- Email: [email protected]
A general Ostrowski-type inequality
ARTICLE IN PRESS
Statistics & Probability Letters 72 (2005) 145–152 www.elsevier.com/locate/stapro
A general Ostrowski-type inequality$ J. de la Cala,, J. Ca´rcamob a
Departamento de Matema´tica Aplicada y Estadı´stica e Investigacio´n Operativa, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain b Departamento de Economı´a Aplicada V, E.U. de Estudios Empresariales, Universidad del Paı´s Vasco, Elcano 21, 48008 Bilbao, Spain Received 27 February 2004; received in revised form 27 July 2004; accepted 20 December 2004
Abstract We obtain an integral representation for the expectation which generalizes a well-known formula. As a consequence, we establish an estimate for the difference of two expectations which is optimal in a specific sense and is general enough to include as particular cases many of the Ostrowski-type inequalities existing in the literature. Other consequences concerning inequalities and stochastic orders are also discussed. r 2005 Elsevier B.V. All rights reserved. Keywords: Ostrowski inequality; Random variable; Distribution function; Expectation; Lebesgue integral; Absolutely continuous function; Convex function
1. Introduction and main results As it is well known, if X is a real integrable random variable with distribution function F, then we have Z
1
FðuÞ du þ
EX ¼ 1
$
Z
0
ð1 F ðuÞÞ du. 0
Research supported by the Spanish MCYT, Proyecto BFM2002-04163-C02-02, and by FEDER.
Corresponding author. Tel.: +34 94 601 2654; fax: +34 94 601 3500.
E-mail addresses: [email protected] (J. de la Cal), [email protected] (J. Ca´rcamo). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.12.013
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
146
Even, when X takes values in an interval I, we can write Z Z F ðuÞ du þ ð1 FðuÞÞ du, EX ¼ a I a
(1)
Iþ a
where a is any fixed point of I, and I a :¼I \ ð1; a;
Iþ a :¼I \ ða; 1Þ.
As a consequence, if Y is another I-valued integrable random variable with distribution function G, then we have Z ðFðuÞ GðuÞÞ du. (2) EY EX ¼ I
In this paper, we generalize formulae (1) and (2) by giving integral representations for Ef ðX Þ and Ef ðY Þ Ef ðX Þ; when f 2 LðIÞ:¼ the space of all real locally absolutely continuous functions on I. Thus, f 2 LðIÞ means that the derivative f 0 exists almost everywhere, is locally integrable (i.e., integrable on each compact subinterval of I), and we can write, for all x 2 I; Z a Z x 0 þ f ðxÞ ¼ f ðaÞ 1I a ðxÞ f ðuÞ du þ 1I a ðxÞ f 0 ðuÞ du, (3) x
a
where a is any fixed point of I, and 1A stands for the indicator function of the set A. The aforementioned representations are expressed in terms of the principal value of an integral. We recall that, if g is a real locally integrable function on I, the principal value of the integral of g (on I) is defined by Z Z gðuÞ du, (4) gðuÞ du:¼ lim I
n!1
I
I
In
S where ðI n Þ is any increasing sequence of compact subintervals of I such that n I n ¼ I; provided that the (finite or infinite) limit exists and does not depend upon the particular sequence ðI n Þ involved. It is clear that Z Z gðuÞ du, gðuÞ du ¼ whenever g is integrable on I or g is nonnegative (nonpositive) a.e. on I. Theorem 1. Let X and Y be I-valued random variables with distribution functions F and G, respectively, and let f 2 LðIÞ be such that f ðX Þ and f ðY Þ are integrable. Then, we have Z Z 0 (5) Ef ðX Þ ¼ f ðaÞ f ðuÞF ðuÞ du þ f 0 ðuÞð1 FðuÞÞ du I a
Iþ a
(where a is any fixed point of I), and Z Ef ðY Þ Ef ðX Þ ¼ f 0 ðuÞðF ðuÞ GðuÞÞ du.
(6)
I
Remark 1. In the setting of the preceding theorem, the principal value cannot be, in general, replaced by a proper integral. Take, for instance, f ðxÞ:¼ x cos x þ sin x; and let X be a random
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
147
variable having the probability density fðxÞ:¼ð2=x3 Þ1½1;1Þ ðxÞ: Then, both X and f ðX Þ are integrable, but the function x7!f 0 ðxÞð1 F ðxÞÞ ¼ sin x=x is not Lebesgue integrable on I:¼½1; 1Þ: For 1ppp1; we denote by Lp ðIÞ the space of all real measurable functions g on I such that jgjp is Lebesgue integrable on I, endowed with the usual norm Z 1=p p jgðtÞj dt ð1ppo1Þ; kgk1 :¼ ess sup jgðxÞj. kgkp :¼ x2I
I
In this setting, we also denote by q the conjugate of p, i.e., q:¼p=ðp 1Þ (with 01 :¼1). As a consequence of Theorem 1, we obtain the following. Corollary 1. Let X and Y be two I-valued random variables having distribution functions F and G, respectively, and let 1ppp1: If F G 2 Lq ðIÞ; then, for each f 2 LðIÞ such that f 0 2 Lp ðIÞ and f ðX Þ and f ðY Þ are integrable, we have jEf ðY Þ Ef ðX Þjpkf 0 kp kF Gkq .
(7)
Moreover, when 1opp1; the equality holds in (7) if and only if f 0 ¼ C signðF GÞjF Gj1=ðp1Þ
a:e:;
for some constant C. Remark 2. In particular, when X is degenerate at x 2 I; we have (take f ðÞ:¼j xj) EjY xj ¼ kF Gk1 . The proofs of Theorem 1 and Corollary 1 are given in the next section. Corollary 1 may be useful in several mathematical fields. In the present paper, we emphasize the fact that it can be viewed as a general Ostrowski-type inequality. Actually, by choosing (the distributions of) X and Y appropriately, we can readily obtain many of the Ostrowski-type inequalities existing in the literature, or generate new ones. We illustrate this point by a few but significant examples collected in Section 3. Apart from Corollary 1, Theorem 1 also has other interesting implications, some of which are considered in Section 4.
2. Proofs of the main results Proof of Theorem 1. It is clear that (6) directly follows from (5). To show (5), we first assume that I is compact. Then, by (3) and Fubini’s theorem, we have Z f ðtÞ dFðtÞ E½ f ðX Þ1I a ðX Þ ¼ I a
¼ f ðaÞPðX 2
I aÞ
Z Z I a
x
a
f ðuÞ du dF ðtÞ 0
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
148
¼ f ðaÞPðX 2 I aÞ ¼ f ðaÞPðX 2
I aÞ
E½ f ðX Þ1 ðX Þ ¼ f ðaÞPðX 2 Iþ a
Iþ aÞ
I a
dFðtÞ f 0 ðuÞ du
I u
Z
f 0 ðuÞF ðuÞ du,
and, analogously,
!
Z
Z
I a
Z
f 0 ðuÞð1 F ðuÞÞ du.
þ Iþ a
Since Ef ðX Þ ¼ E½f ðX Þ1I a ðX Þ þ E½ f ðX Þ1I þa ðX Þ, the conclusion (5) follows in this case.SIf I is not compact, let ðI n Þ be an increasing sequence of compact subintervals of I a such that n I n ¼ I a : We have, by dominated convergence and the compact case, E½ f ðX Þ1I a ðX Þ ¼ lim E½ f ðX Þ1I n ðX Þ n!1
¼ f ðaÞPðX 2
I aÞ
Z
f 0 ðuÞFðuÞ du.
lim
n!1
In
This shows that the principal value Z f 0 ðuÞF ðuÞ du I a
exists (and is finite), and we have
Z E½ f ðX Þ1I a ðX Þ ¼ f ðaÞPðX 2 I Þ a
Similarly, we can show that E½ f ðX Þ1I þa ðX Þ ¼ f ðaÞPðX 2
Iþ aÞ
f 0 ðuÞF ðuÞ du. I a
Z þ
f 0 ðuÞð1 F ðuÞÞ du, Iþ a
and the proof of the theorem is complete.
&
Proof of Corollary 1. We have from (6) and Ho¨lder inequality Z jEf ðY Þ Ef ðX Þjp jf 0 ðxÞj jFðxÞ GðxÞj dx I
pkf 0 kp kF Gkq ,
ð8Þ
showing (7). Moreover, the first inequality in (8) becomes an equality if and only if f 0 and F G have the same sign or opposite sign a.e., while, in the case that 1opp1; the second inequality in (8) becomes an equality if and only if jf 0 j ¼ CjF Gj1=ðp1Þ a.e., for some constant CX0: This completes the proof of the corollary. &
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
149
3. Examples In this section, we show that many of the Ostrowski-type inequalities existing in the literature are nothing but simple particular cases of Corollary 1. Example 1. If Y is uniformly distributed on ½a; b; and X is degenerate at x 2 ½a; b; we have Z b 1 f ðtÞ dt f ðxÞ, jEf ðY Þ Ef ðX Þj ¼ ba a jF ðtÞ GðtÞj ¼
ta bt 1½a;xÞ ðtÞ þ 1½x;bÞ ðtÞ, ba ba
and 8 qþ1 qþ1 1=q > ðx aÞ þ ðb xÞ > > < ðq þ 1Þðb aÞq kF Gkq ¼ > maxðx a; b xÞ > > : ba
if 1pqo1; if q ¼ 1:
The corresponding version of (7) was obtained by Fink (1992). For p ¼ 1 (q ¼ 1), it becomes the classical Ostrowski’s inequality (Ostrowski, 1938). Remark 3. The application of the preceding example to the case in which f ðxÞ:¼PðZpxÞ is the distribution function of a random variable Z taking values in ½a; b and having a density f 0 2 Lp ½a; b gives an inequality of Brnetic´ and Pecˇaric´ (2000). Note that, in such a case, we have by (1) Z b f ðtÞ dt ¼ b EZ. a
Example 2. Let Y be uniformly distributed on ½a; b; and let X be the discrete random variable with probability distribution given by PðX ¼ xi Þ ¼ ðaiþ1 ai Þ=ðb aÞ;
i ¼ 0; 1; . . . ; k,
where a¼:x0 ox1 o oxk1 oxk :¼b is a division of the interval ½a; b; and the ai (i ¼ 0; . . . ; k þ 1) are k þ 2 points so that a0 :¼a; ai 2 ½xi1 ; xi (i ¼ 1; . . . ; k), and akþ1 :¼b: We have Z k X 1 b jEf ðY Þ Ef ðX Þj ¼ f ðxÞ dx ða a Þf ðx Þ iþ1 i i , b a a i¼0
jF ðtÞ GðtÞj ¼
k1 h X aiþ1 t i¼0
ba
1½xi ;aiþ1 Þ ðtÞ þ
i t aiþ1 1½aiþ1 ;xiþ1 Þ ðtÞ , ba
ARTICLE IN PRESS 150
J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
and 8 1=q > Pk1 ðaiþ1 xi Þqþ1 þ ðxiþ1 aiþ1 Þqþ1 > > < i¼0 ðq þ 1Þðb aÞq kF Gkq ¼ > max0pipk1 maxðaiþ1 xi ; xiþ1 aiþ1 Þ > > : ba
if 1pqo1; if q ¼ 1:
The corresponding Ostrowski-type inequalities were first obtained by Dragomir (2001a,b). Remark 4. For k ¼ 1; the preceding example becomes Example 1. It should also be observed that recently published inequalities such as Yang (2003, Theorem 3.1) and O¨zdemir (2003, Theorem 2) are nothing but simple particular cases of Example 2. Example 3. Let X and Y be uniformly distributed on the intervals ½a; b and ½c; d; respectively, where apcodpb and b a d þ c40: Then, we have Z b Z d 1 1 f ðxÞ dx f ðxÞ dx, jEf ðY Þ Ef ðX Þj ¼ ba a d c c jF ðtÞ GðtÞj ¼
t a t c ta 1½a;cÞ ðtÞ þ 1½c;gÞ ðtÞ ba ba d c t c t a bt 1½d;bÞ ðtÞ, þ 1½g;dÞ ðtÞ þ d c ba ba
where g:¼ðcb adÞ=ðb a d þ cÞ is the solution to the equation ta tc ¼ , ba d c and 8 1=q > ðc aÞqþ1 þ ðb dÞqþ1 > > < ðq þ 1Þðb a d þ cÞðb aÞq1 kF Gkq ¼ > maxðc a; b dÞ > > : ba
if 1pqo1; if q ¼ 1:
This example extends Theorem 3 in Matic´ and Pecˇaric´ (2001). Our last example is on an unbounded interval. Example 4. Assume that Y has the exponential distribution with parameter 1, and X is degenerate at xX0: Then Z 1 t e f ðtÞ dt f ðxÞ, jEf ðY Þ Ef ðX Þj ¼ 0
jF ðtÞ GðtÞj ¼ ð1 et Þ1½0;xÞ ðtÞ þ et 1½x;1Þ ðtÞ,
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
151
and ( kF Gkq ¼
2 ex þ x 1 x
if q ¼ 1; x
maxðe ; 1 e Þ
if q ¼ 1:
(9)
Observe that the right-hand side in (9), as a function of x, attains its minimum at x:¼ log 2 (the median of the exponential distribution), and we have (under the appropriate conditions on f) Z 1 t p log 2kf 0 k1 . e f ðtÞ dt f ðlog 2Þ 0
Remark 5. The phenomenon observed in the last example is just a particular case of the following general fact: Let G be the distribution function of an integrable I-valued random variable Y, and, for x 2 I; let F x be the distribution function degenerate at x. Then, for all 1pqp1; we have inf kF x Gkq ¼ kF m Gkq , x2I
where m is a median of G. Actually, when 1pqo1; the derivative of the function cðxÞ:¼kF x Gkqq is given by c0 ðxÞ ¼ ½GðxÞq ½1 GðxÞq a.e., and this implies that c is nonincreasing on I m and nondecreasing on I þ m : In the case q ¼ 1; the proof of the claim is also straightforward.
4. Further remarks In this section, we develop some additional implications of Theorem 1 concerning inequalities and stochastic orders. First of all, we explain the terminology to be used. We say that the real function j on I pivots on a 2 I if it fulfils that ðx aÞjðxÞX0;
x 2 I.
On the other hand, we recall that, for a given function f 2 LðIÞ; the derivative may not exist at the points of a set A having zero measure. By choosing the value f 0 ðxÞ at each exceptional point x 2 A (in the most convenient way for our purposes), we have a version of the (Radon–Nikodym) derivative of f. In what follows, when we say f 0 fulfils property P, we mean there exists a version of the derivative of f fulfilling property P. We assert the following: Corollary 2. Let X and Y be two integrable I-valued random variables with distribution functions F and G, respectively, and let f 2 LðIÞ be such that f ðX Þ and f ðY Þ are integrable. If both F G and f 0 f 0 ðaÞ pivot on a 2 I; then Ef ðY Þ Ef ðX ÞXf 0 ðaÞðEY EX Þ.
(10)
Proof. From the assumptions on F G and f 0 ; we have ðf 0 ðxÞ f 0 ðaÞÞðFðxÞ GðxÞÞX0;
x 2 I,
(11)
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
152
and we conclude from (2) and (6), Ef ðY Þ Ef ðX Þ f 0 ðaÞðEY EX Þ Z ðf 0 ðxÞ f 0 ðaÞÞðF ðxÞ GðxÞÞ dxX0: ¼
&
ð12Þ
I
Remark 6. Observe that, in view of (11), the integral in (12) is a proper one. In connection with Remark 1, it is also worth noting that, if X and f ðX Þ are integrable, and f 0 f 0 ðaÞ pivots on a 2 I; then the principal values in (5) actually are proper integrals. This follows from the fact that Z Z Z f 0 ðuÞF ðuÞ du ¼ ðf 0 ðuÞ f 0 ðaÞÞF ðuÞ du þ f 0 ðaÞ FðuÞ du, I a
I a
I a
and the analogous equality for I þ a: Denote by CðIÞ the set of all real continuous convex functions on I, and let C ðIÞ be the set of all functions in CðIÞ which are nondecreasing. It is well known that, if f 2 CðIÞ; then f 2 LðIÞ and f 0 is nondecreasing, so that, for each a 2 I; f 0 f 0 ðaÞ pivots on a. Also, if f 2 C ðIÞ; then f 0 is nonnegative. Therefore, Corollary 2 entails the following known result concerning the convex order and the increasing convex order for random variables (see Shaked and Shanthikumar, 1994, pp. 65 and 91). Corollary 3. Let X and Y be two integrable I-valued random variables with distribution functions F and G, respectively. If F G pivots on some a 2 I; and EX ¼ EY (resp. EX pEY ), then we have Ef ðY ÞXEf ðX Þ, for all f 2 CðIÞ (resp. for all f 2 C ðIÞ) such that f ðX Þ and f ðY Þ are integrable. It should also be observed that, if X :¼EY ; then F G obviously pivots on EY 2 I: Therefore, the preceding corollary includes Jensen’s inequality as a particular case.
References Brnetic´, I., Pecˇaric´, J., 2000. On an Ostrowski type inequality for a random variable. Math. Inequal. Appl. 3, 143–145. Dragomir, S.S., 2001a. A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to L1 ½a; b and applications in numerical integration. J. Comput. Anal. Appl. 3, 343–360. Dragomir, S.S., 2001b. A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to Lp ½a; b and applications in numerical integration. J. Math. Anal. Appl. 255, 605–626. Fink, A.M., 1992. Bounds on the deviation of a function from its averages. Czechoslovak Math. J. 42, 289–310. Matic´, M., Pecˇaric´, J., 2001. Two-point Ostrowski inequality. Math. Inequal. Appl. 2, 215–221. Ostrowski, A., 1938. U¨ber die Absolutabweichung einer differentierbaren Funktionen von ihren Integralmittelwort. Comment. Math. Helv. 10, 226–227. O¨zdemir, M.E., 2003. A theorem on mappings with bounded derivatives with applications to quadrature rules and means. Appl. Math. Comput. 138, 425–434. Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, Boston. Yang, X., 2003. Refinement of Ho¨lder inequality and application to Ostrowski inequality. Appl. Math. Comput. 138, 455–461.
Statistics & Probability Letters 72 (2005) 145–152 www.elsevier.com/locate/stapro
A general Ostrowski-type inequality$ J. de la Cala,, J. Ca´rcamob a
Departamento de Matema´tica Aplicada y Estadı´stica e Investigacio´n Operativa, Facultad de Ciencia y Tecnologı´a, Universidad del Paı´s Vasco, Apartado 644, 48080 Bilbao, Spain b Departamento de Economı´a Aplicada V, E.U. de Estudios Empresariales, Universidad del Paı´s Vasco, Elcano 21, 48008 Bilbao, Spain Received 27 February 2004; received in revised form 27 July 2004; accepted 20 December 2004
Abstract We obtain an integral representation for the expectation which generalizes a well-known formula. As a consequence, we establish an estimate for the difference of two expectations which is optimal in a specific sense and is general enough to include as particular cases many of the Ostrowski-type inequalities existing in the literature. Other consequences concerning inequalities and stochastic orders are also discussed. r 2005 Elsevier B.V. All rights reserved. Keywords: Ostrowski inequality; Random variable; Distribution function; Expectation; Lebesgue integral; Absolutely continuous function; Convex function
1. Introduction and main results As it is well known, if X is a real integrable random variable with distribution function F, then we have Z
1
FðuÞ du þ
EX ¼ 1
$
Z
0
ð1 F ðuÞÞ du. 0
Research supported by the Spanish MCYT, Proyecto BFM2002-04163-C02-02, and by FEDER.
Corresponding author. Tel.: +34 94 601 2654; fax: +34 94 601 3500.
E-mail addresses: [email protected] (J. de la Cal), [email protected] (J. Ca´rcamo). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2004.12.013
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
146
Even, when X takes values in an interval I, we can write Z Z F ðuÞ du þ ð1 FðuÞÞ du, EX ¼ a I a
(1)
Iþ a
where a is any fixed point of I, and I a :¼I \ ð1; a;
Iþ a :¼I \ ða; 1Þ.
As a consequence, if Y is another I-valued integrable random variable with distribution function G, then we have Z ðFðuÞ GðuÞÞ du. (2) EY EX ¼ I
In this paper, we generalize formulae (1) and (2) by giving integral representations for Ef ðX Þ and Ef ðY Þ Ef ðX Þ; when f 2 LðIÞ:¼ the space of all real locally absolutely continuous functions on I. Thus, f 2 LðIÞ means that the derivative f 0 exists almost everywhere, is locally integrable (i.e., integrable on each compact subinterval of I), and we can write, for all x 2 I; Z a Z x 0 þ f ðxÞ ¼ f ðaÞ 1I a ðxÞ f ðuÞ du þ 1I a ðxÞ f 0 ðuÞ du, (3) x
a
where a is any fixed point of I, and 1A stands for the indicator function of the set A. The aforementioned representations are expressed in terms of the principal value of an integral. We recall that, if g is a real locally integrable function on I, the principal value of the integral of g (on I) is defined by Z Z gðuÞ du, (4) gðuÞ du:¼ lim I
n!1
I
I
In
S where ðI n Þ is any increasing sequence of compact subintervals of I such that n I n ¼ I; provided that the (finite or infinite) limit exists and does not depend upon the particular sequence ðI n Þ involved. It is clear that Z Z gðuÞ du, gðuÞ du ¼ whenever g is integrable on I or g is nonnegative (nonpositive) a.e. on I. Theorem 1. Let X and Y be I-valued random variables with distribution functions F and G, respectively, and let f 2 LðIÞ be such that f ðX Þ and f ðY Þ are integrable. Then, we have Z Z 0 (5) Ef ðX Þ ¼ f ðaÞ f ðuÞF ðuÞ du þ f 0 ðuÞð1 FðuÞÞ du I a
Iþ a
(where a is any fixed point of I), and Z Ef ðY Þ Ef ðX Þ ¼ f 0 ðuÞðF ðuÞ GðuÞÞ du.
(6)
I
Remark 1. In the setting of the preceding theorem, the principal value cannot be, in general, replaced by a proper integral. Take, for instance, f ðxÞ:¼ x cos x þ sin x; and let X be a random
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
147
variable having the probability density fðxÞ:¼ð2=x3 Þ1½1;1Þ ðxÞ: Then, both X and f ðX Þ are integrable, but the function x7!f 0 ðxÞð1 F ðxÞÞ ¼ sin x=x is not Lebesgue integrable on I:¼½1; 1Þ: For 1ppp1; we denote by Lp ðIÞ the space of all real measurable functions g on I such that jgjp is Lebesgue integrable on I, endowed with the usual norm Z 1=p p jgðtÞj dt ð1ppo1Þ; kgk1 :¼ ess sup jgðxÞj. kgkp :¼ x2I
I
In this setting, we also denote by q the conjugate of p, i.e., q:¼p=ðp 1Þ (with 01 :¼1). As a consequence of Theorem 1, we obtain the following. Corollary 1. Let X and Y be two I-valued random variables having distribution functions F and G, respectively, and let 1ppp1: If F G 2 Lq ðIÞ; then, for each f 2 LðIÞ such that f 0 2 Lp ðIÞ and f ðX Þ and f ðY Þ are integrable, we have jEf ðY Þ Ef ðX Þjpkf 0 kp kF Gkq .
(7)
Moreover, when 1opp1; the equality holds in (7) if and only if f 0 ¼ C signðF GÞjF Gj1=ðp1Þ
a:e:;
for some constant C. Remark 2. In particular, when X is degenerate at x 2 I; we have (take f ðÞ:¼j xj) EjY xj ¼ kF Gk1 . The proofs of Theorem 1 and Corollary 1 are given in the next section. Corollary 1 may be useful in several mathematical fields. In the present paper, we emphasize the fact that it can be viewed as a general Ostrowski-type inequality. Actually, by choosing (the distributions of) X and Y appropriately, we can readily obtain many of the Ostrowski-type inequalities existing in the literature, or generate new ones. We illustrate this point by a few but significant examples collected in Section 3. Apart from Corollary 1, Theorem 1 also has other interesting implications, some of which are considered in Section 4.
2. Proofs of the main results Proof of Theorem 1. It is clear that (6) directly follows from (5). To show (5), we first assume that I is compact. Then, by (3) and Fubini’s theorem, we have Z f ðtÞ dFðtÞ E½ f ðX Þ1I a ðX Þ ¼ I a
¼ f ðaÞPðX 2
I aÞ
Z Z I a
x
a
f ðuÞ du dF ðtÞ 0
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
148
¼ f ðaÞPðX 2 I aÞ ¼ f ðaÞPðX 2
I aÞ
E½ f ðX Þ1 ðX Þ ¼ f ðaÞPðX 2 Iþ a
Iþ aÞ
I a
dFðtÞ f 0 ðuÞ du
I u
Z
f 0 ðuÞF ðuÞ du,
and, analogously,
!
Z
Z
I a
Z
f 0 ðuÞð1 F ðuÞÞ du.
þ Iþ a
Since Ef ðX Þ ¼ E½f ðX Þ1I a ðX Þ þ E½ f ðX Þ1I þa ðX Þ, the conclusion (5) follows in this case.SIf I is not compact, let ðI n Þ be an increasing sequence of compact subintervals of I a such that n I n ¼ I a : We have, by dominated convergence and the compact case, E½ f ðX Þ1I a ðX Þ ¼ lim E½ f ðX Þ1I n ðX Þ n!1
¼ f ðaÞPðX 2
I aÞ
Z
f 0 ðuÞFðuÞ du.
lim
n!1
In
This shows that the principal value Z f 0 ðuÞF ðuÞ du I a
exists (and is finite), and we have
Z E½ f ðX Þ1I a ðX Þ ¼ f ðaÞPðX 2 I Þ a
Similarly, we can show that E½ f ðX Þ1I þa ðX Þ ¼ f ðaÞPðX 2
Iþ aÞ
f 0 ðuÞF ðuÞ du. I a
Z þ
f 0 ðuÞð1 F ðuÞÞ du, Iþ a
and the proof of the theorem is complete.
&
Proof of Corollary 1. We have from (6) and Ho¨lder inequality Z jEf ðY Þ Ef ðX Þjp jf 0 ðxÞj jFðxÞ GðxÞj dx I
pkf 0 kp kF Gkq ,
ð8Þ
showing (7). Moreover, the first inequality in (8) becomes an equality if and only if f 0 and F G have the same sign or opposite sign a.e., while, in the case that 1opp1; the second inequality in (8) becomes an equality if and only if jf 0 j ¼ CjF Gj1=ðp1Þ a.e., for some constant CX0: This completes the proof of the corollary. &
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
149
3. Examples In this section, we show that many of the Ostrowski-type inequalities existing in the literature are nothing but simple particular cases of Corollary 1. Example 1. If Y is uniformly distributed on ½a; b; and X is degenerate at x 2 ½a; b; we have Z b 1 f ðtÞ dt f ðxÞ, jEf ðY Þ Ef ðX Þj ¼ ba a jF ðtÞ GðtÞj ¼
ta bt 1½a;xÞ ðtÞ þ 1½x;bÞ ðtÞ, ba ba
and 8 qþ1 qþ1 1=q > ðx aÞ þ ðb xÞ > > < ðq þ 1Þðb aÞq kF Gkq ¼ > maxðx a; b xÞ > > : ba
if 1pqo1; if q ¼ 1:
The corresponding version of (7) was obtained by Fink (1992). For p ¼ 1 (q ¼ 1), it becomes the classical Ostrowski’s inequality (Ostrowski, 1938). Remark 3. The application of the preceding example to the case in which f ðxÞ:¼PðZpxÞ is the distribution function of a random variable Z taking values in ½a; b and having a density f 0 2 Lp ½a; b gives an inequality of Brnetic´ and Pecˇaric´ (2000). Note that, in such a case, we have by (1) Z b f ðtÞ dt ¼ b EZ. a
Example 2. Let Y be uniformly distributed on ½a; b; and let X be the discrete random variable with probability distribution given by PðX ¼ xi Þ ¼ ðaiþ1 ai Þ=ðb aÞ;
i ¼ 0; 1; . . . ; k,
where a¼:x0 ox1 o oxk1 oxk :¼b is a division of the interval ½a; b; and the ai (i ¼ 0; . . . ; k þ 1) are k þ 2 points so that a0 :¼a; ai 2 ½xi1 ; xi (i ¼ 1; . . . ; k), and akþ1 :¼b: We have Z k X 1 b jEf ðY Þ Ef ðX Þj ¼ f ðxÞ dx ða a Þf ðx Þ iþ1 i i , b a a i¼0
jF ðtÞ GðtÞj ¼
k1 h X aiþ1 t i¼0
ba
1½xi ;aiþ1 Þ ðtÞ þ
i t aiþ1 1½aiþ1 ;xiþ1 Þ ðtÞ , ba
ARTICLE IN PRESS 150
J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
and 8 1=q > Pk1 ðaiþ1 xi Þqþ1 þ ðxiþ1 aiþ1 Þqþ1 > > < i¼0 ðq þ 1Þðb aÞq kF Gkq ¼ > max0pipk1 maxðaiþ1 xi ; xiþ1 aiþ1 Þ > > : ba
if 1pqo1; if q ¼ 1:
The corresponding Ostrowski-type inequalities were first obtained by Dragomir (2001a,b). Remark 4. For k ¼ 1; the preceding example becomes Example 1. It should also be observed that recently published inequalities such as Yang (2003, Theorem 3.1) and O¨zdemir (2003, Theorem 2) are nothing but simple particular cases of Example 2. Example 3. Let X and Y be uniformly distributed on the intervals ½a; b and ½c; d; respectively, where apcodpb and b a d þ c40: Then, we have Z b Z d 1 1 f ðxÞ dx f ðxÞ dx, jEf ðY Þ Ef ðX Þj ¼ ba a d c c jF ðtÞ GðtÞj ¼
t a t c ta 1½a;cÞ ðtÞ þ 1½c;gÞ ðtÞ ba ba d c t c t a bt 1½d;bÞ ðtÞ, þ 1½g;dÞ ðtÞ þ d c ba ba
where g:¼ðcb adÞ=ðb a d þ cÞ is the solution to the equation ta tc ¼ , ba d c and 8 1=q > ðc aÞqþ1 þ ðb dÞqþ1 > > < ðq þ 1Þðb a d þ cÞðb aÞq1 kF Gkq ¼ > maxðc a; b dÞ > > : ba
if 1pqo1; if q ¼ 1:
This example extends Theorem 3 in Matic´ and Pecˇaric´ (2001). Our last example is on an unbounded interval. Example 4. Assume that Y has the exponential distribution with parameter 1, and X is degenerate at xX0: Then Z 1 t e f ðtÞ dt f ðxÞ, jEf ðY Þ Ef ðX Þj ¼ 0
jF ðtÞ GðtÞj ¼ ð1 et Þ1½0;xÞ ðtÞ þ et 1½x;1Þ ðtÞ,
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
151
and ( kF Gkq ¼
2 ex þ x 1 x
if q ¼ 1; x
maxðe ; 1 e Þ
if q ¼ 1:
(9)
Observe that the right-hand side in (9), as a function of x, attains its minimum at x:¼ log 2 (the median of the exponential distribution), and we have (under the appropriate conditions on f) Z 1 t p log 2kf 0 k1 . e f ðtÞ dt f ðlog 2Þ 0
Remark 5. The phenomenon observed in the last example is just a particular case of the following general fact: Let G be the distribution function of an integrable I-valued random variable Y, and, for x 2 I; let F x be the distribution function degenerate at x. Then, for all 1pqp1; we have inf kF x Gkq ¼ kF m Gkq , x2I
where m is a median of G. Actually, when 1pqo1; the derivative of the function cðxÞ:¼kF x Gkqq is given by c0 ðxÞ ¼ ½GðxÞq ½1 GðxÞq a.e., and this implies that c is nonincreasing on I m and nondecreasing on I þ m : In the case q ¼ 1; the proof of the claim is also straightforward.
4. Further remarks In this section, we develop some additional implications of Theorem 1 concerning inequalities and stochastic orders. First of all, we explain the terminology to be used. We say that the real function j on I pivots on a 2 I if it fulfils that ðx aÞjðxÞX0;
x 2 I.
On the other hand, we recall that, for a given function f 2 LðIÞ; the derivative may not exist at the points of a set A having zero measure. By choosing the value f 0 ðxÞ at each exceptional point x 2 A (in the most convenient way for our purposes), we have a version of the (Radon–Nikodym) derivative of f. In what follows, when we say f 0 fulfils property P, we mean there exists a version of the derivative of f fulfilling property P. We assert the following: Corollary 2. Let X and Y be two integrable I-valued random variables with distribution functions F and G, respectively, and let f 2 LðIÞ be such that f ðX Þ and f ðY Þ are integrable. If both F G and f 0 f 0 ðaÞ pivot on a 2 I; then Ef ðY Þ Ef ðX ÞXf 0 ðaÞðEY EX Þ.
(10)
Proof. From the assumptions on F G and f 0 ; we have ðf 0 ðxÞ f 0 ðaÞÞðFðxÞ GðxÞÞX0;
x 2 I,
(11)
ARTICLE IN PRESS J. de la Cal, J. Ca´rcamo / Statistics & Probability Letters 72 (2005) 145–152
152
and we conclude from (2) and (6), Ef ðY Þ Ef ðX Þ f 0 ðaÞðEY EX Þ Z ðf 0 ðxÞ f 0 ðaÞÞðF ðxÞ GðxÞÞ dxX0: ¼
&
ð12Þ
I
Remark 6. Observe that, in view of (11), the integral in (12) is a proper one. In connection with Remark 1, it is also worth noting that, if X and f ðX Þ are integrable, and f 0 f 0 ðaÞ pivots on a 2 I; then the principal values in (5) actually are proper integrals. This follows from the fact that Z Z Z f 0 ðuÞF ðuÞ du ¼ ðf 0 ðuÞ f 0 ðaÞÞF ðuÞ du þ f 0 ðaÞ FðuÞ du, I a
I a
I a
and the analogous equality for I þ a: Denote by CðIÞ the set of all real continuous convex functions on I, and let C ðIÞ be the set of all functions in CðIÞ which are nondecreasing. It is well known that, if f 2 CðIÞ; then f 2 LðIÞ and f 0 is nondecreasing, so that, for each a 2 I; f 0 f 0 ðaÞ pivots on a. Also, if f 2 C ðIÞ; then f 0 is nonnegative. Therefore, Corollary 2 entails the following known result concerning the convex order and the increasing convex order for random variables (see Shaked and Shanthikumar, 1994, pp. 65 and 91). Corollary 3. Let X and Y be two integrable I-valued random variables with distribution functions F and G, respectively. If F G pivots on some a 2 I; and EX ¼ EY (resp. EX pEY ), then we have Ef ðY ÞXEf ðX Þ, for all f 2 CðIÞ (resp. for all f 2 C ðIÞ) such that f ðX Þ and f ðY Þ are integrable. It should also be observed that, if X :¼EY ; then F G obviously pivots on EY 2 I: Therefore, the preceding corollary includes Jensen’s inequality as a particular case.
References Brnetic´, I., Pecˇaric´, J., 2000. On an Ostrowski type inequality for a random variable. Math. Inequal. Appl. 3, 143–145. Dragomir, S.S., 2001a. A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to L1 ½a; b and applications in numerical integration. J. Comput. Anal. Appl. 3, 343–360. Dragomir, S.S., 2001b. A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to Lp ½a; b and applications in numerical integration. J. Math. Anal. Appl. 255, 605–626. Fink, A.M., 1992. Bounds on the deviation of a function from its averages. Czechoslovak Math. J. 42, 289–310. Matic´, M., Pecˇaric´, J., 2001. Two-point Ostrowski inequality. Math. Inequal. Appl. 2, 215–221. Ostrowski, A., 1938. U¨ber die Absolutabweichung einer differentierbaren Funktionen von ihren Integralmittelwort. Comment. Math. Helv. 10, 226–227. O¨zdemir, M.E., 2003. A theorem on mappings with bounded derivatives with applications to quadrature rules and means. Appl. Math. Comput. 138, 425–434. Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, Boston. Yang, X., 2003. Refinement of Ho¨lder inequality and application to Ostrowski inequality. Appl. Math. Comput. 138, 455–461.