International Journal of Analysis and Applications ISSN 2291-8639 Volume 11, Number 2 (2016), 146-156 http://www.etamaths.com SOME PERTURBED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE FIRST DERIVATIVES ARE OF BOUNDED VARIATION HÜSEYIN BUDAK∗ AND MEHMET ZEKI SARIKAYA Abstract. The main aim of this paper is to establish some new perturbed Ostrowski type integral inequalities for functions whose first derivatives are of bounded variation. Some perturbed Ostrowski type inequalities for Lipschitzian and monotonic mappings are also obtained. 1. Introduction In 1938, Ostrowski [20] established a following useful inequality: Theorem 1. Let f : [a,b] → R be a differentiable mapping on (a,b) whose derivative f′ : (a,b) → R is bounded on (a,b) , i.e. ‖f′‖∞ := sup t∈(a,b) |f′(t)| < ∞. Then, we have the inequality (1.1) ∣∣∣∣∣∣f(x) − 1b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ [ 1 4 + ( x− a+b 2 )2 (b−a)2 ] (b−a)‖f′‖∞ , for all x ∈ [a,b]. The constant 1 4 is the best possible. The following definitions will be frequently used to prove our results. Definition 1. Let P : a = x0 < x1 < ... < xn = b be any partition of [a,b] and let ∆f(xi) = f(xi+1) −f(xi), then f is said to be of bounded variation if the sum m∑ i=1 |∆f(xi)| is bounded for all such partitions. Definition 2. Let f be of bounded variation on [a,b], and ∑ ∆f (P) denotes the sum n∑ i=1 |∆f(xi)| corresponding to the partition P of [a,b]. The number b∨ a (f) := sup {∑ ∆f (P) : P ∈ P ([a,b]) } , is called the total variation of f on [a,b] . Here P([a,b]) denotes the family of partitions of [a,b] . In [14], Dragomir proved the following Ostrowski type inequalities related functions of bounded variation: Theorem 2. Let f : [a,b] → R be a mapping of bounded variation on [a,b] . Then (1.2) ∣∣∣∣∣∣ b∫ a f(t)dt− (b−a) f(x) ∣∣∣∣∣∣ ≤ [ 1 2 (b−a) + ∣∣∣∣x− a + b2 ∣∣∣∣ ] b∨ a (f) holds for all x ∈ [a,b] . The constant 1 2 is the best possible. 2010 Mathematics Subject Classification. 26D15, 26A45, 26D10. Key words and phrases. bounded variation; Perturbed Ostrowski type inequalities; Riemann-Stieltjes integrals. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 146 SOME PERTURBED OSTROWSKI TYPE INEQUALITIES 147 In the past, many authors have worked on Ostrowski type inequalities for function of bounded variation, see for example ([1]-[4],[6]-[9],[11]-[16],[19]). For a function of bounded variation v : [a,b] → C. we define the Cumulative Variation Function (CVF) V : [a; b] → [0,∞) by V (t) := t∨ a (v), the total variation of v on the interval [a,t] with t ∈ [a,b]. It is know that the CVF is monotonic nondecreasing on [a,b] and is continuous in a point c ∈ [a,b] if and only if the generating function v is continuing in that point. If v is Lipschitzian with the constant L > 0, i.e. |v(t) −v(s)| ≤ L |t−s| , for any t,s ∈ [a,b] , then V is also Lipschitzian with the same constant. A simple proof of the following Lemma was given in [15]. Lemma 1. Let f,u : [a,b] → C. If f is continuous on [a,b] and u is of bounded variation on [a,b] , then the Riemann-Stieltjes integral b∫ a f(t)du(t) exist and (1.3) ∣∣∣∣∣∣ b∫ a f(t)du(t) ∣∣∣∣∣∣ ≤ b∫ a |f(t)|d ( t∨ a (u) ) ≤ max t∈[a,b] |f(t)| b∨ a (u). In [8], authors gave the following Ostrowski type inequality for mapping whoose first derivatives are of bounded variation: Theorem 3. Let f : [a,b] → R be such that f′ is a continuous function of bounded variation on [a,b] . Then we have the inequality∣∣∣∣∣∣ 1b−a b∫ a f(t)dt− 1 2 [f(x) + f(a + b−x)] + 1 2 ( x− 3a + b 4 ) [f′(x) −f′(a + b−x)] ∣∣∣∣ ≤ 1 16 [ 5 (x−a)2 − 2 (x−a) (b−x) + (b−x)2 b−a + 4 ∣∣∣∣x− 3a + b4 ∣∣∣∣ ] b∨ a (f′) for any x ∈ [ a, a+b 2 ] . For recent related results, see [5],[7] and [9]. Moreover, Dragomir proved some perturbed Ostrowski type inequalities for functions of bounded variation in [17, 18]. The aim of this paper is to obtain new perturbed Ostrowski type inequalities for mappings whose first derivatives are of bounded variation. 2. Some Identities Before we start our main results, we state and prove following lemma: Lemma 2. Let f : [a,b] → C be a twice differantiable function on (a,b) . Then for any λ1(x) and λ2(x) complex number the following identity holds( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(2.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ] = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λ1(x)t] + 1 b−a b∫ x (t− b)2 d [f′(t) −λ2(x)t]   , 148 BUDAK AND SARIKAYA where the integrals in the right hand side are taken in the Riemann-Stieltjes sense. Proof. Using the integration by parts for Riemann-Stieltjes, we have x∫ a (t−a)2 d [f′(t) −λ1(x)t](2.2) = x∫ a (t−a)2 df′(t) −λ1(x) x∫ a (t−a)2 dt = (t−a)2 f′(t) ∣∣∣x a − 2 x∫ a (t−a) f′(t)dt− λ1(x) 3 (t−a)3 ∣∣∣∣x a = (x−a)2 f′(x) − 2  (t−a) f(t)|xa − x∫ a f(t)dt  − λ1(x) 3 (x−a)3 = (x−a)2 f′(x) − 2 (x−a) f(x) + 2 x∫ a f(t)dt− λ1(x) 3 (x−a)3 and b∫ x (t− b)2 d [f′(t) −λ2(x)t](2.3) = b∫ x (t− b)2 df′(t) −λ2(x) b∫ x (t− b)2 dt = (t− b)2 f′(t) ∣∣∣b x − 2 b∫ x (t− b) f′(t)dt− λ1(x) 3 (t− b)3 ∣∣∣∣b x = −(b−x)2 f′(x) − 2  (t− b) f(t)|bx − b∫ x f(t)dt  − λ2(x) 3 (b−x)3 = (b−x)2 f′(x) − 2 (b−x) f(x) + 2 b∫ x f(t)dt− λ1(x) 3 (x−a)3 . If we add the equality (2.2) and (2.3) and devide by 2(b−a), we obtain required identity. � Corollary 1. Under assumption of Lemma 2 with λ1(x) = λ2(x) = λ(x), we have ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ] (2.4) = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λ(x)t] + 1 b−a b∫ x (t− b)2 d [f′(t) −λ(x)t]   for all x ∈ [a,b] . SOME PERTURBED OSTROWSKI TYPE INEQUALITIES 149 Remark 1. If we choose λ(x) = 0 in (2.4), then we have the following identity ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(2.5) = 1 2   1 b−a x∫ a (t−a)2 df′(t) + 1 b−a b∫ x (t− b)2 df′(t)   for all x ∈ [a,b] . Corollary 2. Under assumption of Lemma 2 with λ1(x) = λ1 ∈ C and λ2(x) = λ2 ∈ C, we get ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(2.6) − 1 6(b−a) [ λ1(x−a)3 + λ2(b−x)3 ] = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λ1t] + 1 b−a b∫ x (t− b)2 d [f′(t) −λ2t]   . In particular, taking λ1 = λ2 = λ we have ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ 6(b−a) [ (x−a)3 + (b−x)3 ] (2.7) = 1 2   1 b−a x∫ a (t−a)2 d [f′(t) −λt] + 1 b−a b∫ x (t− b)2 d [f′(t) −λt]   . 3. Inequalities for Functions Whose First Derivatives are of Bounded Variation We denote by ` : [a,b] → [a,b] the identity function, namely `(t) = t for any t ∈ [a,b] . Theorem 4. Let : f : [a,b] → C be a twice differantiable function on I◦ and [a,b] ⊂ I◦. If the first derivative f′ is of bounded variation on [a,b] , then∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(3.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 (b−a)   x∫ a (t−a) ( x∨ t (f′ −λ1(x)`) ) dt + b∫ x (b− t) ( t∨ x (f′ −λ2(x)`) ) dt   ≤ 1 2(b−a) [ (x−a)2 x∨ a (f′ −λ1(x)`) + (b−x)2 b∨ x (f′ −λ2(x)`) ] 150 BUDAK AND SARIKAYA ≤ 1 2(b−a) ×   [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] max { x∨ a (f′ −λ1(x)`), b∨ x (f′ −λ2(x)`) } (b−a)2, max { (x−a)2, (b−x)2 }[ x∨ a (f′ −λ1(x)`) + b∨ x (f′ −λ2(x)`) ] for any x ∈ [a,b] . Proof. Taking modulus (2.1) and applying Lemma 1, we get ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(3.2) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2   1 b−a ∣∣∣∣∣∣ x∫ a (t−a)2 d [f′(t) −λ1(x)t] ∣∣∣∣∣∣ + 1b−a ∣∣∣∣∣∣ b∫ x (t− b)2 d [f′(t) −λ2(x)t] ∣∣∣∣∣∣   ≤ 1 2(b−a)   x∫ a (t−a)2 d ( t∨ a (f′ −λ1(x)`) ) + b∫ x (t− b)2 d ( t∨ a (f′ −λ2(x)`) ) . Integrating by parts in the Riemann-Stieltjes integral, we get x∫ a (t−a)2 d ( t∨ a (f′ −λ1(x)`) ) (3.3) = (t−a)2 t∨ a (f′ −λ1(x)`) ∣∣∣∣∣ x a − 2 x∫ a (t−a) ( t∨ a (f′ −λ1(x)`) ) dt = (x−a)2 x∨ a (f′ −λ1(x)`) − 2 x∫ a (t−a) ( t∨ a (f′ −λ1(x)`) ) dt = 2 x∫ a (t−a) ( x∨ a (f′ −λ1(x)`) ) dt− 2 x∫ a (t−a) ( t∨ a (f′ −λ1(x)`) ) dt = 2 x∫ a (t−a) ( x∨ t (f′ −λ1(x)`) ) dt SOME PERTURBED OSTROWSKI TYPE INEQUALITIES 151 and b∫ x (t− b)2 d ( t∨ a (f′ −λ2(x)`) ) (3.4) = (t− b)2 t∨ a (f′ −λ2(x)`) ∣∣∣∣∣ b x − 2 b∫ x (t− b) ( t∨ a (f′ −λ2(x)`) ) dt = −(x− b)2 x∨ a (f′ −λ2(x)`) − 2 b∫ x (t− b) ( t∨ a (f′ −λ2(x)`) ) dt = −2 b∫ x (b− t) ( x∨ a (f′ −λ2(x)`) ) dt + 2 b∫ x (b− t) ( t∨ a (f′ −λ2(x)`) ) dt = 2 b∫ x (b− t) ( t∨ x (f′ −λ2(x)`) ) dt. If we put the identities (3.3) and (3.4) in (3.2), then we obtain the first inequality in (3.1). Moreover, we have, (3.5) x∫ a (t−a) ( x∨ t (f′ −λ1(x)`) ) dt ≤ 1 2 (x−a)2 x∨ a (f′ −λ1(x)`) and (3.6) b∫ x (b− t) ( t∨ x (f′ −λ2(x)`) ) dt ≤ 1 2 (b−x)2 b∨ x (f′ −λ2(x)`). With the inequalities (3.5) and (3.6), the proof of Theorem 4 is completed. � Corollary 3. If we chosose λ1(x) = λ2(x) = 0, then we have the following inequality∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt ∣∣∣∣∣∣ ≤ 1 (b−a)   x∫ a (t−a) ( x∨ t (f′) ) dt + b∫ x (b− t) ( t∨ x (f′`) ) dt   ≤ 1 2(b−a) [ (x−a)2 x∨ a (f′) + (b−x)2 b∨ x (f′) ] ≤ b−a 2   [ 1 4 + (x−a+b2 ) 2 (b−a)2 ][ 1 2 b∨ a (f′) + 1 2 ∣∣∣∣ x∨ a (f′) − b∨ x (f′) ∣∣∣∣ ] , [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]2 b∨ a (f′) for all x ∈ [a,b] . 152 BUDAK AND SARIKAYA Corollary 4. Under assumption of Theorem 4 with λ1(x) = λ2(x) = λ(x), we have (3.7) ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ]∣∣∣∣∣∣ ≤ 1 (b−a)   x∫ a (t−a) ( x∨ t (f′ −λ(x)`) ) dt + b∫ x (b− t) ( t∨ x (f′ −λ(x)`) ) dt   ≤ 1 2(b−a) [ (x−a)2 x∨ a (f′ −λ(x)`) + (b−x)2 b∨ x (f′ −λ(x)`) ] ≤ b−a 2   [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] × [ 1 2 b∨ a (f′ −λ(x)`) + 1 2 ∣∣∣∣ x∨ a (f′ −λ(x)`) − b∨ x (f′ −λ(x)`) ∣∣∣∣ ] , [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]2 b∨ a (f′ −λ(x)`) for all x ∈ [a,b] . Corollary 5. If we choose λ(x) = λ and x = a+b 2 in (3.7), then we have the following identity ∣∣∣∣∣∣ 1b−a b∫ a f(t)dt−f(x) − λ(b−a)2 24 ∣∣∣∣∣∣ ≤ 1 (b−a)   a+b 2∫ a (t−a)  a+b2∨ t (f′ −λ`)  dt + b∫ a+b 2 (b− t)   t∨ a+b 2 (f′ −λ`)  dt   ≤ (b−a) 8 b∨ a (f′ −λ(x)`). 4. Inequalities for Functions Whose First Derivatives are Lipschitzian Theorem 5. Let f : [a,b] → C be a twice differantiable function on I◦ and [a,b] ⊂ I◦. If there exist the positive numbers K1(x) and K2(x) such that f ′ −λ1(x)` is Lipschitzian with the constant K1(x) on the interval [a,x] and f′−λ2(x)` is Lipschitzian with the constant K2(x) on the interval [x,b] , then we have for any x ∈ [a,b] ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(4.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ (b−a)2 6 [ K1(x) ( x−a b−a )3 + K2(x) ( b−x b−a )3] SOME PERTURBED OSTROWSKI TYPE INEQUALITIES 153 ≤ (b−a)2 6   [( x−a b−a )3 + ( b−x b−a )3] max{K1(x),K2(x)} , [( x−a b−a )3p + ( b−x b−a )3p]1p [(K1(x)) q + (K1(x)) q ] 1 q p > 1, 1 p + 1 q = 1, [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]3 [K1(x) + K2(x)] . Proof. It is known that, if g : [c,d] → C is Riemann integrable and u : [c,d] → C is Lipschitzian with the constant K > 0, then the Riemann-Stieltje integral d∫ c g(t)du(t) exist and ∣∣∣∣∣∣ d∫ c g(t)du(t) ∣∣∣∣∣∣ ≤ K d∫ c |g(t)|dt. Taking the madulus (2.1), we get∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2(b−a)   ∣∣∣∣∣∣ x∫ a (t−a)2 d [f′(t) −λ1(x)t] ∣∣∣∣∣∣ + ∣∣∣∣∣∣ b∫ x (t− b)2 d [f′(t) −λ2(x)t] ∣∣∣∣∣∣   ≤ 1 2(b−a)  K1(x) x∫ a ∣∣∣(t−a)2∣∣∣dt + K2(x) b∫ x ∣∣∣(t− b)2∣∣∣dt   = (b−a)2 6 [ K1(x) (x−a) 3 + K2(x) (b−x) 3 ] = (b−a)2 6 [ K1(x) ( x−a b−a )3 + K2(x) ( b−x b−a )3] . This completes the proof of first inequality in (4.1). Using the Hölder’s inequality, we have K1(x) ( x−a b−a )3 + K2(x) ( b−x b−a )3 ≤   [( x−a b−a )3 + ( b−x b−a )3] max{K1(x),K2(x)} , [( x−a b−a )3p + ( b−x b−a )3p]1p [(K1(x)) q + (K1(x)) q ] 1 q p > 1, 1 p + 1 q = 1, [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]3 [L1(x) + L2(x)] which completes the proof. � 154 BUDAK AND SARIKAYA Corollary 6. Under assumption of Theorem 5 with K1(x) = K2(x) = K and λ1(x) = λ2(x) = λ(x), we have ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ]∣∣∣∣∣∣(4.2) ≤ 1 6 [ 1 2 + ∣∣∣∣∣x− a+b 2 b−a ∣∣∣∣∣ ]3 K(b−a)2. Corollary 7. If we choose x = a+b 2 and λ(x) = λ ∈ C in (4.2), we get the inequality∣∣∣∣∣∣ 1b−a b∫ a f(t)dt−f ( a + b 2 ) − λ(b−a)2 48 ∣∣∣∣∣∣ ≤ 148K(b−a)2. 5. Inequalities for Mappings Whose First Derivatives are Monotonic Function Theorem 6. Let f : [a,b] → C be a twice differantiable function on I◦ and [a,b] ⊂ I◦. If λ1(x) and λ2(x) are real numbers such that f ′ − λ1(x)` is monotonic nondecreasing on the interval [a,x] and f′ − λ2(x)` is monotonic nondecreasing on the interval [x,b] , then for any x ∈ [a,b] the following inequalities hold: ∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(5.1) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2(b−a) [ (x−a)2 [f′(x) −f′(a) −λ1(x) (x−a)] + (b−x)2 [f′(b) −f′(x) −λ2(x) (b−x)] ] ≤ 1 2(b−a)   [ 1 2 [f′(b) −f′(a) −λ1(x) (x−a) −λ2(x) (b−x)] + ∣∣∣f′(x) − f′(a)+f′(b)2 − 12λ1(x) (x−a) + 12λ2(x) (b−x)∣∣∣] × [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] (b−a)2, max { (x−a)2 , (b−x)2 } × [f′(b) −f′(a) −λ1(x) (x−a) −λ2(x) (b−x)] . Proof. Taking the madulus (2.1), we have∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt(5.2) − 1 2(b−a) [ λ1(x)(x−a)3 + λ2(x)(b−x)3 3 ]∣∣∣∣ ≤ 1 2(b−a)   ∣∣∣∣∣∣ x∫ a (t−a)2 d [f′(t) −λ1(x)t] ∣∣∣∣∣∣ + ∣∣∣∣∣∣ b∫ x (t− b)2 d [f′(t) −λ2(x)t] ∣∣∣∣∣∣   SOME PERTURBED OSTROWSKI TYPE INEQUALITIES 155 Since f′ −λ1(x)` is monotonic nondecreasing on the interval [a,x] , we have x∫ a (t−a)2 d [f′(t) −λ1(x)t](5.3) ≤ (x−a)2 [f′(x) −λ1(x)x−f′(a) + λ1(x)a] = (x−a)2 [f′(x) −f′(a) −λ1(x) (x−a)] and similarly, since f′ −λ2(x)` is monotonic nondecreasing on the interval [x,b] , we have b∫ x (t− b)2 d [f′(t) −λ2(x)t](5.4) ≤ (b−x)2 [f′(b) −λ2(x)b−f′(x) + λ2(x)x] = (b−x)2 [f′(b) −f′(x) −λ2(x) (b−x)] . If we put (5.3) and (5.4) in (5.2), we obtain the first inequality in (5.1). The proofs of last inequalities are obvious, they are omitted. � Corollary 8. Under assumption of Theorem 6 with λ1(x) = λ2(x) = λ(x), we have∣∣∣∣∣∣ ( x− a + b 2 ) f′(x) −f(x) + 1 b−a b∫ a f(t)dt− λ(x) 6(b−a) [ (x−a)3 + (b−x)3 ]∣∣∣∣∣∣(5.5) ≤ 1 2(b−a) [ (x−a)2 [f′(x) −f′(a) −λ(x) (x−a)] + (b−x)2 [f′(b) −f′(x) −λ(x) (b−x)] ] ≤ b−a 2 ×   [ f′(b)−f′(a) 2 − 1 2 λ(x) (b−a) ∣∣∣f′(x) − f′(a)+f′(b)2 −λ(x) (x− a+b2 )∣∣∣ × [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] , [f′(b) −f′(a) −λ(x) (b−a)] [ 1 2 + ∣∣∣x−a+b2b−a ∣∣∣]2 . Corollary 9. If we choose x = a+b 2 and λ(x) = λ in (5.5), we get the inequality∣∣∣∣∣∣ 1b−a b∫ a f(t)dt−f ( a + b 2 ) − λ(b−a)2 48 ∣∣∣∣∣∣ ≤ (b−a)8 [f′(b) −f′(a) −λ (b−a)] . References [1] M. W. Alomari, A Generalization of weighted companion of Ostrowski integral inequality for mappings of bounded variation, RGMIA Research Report Collection, 14(2011), Art. ID 87. [2] M. W. 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Dragomir, Some perturbed Ostrowski type inequalities for functions of bounded variation, RGMIA Research Report Collection, 16(2013), Art. ID 93. [18] S. S. Dragomir, Perturbed companions of Ostrowski’s inequality for functions of bounded variation, RGMIA Re- search Report Collection, 17(2014), Art. ID 1. [19] W. Liu and Y. Sun, A Refinement of the Companion of Ostrowski inequality for functions of bounded variation and Applications, arXiv:1207.3861v1, (2012). [20] A. M. Ostrowski, Über die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv. 10(1938), 226-227. Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-Turkey ∗Corresponding author: hsyn.budak@gmail.com