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A theorem on mappings with bounded derivatives with applications to quadrature rules and means
Applied Mathematics and Computation 138 (2003) 425–434 www.elsevier.com/locate/amc
A theorem on mappings with bounded derivatives with applications to quadrature rules and means € zdemir M. Emin O K.K. Education Faculty, Department of Mathematics, Atat€urk University, Bolumu, 25240 Erzurum, Turkey
Abstract In this paper, we establish a new inequality of Theorem 2 [Appl. Math. Lett. 13 (2000) 19] (Dragomirs integral inequality) for functions whose derivatives are bounded. This has immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid, mid-point. Some natural applications to special means of real numbers are given. For several recent results concerning Dragomirs integral inequality (see [Appl. Math. Lett. 13 (2000) 19] where further references are listed). Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Dragomirs inequality; Quadrature formulas; Numerical integration; Bounded mapping; Trapezoid and mid-point inequality
1. Introduction Theorem 1 (Theorem 2). Let f : ½a; b ! R be continuous on ½a; b, differentiable on ða; bÞ and whose derivative f 0 : ða; bÞ ! R is bounded on ða; bÞ. Denote kf 0 k1 :¼ sup jf 0 ðxÞj < 1. Then t2ða;bÞ
€ zdemir). E-mail address: [email protected] (M.E. O 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 4 6 - 7
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
426
Z
f ðaÞ þ f ðbÞ h ðb aÞ f ðtÞ dt f ðxÞð1 hÞ þ 2 a " 2 # 0 1 aþb 2 2 2 f 6 ðb aÞ ðh þ ðh 1Þ Þ þ x 1 4 2
b
ð1:1Þ
for all h 2 ½0; 1. € zdemir has obtained inequalities for the logarithIn the recent paper [2], O mic, arithmetic, p-logarithmic means. The aim of this paper is to establish some generalizations of (1.1) and Ostrowskis integral inequality [1, p. 469] and to apply these generalizations to obtain inequalities in numerical integration and mean inequalities.
2. The Results The following generalization of Dragomirs inequality [5] (with Cerone and Roumeliotis) holds. Theorem 2. f : ½ab ! R be continuous on ½a; b, and f : ða; bÞ ! R be differentiable and suppose kf 0 k1 :¼ sup jf 0 ðxÞj < 1. a 6¼ b; a; b 2 Z þ or a; b 2 Z . t2ða;bÞ Then Z
b a
2 f ðaÞ þ f ðbÞ h ðb aÞ f ðtÞ dt f ðxÞ 1 h þ n n
" 6
aþb x 2
2
þ
# 1 1 2 0 þ 2 ðhð2h nÞÞ ðb aÞ f 1 4 n
for all h 2 ½0; 1, n P 2 ðn 2 Z þ Þ, and a þ h b a 6 x 6 b h b a . n n 2
Proof. Let us define the mapping S : ½a; b R2 ! R given by ( Sðx; tÞ ¼
t ½a þ hðb aÞ=n;
t 2 ½a; xÞ
t ½b hðb aÞ=n;
t 2 ½ x; b
ð2:1Þ
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
427
integrating by parts, we have Z b Z x b a Sðx; tÞf 0 ðtÞ dt ¼ t aþh f 0 ðtÞ dt n a a Z b b a þ t b h f 0 ðtÞ dt n x Z b 2 f ðaÞ þ f ðbÞ ¼ ðb aÞ 1 h f ðxÞ þ h f ðtÞ dt n n a ð2:2Þ On the other hand, Z b Z 0 6 Sðx; tÞf ðtÞ dt a
¼ f 0
b a
jS ð x; tÞjf 0 ðtÞdt 6 f 0 1
Z x Z t a þ h b a dt þ 1 n
¼ f 0 1 L
a
x
We have that, Z x Z L¼ jt qjdt þ a
b
Z
b
jSðx; tÞjdt a
t b h b a dt n
b
jt pjdt x
2 2 2 2 1 ðx aÞ þ ðb xÞ b a x a b x b a þ h þ h 2 n 2 2 n 2 2 2 2 aþb b a b a ½hð2h nÞ ¼ x þ þ ð2:3Þ 2 2 n2 ¼
and the theorem is thus proved, where q, p are given as following: q¼aþh
b a ; n
p ¼b h
b a n
Remark 1. a. If we choose in (2.1), n ¼ 2, we get Dragomirs (with Cerone and Roumeliotis) integral inequality (1.1). b. If we choose in (2.1), n ¼ 2, h ¼ 0, we get Ostrowskis integral inequality [4]. Corollary 1. If we choose in (2.1) x ¼ ða þ bÞ=2, n ¼ 2 and h ¼ 1, we establish relations between the lemma (2.1) in [2] and the Trapezoid inequality,
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
428
ð b aÞ 2 Z 1 0 ð1 2tÞf ðta þ ð1 tÞbÞdt 2 0 Z b 1 f ðaÞ þ f ðbÞ ¼ f ðtÞ dt ðb aÞ 6 ðb aÞ2 f 0 1 2 4 a
ð2:4Þ
for all t 2 ½a; b; f 0 2 L½a; b (see [6]). Corollary 2. If we choose in (2.1) h ¼ 0, x ¼ ða þ bÞ=2 and n ! 1, we get Z b 1 6 ðb aÞ2 f 0 f ðtÞ dt f ðxÞ ð b aÞ ð2:5Þ 4 1 a
Corollary 3. If we choose in (2.1), h ¼ 1=2 and n ¼ 3, we get the following inequality, Z b 1 7 f ðaÞ þ f ðbÞ f ðxÞ þ f ðtÞ dt ð b aÞ 3 2 2 a ( ) 2 aþb 1 2 6 x þ ðb aÞ f 0 1 ð2:6Þ 2 36 for all x 2 ½ðð5a þ bÞ=6Þ; ðð5b þ aÞ=6Þ, and in particular x is at mid-point, the following mixture of the Trapezoid inequality and mid-point inequality, Z b 1 7 aþb f ðaÞ þ f ðbÞ 5 2 f ðb aÞ f 0 1 f ðtÞ dt þ ðb aÞ 6 3 2 2 2 36 a ð2:7Þ It is interesting to note that the smallest bound for (2.1) and (1.1) is obtained at 1 aþb and n ¼ 2. h¼ ,x¼ 2 2 Corollary 4. Now, for h 2 ½0; 1, if we choose in (2.1), n ! 1, we get a new inequality, # Z b " 2 a þ b 1 2 f ðtÞ dt 6 x þ ðb aÞ f 0 1 2 4 a
3. The results in numerical integration Theorem 3. Let f : ½a; b ! R be a differentiable mapping on ða; bÞ whose derivative is bounded on ða; bÞ. (As theorem is applied to the intervals ½x0 ; x1 ¼ ½a; x1 and ½xn 1 ; xn ¼ ½xn 1 ; b, we need f to be continuous on ½a; b:Þ If
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
429
In ¼ fxi : 0 6 i 6 ng is a partition of ½a; b; a ¼ x0 < x1 < < xn ¼ b, then we have Z b f ðxÞ dx ¼ AT ðIn ; n; d; f Þ þ RT ðIn ; n; n; d; f Þ ð3:1Þ a
where AT ¼ AT ðIn ; n; d; f Þ X n 1 n 1 X 2 f ðxi Þ þ f ðxiþ1 Þ hi f ð ni Þ hi þ d ¼ 1 d n n i¼0 i¼0
ð3:2Þ
d 2 ½0; 1, xi þ dhi 1n 6 ni 6 xiþ1 dhi 1n, i ¼ 0; 1; . . . ; n 1 and the remainder term satisfies the estimation. ( X n 1 0 1 1 f ðni Þh2i þ 2 dð2d nÞ jRT ðIn ; n; d; f ; nÞj 6 f 1 4 n i¼0 ) n 1 2 X xiþ1 þ xi þ ni ð3:3Þ 2 i¼0 Proof. Applying Theorem 2 on the interval ½xi ; xiþ1 , i ¼ 0; 1; . . . ; n 1, we get Z xiþ1 hi 1 2 d f ðni Þ þ f ðxi Þ þ f ðxiþ1 Þ d f ðxÞ dx n n xi
2 1 1 x þ x i iþ1 2 f 0 6 þ dð2d nÞ hi þ ni 1 4 n2 2 for all d 2 ½0; 1 and ni ði ¼ 0; 1; . . . ; n 1Þ as above, summing over i from n 1 and using the triangle inequality we get estimation (3.3). Remark 2. a. If we choose n ¼ 2, we get the Dragomirs integral inequality and all the results concerned with Dragomir. b. If we choose d ¼ 0 and n ¼ 2, then we get the formula Z b f ðxÞ dx ffi Riemann sum þ remainder term ð3:4Þ a
c. Choosing d ¼ 0; 1 in (3.1) respectively, gives Z b n 1 X f ðxÞ dx ¼ f ðni Þhi þ RT ðIn ; n; n; 0; f Þ a
i¼0
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
430
Z
b
f ðxÞ dx ¼
a
n 1 n 1 X 2 X f ðxi Þ þ f ðxiþ1 Þ hi 1 f ðni Þ hi þ n i¼0 n i¼0 þ RT ðIn ; n; n; 1; f Þ
where n 1 X jRT ðIn ; n; n; 0; f Þj 6 f 0 1 i¼0
h2i xi þ xiþ1 2 þ ni 4 2
and n 1 X jRT ðIn ; n; n; 1; f Þj 6 f 0 1 i¼0
1 1 xi þ xiþ1 2 2 þ ð2 nÞ hi þ ni 4 n2 2
If now we let n ! 1 with max hi ! 0 then jRT ðIn ; n; n; 0; f Þj, jRT ðIn ; n; n; 1; f Þj ! 0 and we get the usual Riemann formula for the integral Z b n 1 X f ðxÞ dx ¼ lim f ðni Þ hi ð3:5Þ n!1
a
i¼0
d. Choosing d=n ¼ 1=3 in (3.1), we get Z b f ðxÞ dx ¼ BT ðIn ; n; f Þ þ QT ðIn ; n; f Þ
ð3:6Þ
a
where BT ðIn ; n; f Þ ¼
n 1 n 1 1X 1X f ðni Þ hi þ ð f ðxi Þ þ f ðxiþ1 ÞÞhi 3 i¼0 3 i¼0
BT ðIn ; n; f Þ ¼
n 1 1X ½ f ðni Þ þ f ðxi Þ þ f ðxiþ1 Þhi 3 i¼0
or
ni 2
2xi þ xiþ1 2xiþ1 þ xi ; 3 3
and the remainder term satisfies the estimation " # n 1 n 1 X 0 5 X xi þ xiþ1 2 2 h þ ni ð3:7Þ jQT ðIn ; n; f Þj 6 f 1 36 i¼0 i 2 i¼0 Rb In particular, if we choose ni ¼ ðxi þ xiþ1 Þ=2, we have a f ðxÞ dx ¼ BT ðIn ; f Þ þ QT ðIn ; f Þ, where QT ðIn ; f Þ satisfies the estimation,
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O n 1 X 5 f 0 hi and 1 36 i¼0 n 1 h i 1X xi þ xiþ1 f BT ðIn ; f Þ ¼ þ ð f ðxi Þ þ f ðxiþ1 ÞÞ hi 3 i¼0 2
431
jQT ðIn ; f Þj 6
ð3:8Þ
Finally, we have all other generalizations of Dragomirs inequalities (so Ostrowskis inequality). Corollary 5. If we multiple with ðb aÞ=2 the inequality in (2.4), we have the following inequality ð b aÞ 3 Z 1 1 0 ð1 2tÞf ðta þ ð1 tÞbÞ dt 6 ðb aÞ3 f 0 1 ð3:9Þ 8 4 0 In particular, if we choose t 2 ½0; 1; xiþ1 xi ¼ hi ; i ¼ 0; 1; 2; . . . ; n 1; In : a ¼ x0 < x1 < x2 < < xn 1 < xn ¼ b; ni ¼ txi þ ð1 tÞxiþ1 we have the inequality 3Z 1 Z hi f ðxi Þ þ f ðxiþ1 Þ 1 1 b 0 hi ð1 2tÞf ðni Þ dt ¼ f ðtÞ dt 6 h3i f 0 1 4 4 2 a 8 0 That is, the inequality concerned with derivative of f 3Z 1 hi 1 3 0 0 6 h f ð 1 2tÞf ð n Þ dt i 4 8 i 1 0
ð3:10Þ
4. Applications to inequality (2.1) for special means The following definitions of the arithmetic [6], geometric, identric and harmonic means of two positive numbers are classical: pffiffiffiffiffi aþb A ¼ Aða; bÞ ¼ ; G ¼ Gða; bÞ ¼ ab; 2 2 H ¼ H ða; bÞ ¼ 1=a þ 1=b 8 > < a; ð1=ðb aÞÞ if a ¼ b; b a>b I ¼ Iða; bÞ ¼ 1 b ; if a 6¼ b; > e aa :
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
432
In addition the logarithmic mean is defined by L ¼ Lða; bÞ ¼
b a log b log a
the definition being complete by defining L ¼ ða; aÞ ¼ a ða ¼ bÞ. Further we will define log a log b ; R ¼ Rða; bÞ ¼ G a1=b ; b1=a equivalently log Rða; bÞ ¼ A b a Now, let us reconsider inequality (2.1) in the following equivalent form: Z b 1 2 h f ð x Þ þ f ð aÞ þ f ð bÞ h 1 f ðtÞ dt n n b a a " 2 # 0 x aþb 1 1 2 f 6 ðb aÞ ð4:1Þ þ 2 hð2h nÞ þ 1 4 n b a for all h 2 ½0; 1, n P 2, n 2 Z þ , x 2 ½a; b such that b a b a aþh 6x6b h n n 1. Consider the mapping f : ½a; b ! R, f ð xÞ ¼ log x=x and 0 < a < b. We have f ðaÞ þ f ðbÞ 1 log a log b ¼ þ n n a b 1 1 1 1 log a log b ¼ þ ½ð log a þ ln bÞ þ n a b n b a 4 2 2 ¼ H 1 log G log R ¼ 2H 1 log G log R n n n and 1 b a
Z
b
f ðxÞ dx ¼ L 1 ða; bÞ log Gða; bÞ
a
kf 0 k1 ¼ a12 jlog a 1j and then by (4.1), we deduce, for all h 2 ½0; 1, n P 2 ðn 2 Z þ Þ and a þ hððb aÞ=nÞ 6 x 6 b hððb aÞ=nÞ that 1 2 h log x þ 2 2H 1 log G log R h L 1 log G n x n ) ( 2 1 1 1 ð x AÞ 6 2 j log a 1j ðb aÞ þ 2 hð2h nÞ þ a 4 n b a
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
or
433
1 2 h HL þ 2 ðð2LÞð log GÞ þ ð HLÞ log RÞ h x H log G x n n log x log x ( ) 2 x 1 1 ð x AÞ 6 2 HLj log a 1j ðb aÞ þ 2 hð2h nÞ þ a log x 4 n b a ð4:2Þ pffiffiffi 2. Consider the mapping f : ½0; 1Þ ! R; f ðxÞ ¼ ln x, 0 < a < b. We have pffiffiffii 1 f ðaÞ þ f ðbÞ 1 h pffiffiffi ¼ ln a þ ln b ¼ ln G n n n
and 1 b a
Z a
b
1 f ðxÞ dx ¼ b a
Z
b a
pffiffiffi ln x dx ¼
1 2ð b aÞ
Z
ln x dx ¼
1 ln I 2
then by (4.1), we deduce, pffiffiffi 1 2 h ln x þ h ln G 1 ln I n n 2 ( ) 1 1 1 ð x AÞ 2 6 ðb aÞ þ hð2h nÞ þ 2a 4 n2 b a or ( ) pffiffiffið1 ð2=nÞhÞ ðh=nÞ ! 2 x G 1 1 1 ð x AÞ ðb aÞ þ hð2h nÞ þ 6 ln 2a 4 n2 I 1=2 b a In next work, I will write many inequalities concerned with my paper [3].
Acknowledgements I would like to thank Ervın Y. Rodin. In addition, I would also like to thank Mr. S.S. Dragomir for providing us the necessary documents for this study.
References [1] D.S. Mitronovic, J.E. Pecaric, A.M. Fink, Inequalities for Functions and Their Integral and Derivatives, Kluwer Academic, Dordrecht, 1994. € zdemir, On some inequalities of Hermite–Hadamard integral inequality type for differ[2] M.E. O entiable convex functions and applications to special means of real numbers, to appear.
434
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
€ zdemir, R. Ocak, U.S. Kirmaci, The undetachable representations of the groups Slð2; !Þ, [3] M.E. O ^ ¼ Glðn; !Þ=K2 , Int. J. Appl. Math. 3 (2) (2000) 171–179. K2 , R ¼ Glðn; !Þ=Slðn; !Þ, G [4] S.S. Dragomir, S. Wang, Applications of Ostrowskis inequality to the estimation of errror bounds for some special means for some numerical quadrature rules, Appl. Math. Lett. 30 (11) (1998) 105–109. [5] S.S. Dragomir, P. Cerone, J. Roumeliotis, A new generalization of Ostrowskis integral inequality for mappings whose derivatives are bounded and applications and numerical integration and for special means, Appl. Math. Lett. 13 (2000) 19–25. [6] S.S. Dragomir, Two new inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998) 91–95.
A theorem on mappings with bounded derivatives with applications to quadrature rules and means € zdemir M. Emin O K.K. Education Faculty, Department of Mathematics, Atat€urk University, Bolumu, 25240 Erzurum, Turkey
Abstract In this paper, we establish a new inequality of Theorem 2 [Appl. Math. Lett. 13 (2000) 19] (Dragomirs integral inequality) for functions whose derivatives are bounded. This has immediate applications in numerical integration where new estimates are obtained for the remainder term of the trapezoid, mid-point. Some natural applications to special means of real numbers are given. For several recent results concerning Dragomirs integral inequality (see [Appl. Math. Lett. 13 (2000) 19] where further references are listed). Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Dragomirs inequality; Quadrature formulas; Numerical integration; Bounded mapping; Trapezoid and mid-point inequality
1. Introduction Theorem 1 (Theorem 2). Let f : ½a; b ! R be continuous on ½a; b, differentiable on ða; bÞ and whose derivative f 0 : ða; bÞ ! R is bounded on ða; bÞ. Denote kf 0 k1 :¼ sup jf 0 ðxÞj < 1. Then t2ða;bÞ
€ zdemir). E-mail address: [email protected] (M.E. O 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 2 ) 0 0 1 4 6 - 7
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
426
Z
f ðaÞ þ f ðbÞ h ðb aÞ f ðtÞ dt f ðxÞð1 hÞ þ 2 a " 2 # 0 1 aþb 2 2 2 f 6 ðb aÞ ðh þ ðh 1Þ Þ þ x 1 4 2
b
ð1:1Þ
for all h 2 ½0; 1. € zdemir has obtained inequalities for the logarithIn the recent paper [2], O mic, arithmetic, p-logarithmic means. The aim of this paper is to establish some generalizations of (1.1) and Ostrowskis integral inequality [1, p. 469] and to apply these generalizations to obtain inequalities in numerical integration and mean inequalities.
2. The Results The following generalization of Dragomirs inequality [5] (with Cerone and Roumeliotis) holds. Theorem 2. f : ½ab ! R be continuous on ½a; b, and f : ða; bÞ ! R be differentiable and suppose kf 0 k1 :¼ sup jf 0 ðxÞj < 1. a 6¼ b; a; b 2 Z þ or a; b 2 Z . t2ða;bÞ Then Z
b a
2 f ðaÞ þ f ðbÞ h ðb aÞ f ðtÞ dt f ðxÞ 1 h þ n n
" 6
aþb x 2
2
þ
# 1 1 2 0 þ 2 ðhð2h nÞÞ ðb aÞ f 1 4 n
for all h 2 ½0; 1, n P 2 ðn 2 Z þ Þ, and a þ h b a 6 x 6 b h b a . n n 2
Proof. Let us define the mapping S : ½a; b R2 ! R given by ( Sðx; tÞ ¼
t ½a þ hðb aÞ=n;
t 2 ½a; xÞ
t ½b hðb aÞ=n;
t 2 ½ x; b
ð2:1Þ
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
427
integrating by parts, we have Z b Z x b a Sðx; tÞf 0 ðtÞ dt ¼ t aþh f 0 ðtÞ dt n a a Z b b a þ t b h f 0 ðtÞ dt n x Z b 2 f ðaÞ þ f ðbÞ ¼ ðb aÞ 1 h f ðxÞ þ h f ðtÞ dt n n a ð2:2Þ On the other hand, Z b Z 0 6 Sðx; tÞf ðtÞ dt a
¼ f 0
b a
jS ð x; tÞjf 0 ðtÞdt 6 f 0 1
Z x Z t a þ h b a dt þ 1 n
¼ f 0 1 L
a
x
We have that, Z x Z L¼ jt qjdt þ a
b
Z
b
jSðx; tÞjdt a
t b h b a dt n
b
jt pjdt x
2 2 2 2 1 ðx aÞ þ ðb xÞ b a x a b x b a þ h þ h 2 n 2 2 n 2 2 2 2 aþb b a b a ½hð2h nÞ ¼ x þ þ ð2:3Þ 2 2 n2 ¼
and the theorem is thus proved, where q, p are given as following: q¼aþh
b a ; n
p ¼b h
b a n
Remark 1. a. If we choose in (2.1), n ¼ 2, we get Dragomirs (with Cerone and Roumeliotis) integral inequality (1.1). b. If we choose in (2.1), n ¼ 2, h ¼ 0, we get Ostrowskis integral inequality [4]. Corollary 1. If we choose in (2.1) x ¼ ða þ bÞ=2, n ¼ 2 and h ¼ 1, we establish relations between the lemma (2.1) in [2] and the Trapezoid inequality,
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
428
ð b aÞ 2 Z 1 0 ð1 2tÞf ðta þ ð1 tÞbÞdt 2 0 Z b 1 f ðaÞ þ f ðbÞ ¼ f ðtÞ dt ðb aÞ 6 ðb aÞ2 f 0 1 2 4 a
ð2:4Þ
for all t 2 ½a; b; f 0 2 L½a; b (see [6]). Corollary 2. If we choose in (2.1) h ¼ 0, x ¼ ða þ bÞ=2 and n ! 1, we get Z b 1 6 ðb aÞ2 f 0 f ðtÞ dt f ðxÞ ð b aÞ ð2:5Þ 4 1 a
Corollary 3. If we choose in (2.1), h ¼ 1=2 and n ¼ 3, we get the following inequality, Z b 1 7 f ðaÞ þ f ðbÞ f ðxÞ þ f ðtÞ dt ð b aÞ 3 2 2 a ( ) 2 aþb 1 2 6 x þ ðb aÞ f 0 1 ð2:6Þ 2 36 for all x 2 ½ðð5a þ bÞ=6Þ; ðð5b þ aÞ=6Þ, and in particular x is at mid-point, the following mixture of the Trapezoid inequality and mid-point inequality, Z b 1 7 aþb f ðaÞ þ f ðbÞ 5 2 f ðb aÞ f 0 1 f ðtÞ dt þ ðb aÞ 6 3 2 2 2 36 a ð2:7Þ It is interesting to note that the smallest bound for (2.1) and (1.1) is obtained at 1 aþb and n ¼ 2. h¼ ,x¼ 2 2 Corollary 4. Now, for h 2 ½0; 1, if we choose in (2.1), n ! 1, we get a new inequality, # Z b " 2 a þ b 1 2 f ðtÞ dt 6 x þ ðb aÞ f 0 1 2 4 a
3. The results in numerical integration Theorem 3. Let f : ½a; b ! R be a differentiable mapping on ða; bÞ whose derivative is bounded on ða; bÞ. (As theorem is applied to the intervals ½x0 ; x1 ¼ ½a; x1 and ½xn 1 ; xn ¼ ½xn 1 ; b, we need f to be continuous on ½a; b:Þ If
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
429
In ¼ fxi : 0 6 i 6 ng is a partition of ½a; b; a ¼ x0 < x1 < < xn ¼ b, then we have Z b f ðxÞ dx ¼ AT ðIn ; n; d; f Þ þ RT ðIn ; n; n; d; f Þ ð3:1Þ a
where AT ¼ AT ðIn ; n; d; f Þ X n 1 n 1 X 2 f ðxi Þ þ f ðxiþ1 Þ hi f ð ni Þ hi þ d ¼ 1 d n n i¼0 i¼0
ð3:2Þ
d 2 ½0; 1, xi þ dhi 1n 6 ni 6 xiþ1 dhi 1n, i ¼ 0; 1; . . . ; n 1 and the remainder term satisfies the estimation. ( X n 1 0 1 1 f ðni Þh2i þ 2 dð2d nÞ jRT ðIn ; n; d; f ; nÞj 6 f 1 4 n i¼0 ) n 1 2 X xiþ1 þ xi þ ni ð3:3Þ 2 i¼0 Proof. Applying Theorem 2 on the interval ½xi ; xiþ1 , i ¼ 0; 1; . . . ; n 1, we get Z xiþ1 hi 1 2 d f ðni Þ þ f ðxi Þ þ f ðxiþ1 Þ d f ðxÞ dx n n xi
2 1 1 x þ x i iþ1 2 f 0 6 þ dð2d nÞ hi þ ni 1 4 n2 2 for all d 2 ½0; 1 and ni ði ¼ 0; 1; . . . ; n 1Þ as above, summing over i from n 1 and using the triangle inequality we get estimation (3.3). Remark 2. a. If we choose n ¼ 2, we get the Dragomirs integral inequality and all the results concerned with Dragomir. b. If we choose d ¼ 0 and n ¼ 2, then we get the formula Z b f ðxÞ dx ffi Riemann sum þ remainder term ð3:4Þ a
c. Choosing d ¼ 0; 1 in (3.1) respectively, gives Z b n 1 X f ðxÞ dx ¼ f ðni Þhi þ RT ðIn ; n; n; 0; f Þ a
i¼0
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
430
Z
b
f ðxÞ dx ¼
a
n 1 n 1 X 2 X f ðxi Þ þ f ðxiþ1 Þ hi 1 f ðni Þ hi þ n i¼0 n i¼0 þ RT ðIn ; n; n; 1; f Þ
where n 1 X jRT ðIn ; n; n; 0; f Þj 6 f 0 1 i¼0
h2i xi þ xiþ1 2 þ ni 4 2
and n 1 X jRT ðIn ; n; n; 1; f Þj 6 f 0 1 i¼0
1 1 xi þ xiþ1 2 2 þ ð2 nÞ hi þ ni 4 n2 2
If now we let n ! 1 with max hi ! 0 then jRT ðIn ; n; n; 0; f Þj, jRT ðIn ; n; n; 1; f Þj ! 0 and we get the usual Riemann formula for the integral Z b n 1 X f ðxÞ dx ¼ lim f ðni Þ hi ð3:5Þ n!1
a
i¼0
d. Choosing d=n ¼ 1=3 in (3.1), we get Z b f ðxÞ dx ¼ BT ðIn ; n; f Þ þ QT ðIn ; n; f Þ
ð3:6Þ
a
where BT ðIn ; n; f Þ ¼
n 1 n 1 1X 1X f ðni Þ hi þ ð f ðxi Þ þ f ðxiþ1 ÞÞhi 3 i¼0 3 i¼0
BT ðIn ; n; f Þ ¼
n 1 1X ½ f ðni Þ þ f ðxi Þ þ f ðxiþ1 Þhi 3 i¼0
or
ni 2
2xi þ xiþ1 2xiþ1 þ xi ; 3 3
and the remainder term satisfies the estimation " # n 1 n 1 X 0 5 X xi þ xiþ1 2 2 h þ ni ð3:7Þ jQT ðIn ; n; f Þj 6 f 1 36 i¼0 i 2 i¼0 Rb In particular, if we choose ni ¼ ðxi þ xiþ1 Þ=2, we have a f ðxÞ dx ¼ BT ðIn ; f Þ þ QT ðIn ; f Þ, where QT ðIn ; f Þ satisfies the estimation,
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O n 1 X 5 f 0 hi and 1 36 i¼0 n 1 h i 1X xi þ xiþ1 f BT ðIn ; f Þ ¼ þ ð f ðxi Þ þ f ðxiþ1 ÞÞ hi 3 i¼0 2
431
jQT ðIn ; f Þj 6
ð3:8Þ
Finally, we have all other generalizations of Dragomirs inequalities (so Ostrowskis inequality). Corollary 5. If we multiple with ðb aÞ=2 the inequality in (2.4), we have the following inequality ð b aÞ 3 Z 1 1 0 ð1 2tÞf ðta þ ð1 tÞbÞ dt 6 ðb aÞ3 f 0 1 ð3:9Þ 8 4 0 In particular, if we choose t 2 ½0; 1; xiþ1 xi ¼ hi ; i ¼ 0; 1; 2; . . . ; n 1; In : a ¼ x0 < x1 < x2 < < xn 1 < xn ¼ b; ni ¼ txi þ ð1 tÞxiþ1 we have the inequality 3Z 1 Z hi f ðxi Þ þ f ðxiþ1 Þ 1 1 b 0 hi ð1 2tÞf ðni Þ dt ¼ f ðtÞ dt 6 h3i f 0 1 4 4 2 a 8 0 That is, the inequality concerned with derivative of f 3Z 1 hi 1 3 0 0 6 h f ð 1 2tÞf ð n Þ dt i 4 8 i 1 0
ð3:10Þ
4. Applications to inequality (2.1) for special means The following definitions of the arithmetic [6], geometric, identric and harmonic means of two positive numbers are classical: pffiffiffiffiffi aþb A ¼ Aða; bÞ ¼ ; G ¼ Gða; bÞ ¼ ab; 2 2 H ¼ H ða; bÞ ¼ 1=a þ 1=b 8 > < a; ð1=ðb aÞÞ if a ¼ b; b a>b I ¼ Iða; bÞ ¼ 1 b ; if a 6¼ b; > e aa :
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
432
In addition the logarithmic mean is defined by L ¼ Lða; bÞ ¼
b a log b log a
the definition being complete by defining L ¼ ða; aÞ ¼ a ða ¼ bÞ. Further we will define log a log b ; R ¼ Rða; bÞ ¼ G a1=b ; b1=a equivalently log Rða; bÞ ¼ A b a Now, let us reconsider inequality (2.1) in the following equivalent form: Z b 1 2 h f ð x Þ þ f ð aÞ þ f ð bÞ h 1 f ðtÞ dt n n b a a " 2 # 0 x aþb 1 1 2 f 6 ðb aÞ ð4:1Þ þ 2 hð2h nÞ þ 1 4 n b a for all h 2 ½0; 1, n P 2, n 2 Z þ , x 2 ½a; b such that b a b a aþh 6x6b h n n 1. Consider the mapping f : ½a; b ! R, f ð xÞ ¼ log x=x and 0 < a < b. We have f ðaÞ þ f ðbÞ 1 log a log b ¼ þ n n a b 1 1 1 1 log a log b ¼ þ ½ð log a þ ln bÞ þ n a b n b a 4 2 2 ¼ H 1 log G log R ¼ 2H 1 log G log R n n n and 1 b a
Z
b
f ðxÞ dx ¼ L 1 ða; bÞ log Gða; bÞ
a
kf 0 k1 ¼ a12 jlog a 1j and then by (4.1), we deduce, for all h 2 ½0; 1, n P 2 ðn 2 Z þ Þ and a þ hððb aÞ=nÞ 6 x 6 b hððb aÞ=nÞ that 1 2 h log x þ 2 2H 1 log G log R h L 1 log G n x n ) ( 2 1 1 1 ð x AÞ 6 2 j log a 1j ðb aÞ þ 2 hð2h nÞ þ a 4 n b a
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
or
433
1 2 h HL þ 2 ðð2LÞð log GÞ þ ð HLÞ log RÞ h x H log G x n n log x log x ( ) 2 x 1 1 ð x AÞ 6 2 HLj log a 1j ðb aÞ þ 2 hð2h nÞ þ a log x 4 n b a ð4:2Þ pffiffiffi 2. Consider the mapping f : ½0; 1Þ ! R; f ðxÞ ¼ ln x, 0 < a < b. We have pffiffiffii 1 f ðaÞ þ f ðbÞ 1 h pffiffiffi ¼ ln a þ ln b ¼ ln G n n n
and 1 b a
Z a
b
1 f ðxÞ dx ¼ b a
Z
b a
pffiffiffi ln x dx ¼
1 2ð b aÞ
Z
ln x dx ¼
1 ln I 2
then by (4.1), we deduce, pffiffiffi 1 2 h ln x þ h ln G 1 ln I n n 2 ( ) 1 1 1 ð x AÞ 2 6 ðb aÞ þ hð2h nÞ þ 2a 4 n2 b a or ( ) pffiffiffið1 ð2=nÞhÞ ðh=nÞ ! 2 x G 1 1 1 ð x AÞ ðb aÞ þ hð2h nÞ þ 6 ln 2a 4 n2 I 1=2 b a In next work, I will write many inequalities concerned with my paper [3].
Acknowledgements I would like to thank Ervın Y. Rodin. In addition, I would also like to thank Mr. S.S. Dragomir for providing us the necessary documents for this study.
References [1] D.S. Mitronovic, J.E. Pecaric, A.M. Fink, Inequalities for Functions and Their Integral and Derivatives, Kluwer Academic, Dordrecht, 1994. € zdemir, On some inequalities of Hermite–Hadamard integral inequality type for differ[2] M.E. O entiable convex functions and applications to special means of real numbers, to appear.
434
€ zdemir / Appl. Math. Comput. 138 (2003) 425–434 M.E. O
€ zdemir, R. Ocak, U.S. Kirmaci, The undetachable representations of the groups Slð2; !Þ, [3] M.E. O ^ ¼ Glðn; !Þ=K2 , Int. J. Appl. Math. 3 (2) (2000) 171–179. K2 , R ¼ Glðn; !Þ=Slðn; !Þ, G [4] S.S. Dragomir, S. Wang, Applications of Ostrowskis inequality to the estimation of errror bounds for some special means for some numerical quadrature rules, Appl. Math. Lett. 30 (11) (1998) 105–109. [5] S.S. Dragomir, P. Cerone, J. Roumeliotis, A new generalization of Ostrowskis integral inequality for mappings whose derivatives are bounded and applications and numerical integration and for special means, Appl. Math. Lett. 13 (2000) 19–25. [6] S.S. Dragomir, Two new inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11 (1998) 91–95.