Ostrowski Type Inequalities From A Linear Functional Point of ... - EMIS

Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 1, Issue 2, Article 11, 2000

OSTROWSKI TYPE INEQUALITIES FROM A LINEAR FUNCTIONAL POINT OF VIEW B. GAVREA AND I. GAVREA U NIVERSITY BABE S¸ -B OLYAI C LUJ -NAPOCA , D EPARTMENT OF M ATHEMATICS AND C OMPUTERS , S TR . ˘ M IHAIL KOG ALNICEANU 1, 3400 C LUJ -NAPOCA , ROMANIA [email protected] T ECHNICAL U NIVERSITY C LUJ -NAPOCA , D EPARTMENT OF M ATHEMATICS , S TR . C. DAICOVICIU 15, 3400 C LUJ -NAPOCA , ROMANIA [email protected]

Received 13 Septemer, 1999; accepted 31 January, 2000 Communicated by P. Cerone

A BSTRACT. Inequalities are obtained using P0 -simple functionals. Applications to Lipschitzian mappings are given. Key words and phrases: P0 -simple functionals, inequalities, Lipschitz mappings. 2000 Mathematics Subject Classification. 26D15.

1. I NTRODUCTION Let I be a bounded interval of the real axis. We denote by B(I) the set of all functions which are bounded on [a, b]. Let A be a positive linear functional A : B(I) → R, such that A(e0 ) = 1, where ei : I → R, ei (x) = xi , ∀ x ∈ I, i ∈ N. The following inequality is known in literature as the Grüss inequality for the functional A. Theorem 1.1. Let f, g : I → R be two bounded functions such that m1 ≤ f (x) ≤ M1 and m2 ≤ g(x) ≤ M2 for all x ∈ I, m1 , M1 , m2 and M2 are constants. Then the inequality: 1 (1.1) |A(f g) − A(f )A(g)| ≤ (M1 − m1 )(M2 − m2 ) 4 holds. In 1938 Ostrowski (cf. for example [7, p. 468]) proved the following result: Theorem 1.2. Let f : I → R be continuous on (a, b) whose derivative f 0 : (a, b) → R is bounded on (a, b), i.e. kf 0 k∞ := sup |f 0 (t)| < ∞. t∈(a,b) ISSN (electronic): 1443-5756 c 2000 Victoria University. All rights reserved.

002-99

2

B. G AVREA AND I. G AVREA

Then # " Z b a+b 2 x − 1 1 2 f (x) − (b − a)kf 0 k∞ f (t)dt ≤ + b−a a 4 (b − a)2

(1.2)

for all x ∈ [a, b]. The constant 41 is best. In the recent paper [4] S.S. Dragomir and S. Wang proved the following version of Ostrowski’s inequality. Theorem 1.3. Let f : I → R be a differentiable mapping in the interior of I and a, b ∈ int(I) with a < b. If f 0 ∈ L1 [a, b] and γ ≤ f 0 (x) ≤ Γ for all x ∈ [a, b] then we have the following inequality: Z b 1 f (b) − f (a) a + b 1 ≤ (b − a)(Γ − γ) f (x) − f (t)dt − x − (1.3) 4 b−a b−a 2 a

for all x ∈ [a, b]. The following inequality for mappings with bounded variation can be found in [1]: Theorem 1.4. Let f : I → R be a mapping of bounded variation. Then for all x ∈ [a, b] we have the inequality Z b b a + b _ 1 f, (1.4) f (t)dt − f (x)(b − a) ≤ (b − a) + x − 2 2 a a where

b W

f denotes the total variation of f .

a

The constant 21 is the best possible one. In [2] S.S. Dragomir gave the following result for Lipschitzian mappings: Theorem 1.5. Let f : [a, b] → R be an L-Lipschitzian mapping on [a, b], i.e. |f (x) − f (y)| ≤ L|x − y|, for all x, y ∈ [a, b]. Then we have the inequality " # Z b a+b 2 x − 1 2 (1.5) f (t)dt − f (x)(b − a) ≤ L(b − a)2 + 2 4 (b − a) a for all x ∈ [a, b]. The constant 14 is the best possible one. S.S. Dragomir, P. Cerone, J. Roumeliotis and S. Wang in [3] proved the following theorem: Theorem 1.6. Let f, w : (a, b) ⊆ R → R be so that w(s) ≥ 0 on (a, b), w is integrable on Rb (a, b) and a w(s)ds > 0, f is of r-Hölder type, i.e. (1.6)

|f (x) − f (y)| ≤ H|x − y|r , for all x, y ∈ (a, b)

where H > 0 and r ∈ (0, 1] are given. If w, f ∈ L1 (a, b), then we have the inequality: Z b Z b 1 1 (1.7) w(s)f (s)ds ≤ H R b |x − s|r w(s)ds f (x) − R b w(s)ds a w(s)ds a a a for all x ∈ (a, b). The constant factor 1 in the right hand side cannot be replaced by a smaller one. The aim of this paper is to improve the results from Theorems 1.1 – 1.6 using an unitary method.

J. Ineq. Pure and Appl. Math., 1(2) Art. 11, 2000

http://jipam.vu.edu.au/

O STROWSKI T YPE I NEQUALITIES FROM A L INEAR F UNCTIONAL P OINT OF V IEW

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2. AUXILIARY RESULTS Let X = (X, d) be a compact metric space and C(X) the Banach lattice of real-valued continuous functions on the compact metric space X = (X, d), endowed with the max norm k · kX . For a function f ∈ C(X), the modulus of continuity (with respect to the metric d) is defined by: ω(f ; t) = ωd (f ; t) = sup |f (x) − f (y)|, t ≥ 0. d(x,y)≤t

The least concave majorant of this modulus with respect to the variable t is given by  ;x)  sup (t−x)ω(f ;y)+(y−t)ω(f for 0 ≤ t ≤ d(X);  y−x  0≤x≤t≤y x6=y ω e (f ; t) =    ω(f ; d(X)) for t > d(X), where d(X) < ∞ is the diameter of the compact space X. We denote by LipM α = LipM (α; X) the set of all Lipschitzian functions of order α, α ∈ [0, 1] having the same Lipschitz constant M . That is f ∈ LipM α iff for all x, y ∈ X |f (x) − f (y)| ≤ M dα (x, y). We see that LipM (α; X) = {g ∈ C(X) : ω(g; t) ≤ M tα }. Let I = [a, b] be a compact interval of the real axis, S a subspace of C(I), and A a linear functional defined on S. The following definition was given by T. Popoviciu in [8]. Definition 2.1. The linear functional A defined on the subspace S which contains all polynomials is Pn -simple (n ≥ −1) if (i) A(en+1 ) 6= 0 (ii) for every f ∈ S there are the distinct points t1 , t2 , . . . , tn+2 in [a, b] such that A(f ) = A(en+1 )[t1 , t2 , . . . , tn+2 ; f ], where [t1 , t2 , . . . , tn+2 ; f ] is the divided difference of the function f on the points t1 , t2 , . . . , tn+2 . In [5] the following result is proved. The proof is reproduced here for completeness. Theorem 2.1. Let A be a bounded linear functional, A : C(I) → R. If A is P0 -simple then for all f ∈ C(I) we have kAk 2|A(e1 )| (2.1) |A(f )| ≤ ω e f; . 2 kAk Proof. For g ∈ C 1 (I) we have |A(f )| = |A(f − g) + A(g)| ≤ kAkkf − gk + |A(g)| ≤ kAkkf − gk + |A(e1 )|kg 0 k. From this inequality we obtain |A(f )| ≤

inf (kAkkf − gk + |A(e1 )|kg 0 k)

g∈C 1 (I)

and using the following result (see [10]) t 0 1 inf1 kf − gk + kg k = ω e (f ; t), g∈C (I) 2 2 we obtain the relation (2.1).


t≥0


4


The following result was proved by I. Ra¸sa [9]. Theorem 2.2. Let k be a natural number such that 0 ≤ k ≤ n and A : C (k) [a, b] → R a bounded linear functional, A 6= 0, A(ei ) = 0 for i = 0, 1, . . . , n such that for every f ∈ C (k) [a, b] Pn -nonconcave A(f ) ≥ 0. Then A is Pn -simple. A function f ∈ C (k) [a, b] is called P0 -nonconcave if for any n + 2 points t1 , t2 , . . . , tn+2 ∈ [a, b] the inequality [t1 , t2 , . . . , tn+2 ; f ] ≥ 0 holds. Another criterion for Pn -simple functionals was given by A. Lupa¸s in [6]. He proved that a bounded linear functional A : C[a, b] → R for which A(ek ) = 0, k = 0, 1, . . . , n and A(en+1 ) 6= 0 is Pn -simple if and only if A is Pn -simple on C (n+1) [a, b]. Now we can prove the following result (see also [5]): Theorem 2.3. Let A be a bounded linear functional, A : C(I) → R. If A(e1 ) 6= 0 and the inequality (2.1) holds for any f ∈ C(I) then A is P0 -simple. Proof. We can assume that A(e1 ) > 0. Combining the results of I. Ra¸sa and A. Lupa¸s, it is sufficient, for the proof of the theorem, to show that A(f ) ≥ 0

(2.2)

for every nondecreasing differentiable function f defined on I. For such a function we have |A(f )| ≤ A(e1 )kf 0 k. Let B be the linear functional defined by B(f ) = where Z F (t) =

A(F ) , A(e1 )

t

f (u)du,

f ∈ C[0, 1].

0

The functional B is bounded and for any f ∈ C(I) we have |B(f )| ≤ kf k with B(e0 ) = 1. Let f be a continuous function such that f ≥ 0, f 6= 0. From the inequalities f 0 ≤ e0 − ≤1 kf k we obtain B(f ) f 1− ≤ B e0 − ≤ 1. kf k kf k These inequalities imply that (2.3)

B(f ) ≥ 0.

Further, let f be a differentiable function on I such that f 0 ≥ 0, then, from (2.3) we obtain B(f 0 ) ≥ 0. Since B(f 0 ) = A(f ), the inequality (2.2) is thus proved.




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3. A N I NTEGRAL I NEQUALITY OF O STROWSKI T YPE The following inequality of Ostrowski type holds. Theorem 3.1. Let f be a continuous function on [a, b] and w : (a, b) → R+ an integrable Rb function on (a, b) such that a ω(s)ds = 1. Then for any continuous function f the following inequality: Z x Rx Z b w(t)(x − t)dt a Rx (3.1) f (x) − w(s)f (s)ds ≤ w(t)dt ω e[a,x] f ; w(t) a a a ! Rb Z b w(t)(t − x)dt + w(t)dt ω e[x,b] f ; x R b w(t)dt x x holds, where x is a fixed point in (a, b). Proof. From Theorem 2.3 we get that the linear functionals A1 : C[a, x] → R, defined by

A2 : C[x, b] → R

x

Z

x

Z w(t)dt −

A1 (f ) = f (x)

f (t)w(t)dt

a

a

and

b

Z

b

Z w(t)dt −

A2 (f ) = f (x)

f (t)w(t)dt

x

x

are P0 -simple. It is easy to see that: x

Z kA1 k = 2

Z w(t)dt

and

kA2 k = 2

a

From the inequality: Z Z b f (x) − w(s)f (s)ds ≤

w(t)dt. x

x

a

a

b

! Z b |A1 (e1 ) A2 (e1 ) w(t)dt ω e f; R x + w(t)dt ω e f; R b w(t)dt w(t)dt x a x

and from the results Z |A1 (e1 )| =

x

Z w(t)(x − t)dt

and

|A2 (e1 )| =

a

b

w(t)(t − x)dt, x

(3.1) follows.

Corollary 3.2. Let f be a continuous function on [a, b], such that f ∈ LipM1 (α, [a, x]) and f ∈ LipM2 (β; [x, b]). Then Z x 1−α Z x α Z b (3.2) f (x) − w(s)f (s)ds ≤ M1 w(t)dt w(t)(x − t)dt a

a

a

Z + M2

b

1−β Z b β w(t)dt w(t)(t − x)dt .

x

x

Proof. The proof follows from the inequality (3.1) and the fact that ω e1 (g; t) ≤ M tr for any function g, g ∈ LipM (α, [c, d]), where ω e1 is taken on the interval [c, d].

Corollary 3.2 is an improvement of the result of Theorem 1.6.



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1 Remark 3.1. In the particular case when w(t) = b−a the inequality (3.2) becomes: R b f (s)ds (x − a)α+1 (b − x)β+1 1 a (3.3) + M2 f (x) − ≤ M1 α β b−a 2 2 b−a # " 2 x − a+b 1 2 ≤ max(M1 , M2 ) + (b − a). 4 (b − a)2

Inequality (3.3) improves the inequality (1.5). Corollary 3.3. Let f : [a, b] → R be continuous on (a, b), whose derivative f 0 : (a, b) → R is bounded on (a, b) and w a function as in Theorem 3.1. Then we have the following inequality: Z x Z b Z b (3.4) w(s)f (s)ds ≤ w(t)(x − t)dt + w(t)(t − x)dt kf 0 k∞ . f (x) − a

a

x

Proof. The above inequality is a consequence of the inequality (3.1) and the fact that ω e (f ; t) ≤ kf 0 k∞ t. The inequality of Ostrowski follows from (3.4) if we consider 1 w(t) = , t ∈ [a, b]. b−a Corollary 3.4. Let f : I → R be a mapping with bounded variation and w a function as in Theorem 3.1. Then for all x ∈ [a, b] we have the inequalities Z b Z x Z b x b _ _ (3.5) w(s)f (s)ds ≤ f w(t)dt + f w(t)dt f (x) − a

(3.6)

a

a

 Z b f (x) − ≤ 1 + w(s)f (s)ds 2 a

x

x

R  Rb x b w(t)dt − w(t)dt a _ x  f. 2 a

Proof. It is clear that (3.7)

ω e [a, x](f ; t) ≤

x _ a

f

and

ω e [x, b](f, t) ≤

b _

f

x

for every positive number t. Thus, inequality (3.5) follows from (3.7). Rb Rx For the proof of the inequality (3.6) we note that, if we suppose a w(t)dt ≤ 12 then x w(t)dt ≥ 1 and vice versa. 2 For definiteness we assume that Z x Z b 1 1 w(t)dt ≤ and w(t)dt ≥ . 2 2 a x We then have Z x Z b Z b x b x b _ _ _ 1_ f w(t)dt + f w(t)dt ≤ f+ f w(t)dt 2 a a x x a x x Z b b b _ 1_ 1 f+ f w(t)dt − = 2 a 2 x x




7

and so (3.8)

Z x _ f a

x

Z b b _ w(t)dt + f w(t)dt ≤

a

x

x

1 + 2

Rb x

w(t)dt − 2

Rx a

w(t)dt

!

b _

f.

a

From the inequalities (3.5) and (3.8), the inequality (3.6) follows.

Remark 3.2. The inequality from Theorem 1.4 follows if we take in (3.6) 1 w(t) = . b−a Theorem 3.5. Let g be a continuous differentiable function on [a, b] such that g(a) = g(b) = 0. Then the inequality Z b 3 3 g(x) (x − a)2 + (b − x)2 1 2 (x − a) + (y − b) 0 (3.9) g(t)dt ≤ ω e g; 2 − b−a 8(b − a) 3 (x − a)2 + (y − b)2 a

holds, where x is an arbitrary point in (a, b). Proof. The following functional A defined on C[a, b] by Z b a+b 1 t− f (t)dt A(f ) = b−a a 2 is a linear bounded functional having its norm equal to b−a . For every increasing function f we 4 have: A(f ) ≥ 0. Using Theorem 2.3, we deduce that the functional A is P0 -simple with (b − a)2 . 12 From Theorem 2.1, we obtain the following inequality: Z b 1 b−a a+b 2 (3.10) t− f (t)dt ≤ ω e f ; (b − a) . b − a 2 8 3 A(e1 ) =

a

Inequality (3.10) holds for every continuous function f . Let us suppose that f is differentiable on [a, b]. From the inequality (3.10) (written for f 0 ) we obtain the following inequality: Z b 1 b−a f (a) + f (b) 2 0 ≤ (3.11) f (t)dt − ω e f ; (b − a) . b − a 2 8 3 a

Now, we can prove the inequality (3.9). We have the following identity: Z b Z x g(x) 1 x−a 1 g(a) + g(x) (3.12) − + g(t)dt = g(t)dt − 2 b−a a b−a x−a a 2 Z b b−x 1 g(b) + g(x) + g(t)dt − . b−a b−x a 2 Using the relations (3.11) and (3.12) we obtain Z b g(x) 1 (3.13) − g(t)dt 2 b−a a (x − a)2 (b − x)2 0 2 0 2 ≤ ω e g ; (x − a) + ω e g ; (b − x) . 8(b − a) 3 8(b − a) 3



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As the function ω e (g 0 ; ·), is concave, then from (3.13) and using Jensen’s inequality, we obtain the inequality (3.9). Corollary 3.6. Let g be a continuous differentiable function on [a, b] such that g(a) = g(b) = 0, then the following inequality # " Z b a+b 2 g(x) x − 1 1 2 (3.14) (b − a)kg 0 k∞ g(t)dt ≤ + 2 − b−a 8 2(b − a) a is valid for all x ∈ [a, b]. Proof. It is well known that (3.15)

ω e (g 0 ; t) ≤ 2kg 0 k∞ ,

for every positive number t. The inequality (3.15) then readily follows from the inequality (3.14).

Remark 3.3. The result from the Theorem 1.3 can be written in terms of ω e using the inequality (3.13) for the function x−a b−x f (b) − f (a). g(x) = f (x) − b−a b−a In [5] the following result was proved: Let A be a linear positive functional A : C[0, 1] → R, A(e0 ) = 1 and ϕ, ϕ : [0, 1] → R a continuous increasing function such that A(e1 ϕ) − A(e1 )A(ϕ) > 0. Then the following Grüss type inequality A(|ϕ − A(ϕ)|) 2(A(e1 ϕ) − A(e1 )A(ϕ)) (3.16) |A(ϕf ) − A(ϕ)A(f )| ≤ ω e f; 2 A(|ϕ − A(ϕ)|) holds. We are interested in the following open problem: Open problem. Let A be a linear positive functional defined on C[0, 1] and f, g be two continuous functions. Do positive numbers δ1 = δ1 (f ) < 1 and δ2 = δ2 (f ) < 1 exist such that 1 |A(f g) − A(f )A(g)| ≤ ω e (f ; δ1 )e ω (f, δ2 )? 4 R EFERENCES [1] S.S. DRAGOMIR, On Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Inequal. Appl. (in press). [2] S.S. DRAGOMIR, On the Ostrowski integral inequality for Lipschitzian mappings and applications, Comput. Math. Appl., 38 (1999), 33–37. [3] S.S. DRAGOMIR, P. CERONE, J. ROUMELIOTIS AND S. WANG, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math. Soc. Sc. Math. Roumanie, 42(90)(4), (1999), 301–314. [4] S.S. DRAGOMIR AND S. WANG, An Inequality of Ostrowski-Grüss’ type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Comput. Math. Appl., 33(11) (1997), 15–20. [5] I. GAVREA, Preservation of Lipschitz constants by linear transformations and global smoothness preservation (submitted). [6] A. LUPAS, ¸ Contribu¸tii la teoria aproxim˘arii prin operatori liniari, Ph. D. Thesis, Cluj-Napoca, Babe¸s-Bolyai University, 1975.




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´ J.E. PECARI ˇ [7] D.S. MITRINOVIC, C´ AND A.M. FINK, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers. [8] T. POPOVICIU, Sur le reste dans certains formules lineaires d’approximation de l’analyse, Mathematica, Cluj., 1(24), 95–142. [9] I. RASA, ¸ Sur les fonctionnelles de la forme simple au sens de T. Popoviciu, L’Anal. Num. et la Theorie de l’Approx., 9 (1980), 261–268. [10] E.M. SEMENOV AND B.S. MITJAGIN, Lack of interpolation of linear operators in spaces of smooth functions, Math. USSR-Izv., 11 (1977), 1229–1266.

