Ostrowski’s inequality on time scales

Applied Mathematics Letters 21 (2008) 404–409 www.elsevier.com/locate/aml

Ostrowski’s inequality on time scales Cheh-Chih Yeh Department of Information Management, Lunghwa University of Science and Technology, Kueishan Taoyuan, 33306, Taiwan, ROC Received 5 March 2007; accepted 5 March 2007

Abstract A time scale version of Ostrowski’s inequality is given as follows: Let f, g ∈ Cr d ([a, b], R) be two linearly independent functions, then for any α ∈ [−1, 1] and any arbitrary x ∈ Cr d ([a, b], R) with Z b a

f (t)x(t)1t = 0,

Z b

(C1 )

g(t)x(t)1t = 1, a

the function αx(t) + (1 − α)y(t),

t ∈ [a, b]

satisfies condition (C1 ) and Z b Z b x 2 (t)1t ≥ [αx(t) + (1 − α)y(t)]2 1t ≥ a

a

A , AB − C 2

where Z b A= a

f 2 (t)1t,

Z b B= a

g 2 (t)1t,

Z b C= a

f (t)g(t)1t

and y(t) =

Ag(t) − B f (t) AB − C 2

on [a, b].

c 2007 Elsevier Ltd. All rights reserved.

Keywords: Time scales; Ostrowski’s inequality

1. Introduction The original renowned Ostrowski inequality [1,3,4,6,7] reads as follows:

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2007.03.024

C.-C. Yeh / Applied Mathematics Letters 21 (2008) 404–409

405

Theorem. Let a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) be two nonproportional sequences of real numbers and |a| =

n X

ai2 ,

|b| =

i=1

n X

bi2 ,

and

i=1

a·b =

n X

ai bi .

i=1

If x = (x1 , x2 , . . . , xn ) is any sequence of real numbers satisfying n X

ai xi = 0,

i=1

n X

bi xi = 1,

(C4 )

i=1

then n X

xi2 ≥

i=1

|a| |a||b| − |a · b|2

with equality if and only if xk =

bk |a| − ak |b| , |a||b| − |a · b|2

k = 1, 2, . . . , n.

The purpose of this note is to establish the time scale version of the above-mentioned Ostrowski inequality. We briefly introduce the time scale calculus and refer the reader to the books of Bohner and Peterson [2] and Lakshmikantham, Sivasundaram and Kaymakcalan [5] for further details. Definition 1.1. A time scale T is an arbitrary nonempty closed subset of the set R of all real numbers. Let T have the topology that it inherits from the standard topology on R. For t ∈ T, if t < sup T, we define the forward jump operator σ : T → T by σ (t) := inf{τ > t : τ ∈ T} ∈ T, while if t > inf T, the backward jump operator ρ : T → T is defined by ρ(t) := sup{τ < t : τ ∈ T} ∈ T. If σ (t) > t, we say t is right scattered, while if ρ(t) < t, we say t is left scattered. If σ (t) = t, we say t is right dense, while if ρ(t) = t, we say t is left dense. Throughout this work, we suppose that: (a) R = (−∞, +∞); (b) T is a time scale; (c) an interval means the intersection of a real interval with the given time scale. Definition 1.2. If f : T → R, then f σ : T → R is defined by f σ (t) = f (σ (t)) for all t ∈ T. Definition 1.3. A mapping f : T → R is called r d-continuous if it satisfies: (A) f is continuous at each right-dense point or maximal element of T, (B) the left-sided limit lims→t − f (s) = f (t − ) exists at each left-dense point t of T. Let Cr d (T, R) := { f | f : T → R is a r d-continuous function} and Tk :=



T − {m}, T,

if T has a left-scattered maximal point m, otherwise.

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C.-C. Yeh / Applied Mathematics Letters 21 (2008) 404–409

Definition 1.4. Assume x : T → R and fix t ∈ Tk , then we define x 1 (t) to be the number (provided it exists) with the property that given any  > 0, there is a neighborhood U of t such that |[x(σ (t)) − x(s)] − x 1 (t)[σ (t) − s]| < |σ (t) − s|, for all s ∈ U . We call x 1 (t) the delta derivative of x(t) at t. It can be shown that if x : T → R is continuous at t ∈ T and t is right scattered, then x 1 (t) =

x(σ (t)) − x(t) . σ (t) − t

Definition 1.5. A function F : T → R is an antiderivative of f : T → R if F 1 (t) = f (t) for all t ∈ Tk . In this case, we define the integral of f by Z t f (τ ) 1τ = F(t) − F(s) s

for s, t ∈ Tk . It follows from Theorem 1.74 of Bohner and Peterson [2] that every r d-continuous function has an antiderivative. 2. Main results We can state and prove our main result as follows. Theorem 1. Let f, g ∈ Cr d ([a, b], R) be two linearly independent functions and Z b Z b Z b A= f 2 (t)1t, B= g 2 (t)1t, C= f (t)g(t)1t. a

a

a

(A) If x ∈ Cr d ([a, b], R) is any function such that Z

b

b

Z

f (t)x(t)1t = 0,

g(t)x(t)1t = 1,

a

(C1 )

a

then Z

b

x 2 (t)1t ≥

a

A AB − C 2

(R1 )

with equality if and only if x(t) =

Ag(t) − B f (t) AB − C 2

a.e. on [a, b].

(C2 )

f (t) (B) If y(t) = Ag(t)−B on [a, b], then for any α ∈ [−1, 1] and any arbitrary x ∈ Cr d ([a, b], R) satisfying condition AB−C 2 (C1 ) , the function

αx(t) + (1 − α)y(t),

t ∈ [a, b]

satisfies condition (C1 ) and Z b Z b 2 x (t)1t ≥ [αx(t) + (1 − α)y(t)]2 1t ≥

A . AB − C 2 a a The first inequality in (R2 ) becomes equality if and only if Z b Z b |α| = 1, or x 2 (t)1t = y 2 (t)1t. a

a

The second inequality in (R2 ) becomes equality if and only if α = 0,

or

x(t) = y(t)

a.e. on [a, b].

(R2 )

407

C.-C. Yeh / Applied Mathematics Letters 21 (2008) 404–409 f (t) Proof. (A) Clearly, AB − C 2 > 0. Let y(t) = Ag(t)−C for t ∈ [a, b]. Then y(t) satisfies (C1 ). In fact, AB−C 2 Z b Z b 1 f (t)y(t)1t = [A f (t)g(t) − C f 2 (t)]1t = 0 2 AB − C a a

and b

Z a

1 g(t)y(t)1t = AB − C 2

b

Z

[Ag 2 (t) − C f (t)g(t)]1t = 1.

a

For any x(t) ∈ Cr d ([a, b], R) satisfying (C1 ), we have Z b Z b 1 x(t)y(t)1t = [Ag(t)x(t) − C f (t)x(t)]1t AB − C 2 a a Z b A = = y 2 (t)1t, AB − C 2 a and moreover, Z b Z 2 [x(t) − y(t)] 1t = a

b

Z

x (t)1t − 2 2

Z

b

x 2 (t)1t −

a b

Z

a

b

Z x(t)y(t)1t +

a

=

b

(1)

y 2 (t)1t

a

y 2 (t)1t ≥ 0.

(2)

a

It follows from (1) and (2) that Z b A . x 2 (t)1t ≥ AB − C 2 a Thus, we obtain the desired result (R1 ). It follows from (2) that equality holds in (R1 ) if and only if x(t) = y(t) a.e. on [a, b], i.e., if and only if (C2 ) holds. Thus, we complete the proof of (A). (B) It follows from (1) and (2) and α ∈ [−1, 1] that Z b Z b Z b x 2 (t)1t ≥ α 2 x 2 (t)1t + (1 − α 2 ) y 2 (t)1t a

a

Z

a b

=

α x (t)1t + [(1 − α) + 2α(1 − α)] 2 2

2

a

Z

b

α x (t)1t + 2α(1 − α) 2 2

b

Z

y 2 (t)1t

x(t)y(t)1t + (1 − α)

2

a

a

Z

b

a

= =

Z

b

Z

b

y 2 (t)1t

a

[αx(t) + (1 − α)y(t)]2 1t.

a

This completes the proof of the first inequality in (R2 ). Next, Z b Z b [αx(t) + (1 − α)y(t)]2 1t = [α 2 x 2 (t) + (1 − 2α + α 2 )y 2 (t) + 2α(1 − α)x(t)y(t)]1t a

a

=α

2

Z

b

[x 2 (t) − 2x(t)y(t) + y 2 (t)]1t

a

+ (1 − 2α)

Z

b

y (t)1t + 2α 2

a

= α2

Z

b

b

x(t)y(t)1t a

[x(t) − y(t)]2 1t +

a

This completes the proof of the second inequality in (R2 ).

Z



A A ≥ . 2 AB − C AB − C 2

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C.-C. Yeh / Applied Mathematics Letters 21 (2008) 404–409

Taking T := R in Theorem 1, we have the following continuous version of Ostrowski’s inequality: Corollary 2. Let f, g ∈ C([a, b], R) be two nonproportional functions and Z b Z b Z b 2 2 f (t)g(t)dt. g (t)dt, C∗ = f (t)dt, B∗ = A∗ = a

a

a

(A∗ ) If x ∈ C([a, b], R) is an arbitrary function such that Z b Z b g(t)x(t)dt = 1, f (t)x(t)dt = 0,

(C3 )

a

a

then b

Z

x 2 (t)dt ≥

a

A∗ A∗ B∗ − C∗2

with equality if and only if x(t) =

A∗ g(t) − B∗ f (t) A∗ B∗ − C∗2

a.e. on [a, b].

f (t) (B∗ ) If the function y(t) = Ag(t)−B on [a, b], then for any α ∈ [−1, 1] and any arbitrary x ∈ C([a, b], R) A∗ B∗ −C∗2 satisfying condition (C3 ) , the function

αx(t) + (1 − α)y(t),

t ∈ [a, b]

satisfies condition (C3 ) and Z b Z b x 2 (t)dt ≥ [αx(t) + (1 − α)y(t)]2 dt ≥ a

a

A∗ . A∗ B∗ − C∗2

Taking T := Z (the set of all integers) in Theorem 1, we have the following Ostrowski inequality, see [4, p. 166], [6] and [7]. Corollary 3. Let a = (a1 , a2 , . . . , an ) and b = (b1 , b2 , . . . , bn ) be two nonproportional sequences of real numbers and n n n X X X |a| = ai2 , |b| = bi2 , and a · b = ai bi . i=1

i=1

i=1

(A∗∗ ) If x = (x1 , x2 , . . . , xn ) is any sequence of real numbers satisfying n X

ai xi = 0,

i=1

n X

bi xi = 1,

(C4 )

i=1

then n X

xi2 ≥

i=1

|a| |a||b| − |a · b|2

with equality if and only if xk =

bk |a| − ak |b| , |a||b| − |a · b|2

k = 1, 2, . . . , n.

bk |a|−ak |b| (B∗∗ ) If yk (t) = |a||b|−|a·b| 2 , k = 1, 2, . . . , n, then for any α ∈ [−1, 1] and real numbers x 1 , x 2 , . . . , x n satisfying (C4 ) , the numbers

αxi + (1 − α)yi ,

i = 1, 2, . . . , n

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C.-C. Yeh / Applied Mathematics Letters 21 (2008) 404–409

satisfy (C4 ) and n n X X xi2 ≥ [αxi + (1 − α)yi ]2 ≥ i=1

i=1

|a| . |a||b| − |a · b|2

Similarly, we can extend the preceding results to complex functions. In this case, Theorem 1 reads as follows: Theorem 4. Let f, g ∈ Cr d ([a, b], C) be two linearly independent functions and Z b Z b Z b f (t)g(t)1t, ¯ A∗ = f (t) f¯(t)1t, B∗ = g(t)g(t)1t, ¯ C∗ = a

a

a

where C is the set of all complex numbers. Then the following two statements hold: (A) If x ∈ Cr d ([a, b], C) is any function such that Z b Z b g(t)x(t)1t ¯ = 1, f (t)x(t)1t ¯ = 0,

(C5 )

a

a

then b

A∗ A∗ B ∗ − |C ∗ |2 a with equality if and only if Z

x(t)x(t)1t ¯ ≥

x(t) =

A∗ g(t) − B ∗ f (t) A∗ B ∗ − |C ∗ |2

a.e. on [a, b].

f (t) (B) If y(t) = AAg(t)−B on [a, b], then for any α ∈ [−1, 1] and any arbitrary x ∈ Cr d ([a, b], C) satisfying ∗ B ∗ −|C ∗ |2 condition (C5 ) , the function ∗

∗

h(t) := αx(t) + (1 − α)y(t),

t ∈ [a, b]

satisfies condition (C5 ) and Z b Z b ¯ x(t)x(t)1t ¯ ≥ h(t)h(t)1t ≥

A∗ . − |C ∗ |2 a a The first inequality in (R3 ) becomes equality if and only if Z b Z b |α| = 1, or x(t)x(t)1t ¯ = y(t) y¯ (t)1t. a

A∗ B ∗

(R3 )

a

The second inequality in (R3 ) becomes equality if and only if α = 0,

or

x(t) = y(t)

a.e. on [a, b].

References [1] M. Bjelica, Refinements of Ostrowski’s and Fan–Todd’s inequalities, in: G.V. Milovanovi´c (Ed.), Recent Progress in Inequalities, Klumer Academic Publishers, Dordrecht, Boston, London, 1998, pp. 445–448. [2] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkh¨auser, Boston, Basel, Berlin, 2001. [3] K. Hu, Some Problems of Analytic Inequalities, Wuhan University Press, 2003. [4] J.C. Kuang, Applied Inequalities, 3rd ed., Shandong Science and Technology, Shandong, China, 2003 (in Chinese). [5] V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Dordrecht, Boston, London, 1996. [6] D.S. Mitrinovi´c, Analytic Inequalities, Springer-Verlag, New York, Heidelberg, Berlin, 1970. [7] A.M. Ostrowski, Vorlesungen¨uber Differential-und Integralrechnung II, Basel, 1951.