup { |f′ (a)| , ∣∣f′(a + 1 2 M (b,a) )∣∣} + sup {∣∣f′(a + 1 2 M (b,a) )∣∣ , |f′ (a + M (b,a))|} ] . 122 QAISAR, HE AND HUSSAIN (3.1) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+M(b,a) 2 ) + f (a + M (b,a)) ] − 1 M(b,a) a+M(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ M(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( sup { |f′ (a)| p p−1 , ∣∣f′(a + 1 2 M (b,a) )∣∣ pp−1 })p−1p + M(b,a) 2 ( 2(1+2p+1) 6p+1(p+1) )1/p ( sup {∣∣f′(a + 1 2 M (b,a) )∣∣ pp−1 , |f′ (a + M (b,a))| pp−1 })p−1p (3.2) ∣∣∣∣∣16 [ f (a) + 4f ( 2a+M(b,a) 2 ) + f (a + M (b,a)) ] − 1 η(b,a) a+M(b,a)∫ a f(x)dx ∣∣∣∣∣ ≤ 5M(b,a) 72 ( sup { |f′ (a)|q , ∣∣f′(a + 1 2 M (b,a) )∣∣q})1q + 5M(b,a) 72 ( sup {∣∣f′(a + 1 2 M (b,a) )∣∣q , |f′ (a + M (b,a))|q})1q . 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