On the weighted integral inequalities for convex function

Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 194–208 DOI: 10.1515/ausm-2015-0006

On the weighted integral inequalities for convex function Mehmet Zeki Sarikaya

Samet Erden

Department of Mathematics, Faculty of Science and Arts, Duzce University, Konuralp Campus, Duzce-Turkey email: [email protected]

Department of Mathematics, Faculty of Science, Bartin University, Bartin-Turkey email: [email protected]

Abstract. In this paper, we establish several weighted inequalities for some differantiable mappings that are connected with the celebrated Hermite-Hadamard-Fejér type and Ostrowski type integral inequalities. The results presented here would provide extensions of those given in earlier works.

1

Introduction

The following result is known in the literature as Ostrowski’s inequality [10]: Theorem 1 Let f : [a, b]→ R be a differentiable mapping on (a, b) whose 0 derivative f : (a, b)→ R is bounded on (a, b), i.e., kf0 k∞ = sup |f0 (t)| < ∞. t∈(a,b)

Then, the inequality: " # Zb a+b 2

0 (x − ) 1 1 2 ≤

f f(x) − f(t)dt + (b − a) ∞ b−a 4 (b − a)2

(1)

a

holds for all x ∈ [a, b]. The constant

1 4

is the best possible.

2010 Mathematics Subject Classification: 26D07, 26D15 Key words and phrases: Ostrowski’s inequality, Montgomery’s identities, convex function, H¨ older inequality

194

On the weighted integral inequalities

195

Inequality (1) has wide applications in numerical analysis and in the theory of some special means; estimating error bounds for some special means, some mid-point, trapezoid and Simpson rules and quadrature rules, etc. Hence inequality (1) has attracted considerable attention and interest from mathematicans and researchers. Due to this, over the years, the interested reader is also refered to ([1]-[7],[12]-[17]) for integral inequalities in several independent variables. In addition, the current approach of obtaining the bounds, for a particular quadrature rule, have depended on the use of Peano kernel. The general approach in the past has involved the assumption of bounded derivatives of degree greater than one. If f : [a, b] → R is differentiable on [a, b] with the first derivative f0 integrable on [a, b], then Montgomery identity holds: Zb Zb 1 f(x) = f(t)dt + P(x, t)f0 (t)dt, b−a a

(2)

a

where P(x, t) is the Peano kernel defined by  t−a   , a≤t 0 with a ≥ 0 are defined by Jαa+ f(x) =

1 Γ (α)

Jαb− f(x) =

1 Γ (α)

and

Zx

(x − t)α−1 f(t)dt, x > a

a

Zb

(t − x)α−1 f(t)dt, x < b

x

respectively. Here, Γ (α) is the Gamma function and J0a+ f(x) = J0b− f(x) = f(x).


199

Corollary 2 Under the same assumptions as in Lemma 2, if we put w(s) = 1, then the following equality holds: [(x − a)α + (b − x)α ] f(x) − Γ (α + 1)Jαx− f(a) − Γ (α + 1)Jαx+ f(b) (11) Zx α

0

Zb

0

(t − a) f (t)dt − (b − t)α f (t)dt.

= a

x

Corollary 3 Under the same assumptions of Corollary 2 with x = a+b 2 , the idendity (11) becomes to the following identity a+b Γ (α + 1) α α f − 1−α J − f(a) + J a+b + f(b) ( 2 ) 2 2 (b − a)α ( a+b 2 )   a+b   b 2   Z   Z 1 α 0 α 0 (b (t − t) f (t)dt − a) f (t)dt − = .  21−α (b − a)α      a a+b 2

Now, by using the above lemma, we prove our main theorems: Theorem 4 Let f : I◦ ⊆ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b and let w : [a, b] → R be continuous on [a, b]. If |f0 | is convex on [a, b], then the following inequality holds:  x α  b α  Z Z  w(s)ds +  w(s)ds  f(x) a x    α−1 α−1 Zx Zt Zb Zb −α  w(s)ds w(t)f(t)dt − α  w(s)ds w(t)f(t)dt a a x t kwkα[a,x],∞ (b − a)(x − a)α+1 (x − a)α+2 0 ≤ − |f (a)| b−a α+1 α+2 kwkα[x,b],∞ (b − x)α+2 0 kwkα[a,x],∞ (x − a)α+2 0 + |f (b)| + |f (a)| b−a α+2 b−a α+2 kwkα[a,x],∞ (b − a)(b − x)α+1 (b − x)α+2 0 + − |f (b)| b−a α+1 α+2

200

M. Z. Sarikaya, S. Erden kwkα[a,b],∞

(b − a)(x − a)α+1 (b − x)α+2 − (x − a)α+2 ≤ + b−a α+1 α+2

α+1 α+2 α+2 (b − a)(b − x) (x − a) − (b − x) 0 |f (b)| + + α+1 α+2

0

|f (a)|

where α > 0 and kwk[a,b],∞ = sup |w(t)| . t∈[a,b]

Proof. We take absolute value of both sizes of (7), we find that  x α  b α  Z Z  w(s)ds +  w(s)ds  f(x) a x    α−1 α−1 Zx Zt Zb Zb −α  w(s)ds w(t)f(t)dt − α  w(s)ds w(t)f(t)dt a a x t     α α Zx Zt Zb Zb 0 0 ≤  w(s)ds  f (t) dt +  w(s)ds  |f (t)|dt a

a

x

Zx ≤

kwkα[a,x],∞

t

Zb 0 0 (t − a) f (t) dt + kwk[x,b],∞ (b − t)α |f (t)|dt α

a

Zx = kwkα[a,x],∞

x

t − a α 0 b−t (t − a) f ( a+ b) dt b−a b−a

a

Zb +

kwkα[x,b],∞

0 b−t t − a (b − t) f ( a+ b) dt b−a b−a α

x 0

Since |f | is convex on [a.b], it follows that  x α  b α  Z Z  w(s)ds +  w(s)ds  f(x) a x   α−1  α−1 Zx Zt Zb Zb w(t)f(t)dt − α  w(s)ds w(t)f(t)dt −α  w(s)ds a a x t

On the weighted integral inequalities Zx ≤

kwkα[a,x],∞

(t − a)

α

201

b − t 0 t − a 0 |f (b)| dt f (a) + b−a b−a

a

Zb

b−t 0 t−a 0 |f (a)| + |f (b)| dt b−a b−a x

kwkα[a,x],∞ (b − a)(x − a)α+1 (x − a)α+2 0 (x − a)α+2 0 |f (a)| + = − |f (b)| b−a α+1 α+2 α+2

kwkα[x,b],∞ (b − x)α+2 0 (b − a)(b − x)α+1 (b − x)α+2 0 f (a) + − |f (b)| + b−a α+2 α+1 α+2 α kwk[a,b],∞ (b − a)(x − a)α+1 (b − x)α+2 − (x − a)α+2 0 ≤ + |f (a)| b−a α+1 α+2

(b − a)(b − x)α+1 (x − a)α+2 − (b − x)α+2 0 + + |f (b)| . α+1 α+2 +

kwkα[x,b],∞

(b − t)α

Hence, the proof of theorem is completed.

Corollary 4 Under the same assumptions as in Theorem 4, if we take w(s) = 1, then the following inequality holds: |[(x − a)α + (b − x)α ] f(x) − Γ (α + 1) [Jαx− f(a) + Jαx+ f(b)]| 1 (b − a)(x − a)α+1 (b − x)α+2 − (x − a)α+2 0 ≤ + |f (a)| b−a α+1 α+2

(b − a)(b − x)α+1 (x − a)α+2 − (b − x)α+2 0 + + |f (b)| . α+1 α+2 Remark 2 If we take x =

a+b 2

in (12), we get

a+b 2α−1 Γ (α + 1) α α f − f(a) + J f(b) J − + a+b a+b ( 2 ) ( 2 ) 2 (b − a)α (b − a) 0 0 ≤ f (a) + f (b) 4 (α + 1) which is proved by Sarikaya and Yildirim in [19].

(12)

202

M. Z. Sarikaya, S. Erden

Corollary 5 Under the same assumptions as in Theorem 4, if we take α = 1, then the following inequality holds:  b  Z Zb  w(s)ds f(x) − w(t)f(t)dt a a kwk[a,b],∞ (b − a)(x − a)2 (b − x)3 − (x − a)3 0 |f (a)| + ≤ b−a 2 3

2 (x − a)3 − (b − x)3 (b − a)(b − x) 0 + + |f (b)| . 2 3 Corollary 6 Under the same assumptions of Corollary 5 with x = get  b  Z Zb  w(s)ds f a + b − w(t)f(t)dt 2 a a  0 0  (b − a)2 kwk[a,b],∞ f (a) + f (b)  . ≤ 4 2

a+b 2 ,

we

(13)

Remark 3 If we take w(s) = 1 in (13), we have  0 0  Zb a+b (b − a) f (a) + f (b) 1 f   − f(t)dt ≤ 2 b−a 4 2 a

which is proved by Kırmacı in [9]. 0

Corollary 7 Under the same assumptions as in Theorem 4, if we put |f (a)| = 0 |f (b)| in (10), then the following inequality holds:  x α  b α  Z Z  w(s)ds +  w(s)ds  f(x) a x  α−1 t α−1 x Z Z Zb Zb     w(s)ds w(t)f(t)dt − α w(s)ds w(t)f(t)dt −α a a x t

On the weighted integral inequalities 0 α f (a) kwk[a,x],∞

0 α f (a) kwk[x,b],∞

(x − a)α+1 + α + 1 α+1 0 α i f (a) kwk[a,b],∞ h ≤ (x − a)α+1 + (b − x)α+1 α+1 ≤

203

(b − x)α+1

Theorem 5 Let f : I◦ ⊆ R → R be a differentiable mapping on I◦ , a, b ∈ I◦ with a < b and let w : [a, b] → R be continuous on [a, b]. If |f0 |q is convex on [a, b], q > 1, then the following inequality holds:  α  b α   α−1 Zx Z Zx Zt  w(s)ds +  w(s)ds  f(x) − α  w(s)ds w(t)f(t)dt a x a a   α−1 1 Zb Zb kwkα αp+1 p (x − a) [a,x],∞ −α  w(s)ds w(t)f(t)dt ≤ 1 αp + 1 q (b − a) x t q1 kwkα (b − a)2 − (b − x)2 0 (x − a)2 0 [x,b],∞ q q |f (a)| + |f (b)| + 1 2 2 (b − a) q 1 q1 2 − (x − a)2 (14) (b − a) (b − x)αp+1 p (b − x)2 0 0 |f (a)|q + |f (b)|q αp + 1 2 2  1 ! kwkα[a,b],∞  (x − a)αp+1 p (b − a)2 − (b − x)2 0 ≤ |f (a)|q 1  αp + 1 2 q (b − a) (x − a)2 0 + |f (b)kq 2 (b − 2

x)2

0

1 p

+

(b − x)αp+1 αp + 1 2

|f (a)|q +

where α > 0,

q1

+

1 q

!1

p

2

(b − a) − (x − a) 0 |f (b)|q 2

!1  q 

= 1, and kwk[a,b],∞ = sup |w(t)| . t∈[a,b]

204


Proof. We take absolute value of (7). Using Holder’s inequality, we find that  x α  α  b Z Z  w(s)ds +  w(s)ds  f(x) x a α−1  α−1 t x Zb Zb Z Z w(t)f(t)dt − α  w(s)ds w(t)f(t)dt −α  w(s)ds x a a t α α Zx Zt Zb Zb 0 0 ≤ w(s)ds |f (t)|dt + w(s)ds |f (t)|dt a a

x t

αp  p1  x  1 q Zx Zt Z 0 q     ≤ |f (t)| dt w(s)ds dt a

a

a

αp  p1  b   q1 Zb Zb Z 0 +  w(s)ds dt  |f (t)|q dt x x t x 1 1 x p q Z Z 0 α αp q ≤ kwk[a,x],∞  |t − a| dt  |f (t)| dt a

a

b  p1  b  q1 Z Z 0 + kwkα[x,b],∞  |b − t|αp dt  |f (t)|q dt x

x

0 q Since f (t) is convex on [a, b] q q q 0 b−t t − a ≤ b − t f0 (a) + t − a f0 (b) f ( a + b) b−a b−a b−a b−a

(15)

From (15), it follows that  x α  b α  Z Z  w(s)ds +  w(s)ds  f(x) a x α−1  α−1 t x Z Z Zb Zb     w(s)ds w(t)f(t)dt −α w(s)ds w(t)f(t)dt − α a a x t


≤

kwkα[a,x],∞ 1

(b − a) q

(x − a)αp+1 αp + 1

p1

205

(b − a)2 − (b − x)2 0 |f (a)|q 2

1 1 α (x − a)2 0 q q kwk[x,b],∞ (b − x)αp+1 p + + f (b) 1 2 αp + 1 (b − a) q !1 q kwkα[a,b],∞ (b − a)2 − (x − a)2 0 (b − x)2 0 q q |f (a)| + |f (b)| ≤ 1 2 2 (b − a) q  !1 !1 q  (x − a)αp+1 p (b − a)2 − (b − x)2 0 2 (x − a) 0 |f (a)|q + |f (b)|q  αp + 1 2 2 αp+1

+

(b − x) αp + 1

!1

p

(b − 2

x)2

0

|f (a)|q +

2

2

(b − a) − (x − a) 0 |f (b)|q 2

!1  q 

which completes the proof.

Corollary 8 Under the same assumptions as in Theorem 4, if we put w(s) = 1, then the following inequality holds: |[(x − a)α + (b − x)α ] f(x) − Γ (α + 1) [Jαx− f(a) + Jαx+ f(b)]| ≤  !1  (x − a)αp+1 p  αp + 1 +

(b − x)αp+1 αp + 1

!1

p

1 1

(b − a) q !1

(b − a)2 − (b − x)2 0 (x − a)2 0 |f (a)|q + |f (b)|q 2 2

(b − a)2 − (x − a)2 0 (b − x)2 0 |f (a)|q + |f (b)|q 2 2

q

(16) !1  q 

.

Remark 4 If we take x = a+b 2 in (16), we have a+b 2α−1 Γ (α + 1) α α f − J − f(a) + J a+b + f(b) a+b ( 2 ) ( 2 ) 2 (b − a)α  !1 !1  0 0 0 0 q q q q q (b − a)  3|f (a)| + |f (b)| |f (a)| + 3|f (b)| q  ≤ + 1 4 4 4(αp + 1) p which is proved by Sarikaya and Yildirim in [19].

206


Corollary 9 Let the conditions of Theorem 5 hold. If we take α = 1 in (14), then the following inequality holds:  b  Z Zb kwk[a,b],∞  w(s)ds f(x) − w(t)f(t)dt ≤ 1 (b − a) q a a  !1 !1 q  (x − a)p+1 p (b − a)2 − (b − x)2 0 (x − a)2 0 q q |f (a)| + |f (b)|  p+1 2 2 !1 !1  q p+1 p 2 2 2 q (b − x) (b − a) − (x − a) 0 (b − x) 0 q + |f (b)| f (a) +  p+1 2 2 Corollary 10 Under the same assumptions of Corollary 9 with x = get  b  Z Zb (b − a)2 kwk[a,b],∞ a + b ≤  w(s)ds f − w(t)f(t)dt 1 1 2 22+ q (p + 1) p a a  !1 !1  0 0  3|f0 (a)|q + |f0 (b)|q q q q |f (a)| + 3|f (b)| q  + .   2 2 Remark 5 If we take w(s) = 1 in (17), we have Zb a+b 1 f − f(t)dt 2 b−a a  !1  3|f0 (a)|q + |f0 (b)|q q (b − a) ≤ + 1 1 2 22+ q (p + 1) p  which is proved by Kırmacı in [9].

0

|f

(a)|q

+ 3|f 2

0

(b)|q

a+b 2 ,

we

(17)

!1  q 


207

References [1] F. Ahmad, N. S. Barnett, S. S. Dragomir, New weighted Ostrowski and Cebysev type inequalities, Nonlinear Anal., 71 (12) (2009), 1408–1412. [2] F. Ahmad, A. Rafiq, N. A. Mir, Weighted Ostrowski type inequality for twice differentiable mappings, Global Journal of Research in Pure and Applied Math., 2 (2) (2006), 147–154. [3] N. S. Barnett, S. S. Dragomir, An Ostrowski type inequality for double integrals and applications for cubature formulae, Soochow J. Math., 27 (1) (2001), 109–114. [4] N. S. Barnett, S. S. Dragomir, C. E. M. Pearce, A Quasi-trapezoid inequality for double integrals, ANZIAM J., 44 (2003), 355–364. [5] S. S. Dragomir, P. Cerone, N. S. Barnett, J. Roumeliotis, An inequlity of the Ostrowski type for double integrals and applications for cubature formulae, Tamsui Oxf. J. Math., 16 (1) (2000), 1–16. [6] S. Hussain, M. A. Latif, M. Alomari, Generalized duble-integral Ostrowski type inequalities on time scales, Appl. Math. Letters, 24 (2011), 1461– 1467. [7] M. E. Kiris, M. Z. Sarikaya, On the new generalization of Ostrowski type inequality for double integrals, International Journal of Modern Mathematical Sciences, 9 (3) 2014, 221–229. ¨ [8] L. Fejér, Uber die Fourierreihen, II. Math. Naturwiss Anz. Ungar. Akad. Wiss., 24 (1906), 369–390. (Hungarian). [9] U. S. Kırmacı, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comp., 147 (2004), 137–146. ¨ [10] A. M. Ostrowski, Uber die absolutabweichung einer differentiebaren funktion von ihrem integralmitelwert, Comment. Math. Helv., 10 (1938), 226– 227. [11] J. Peˇcarić, F. Proschan, Y. L. Tong, Convex functions, partial ordering and statistical applications, Academic Press, New York, 1991.

208


[12] A. Qayyum, A weighted Ostrowski-Gr¨ uss type inequality and applications, Proceeding of the World Cong. on Engineering, 2 (2009), 1–9. [13] A. Rafiq, F. Ahmad, Another weighted Ostrowski-Gr¨ uss type inequality for twice differentiable mappings, Kragujevac Journal of Mathematics, 31 (2008), 43–51. [14] M. Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenianae, Vol. LXXIX, 1 (2010), 129–134. [15] M. Z. Sarikaya, On the Ostrowski type integral inequality for double integrals, Demonstratio Mathematica, Vol. XLV, 3 (2012), 533–540. [16] M. Z. Sarikaya, H. Ogunmez, On the weighted Ostrowski type integral inequality for double integrals, The Arabian Journal for Science and Engineering (AJSE)-Mathematics, 36 (2011), 1153–1160. [17] M. Z. Sarikaya, On the generalized weighted integral inequality for double integrals, Annals of the Alexandru Ioan Cuza University-Mathematics, accepted. [18] M. Z. Sarikaya, On new Hermite Hadamard Fejer Type integral inequalities, Studia Universitatis Babe¸s-Bolyai Mathematica, 57 (3) (2012), 377– 386. [19] M. Z. Sarikaya, H. Yildirim, On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals, Submitted. [20] K-L. Tseng, G-S. Yang, K-C. Hsu, Some inequalities for differentiable mappings and applications to Fejer inequality and weighted trapozidal formula, Taiwanese J. Math, 15 (4) (2011), 1737–1747, [21] C.-L. Wang, X.-H. Wang, On an extension of Hadamard inequality for convex functions, Chin. Ann. Math., 3 (1982), 567–570. [22] S.-H. Wu, On the weighted generalization of the Hermite-Hadamard inequality and its applications, The Rocky Mountain J. of Math., 39 (5) (2009), 1741–1749.

Received: 16 May 2014