Acta Appl Math (2010) 110: 283–288 DOI 10.1007/s10440-008-9407-z
Ostrowski Type Inequality for Double Integrals on Time Scales Umut Mutlu Özkan · Hüseyin Yildirim
Received: 26 October 2008 / Accepted: 2 December 2008 / Published online: 12 December 2008 © Springer Science+Business Media B.V. 2008
Abstract We prove the Ostrowski type inequality for double integrals on time scales and thus unify corresponding continuous and discrete versions from the literature. We also apply the Ostrowski inequality for double integrals to the quantum time scales. Keywords Ostrowski type inequality · Double integral · Time scales Mathematics Subject Classification (2000) 26D15 · 39A10
1 Introduction In 1938, the classical integral inequality established by Ostrowski [9] can be stated as follows. Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b) whose derivative f : (a, b) → R is bounded on (a, b), i.e., f ∞ := supt∈(a,b) |f (t)| < ∞, then b )2 1 (x − a+b 2 f (x) − 1 f (t)dt ≤ + (b − a)f ∞ b−a a 4 (b − a)2 for all x ∈ [a, b]. Many authors have studied the theory of certain integral inequalities on time scales. For example, we refer the reader to [4, 6–8]. In [10] Pachpatte proved the Ostrowski type inequality for double integrals. The aim of this paper is to establish Pachpatte’s inequality on arbitrary time scales.
U.M. Özkan () · H. Yildirim Department of Mathematics, Faculty of Science and Arts, Kocatepe University, 03200 Afyon, Turkey e-mail:
[email protected] H. Yildirim e-mail:
[email protected]
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We first briefly introduce the time scales theory. By a time scale T we mean any closed subset of R with order and topological structure present in canonical way. Since a time scale T may or may not be connected, we need the concept of jump operators. Let t ∈ T, where T is a time scale; then two mappings σ, ρ : T → T satisfying σ (t) = inf{s ∈ T : s > t},
ρ(t) = sup{s ∈ T : s < t}
are called the jump operators. If σ (t) > t , t ∈ T, we say t is right-scattered. If ρ(t) < t , t ∈ T, we say t is left-scattered. If σ (t) = t , t ∈ T, we say t is right-dense. If ρ(t) = t , t ∈ T, we say t is left-dense. A mapping f : T → R is called rd-continuous if (i) f is continuous at each right-dense point or maximal point of T; (ii) at each left-dense point t ∈ T, lim g(s) = g(t − )
s→t −
exists. The set of all rd-continuous functions from T → R is denoted by Crd (T, R). Let T−{m}, if T has a left-scattered maximal point m, Tκ = T, otherwise. If f : T → R is a function, then we define the function f σ : T → R by f σ (t) = f (σ (t)) for all t ∈ T, i.e., f σ = f ◦ σ . Assume that f : T → R and t ∈ Tκ , then we define f (t) to be the number (provided it exists) with the property that for any given any ε > 0, there is a neighborhood U of t such that f (σ (t)) − f (s) − f (t)[σ (t) − s] ≤ ε |σ (t) − s| for all s ∈ U . In this case f (t) is called the delta derivative of f (t) at t . If f is differentiable at each t ∈ T, then f is called delta differentiable on T. A function F : T → R is called an antiderivative of f : T → R if F (t) = f (t) for all t ∈ Tκ , and in this case, we define the integral of f by
b
f (t)t = F (b) − F (a) a
for all a, b ∈ T, and we say that f is integrable on T. Also let us recall some essentials about partial derivatives on time scales: Let T1 and T2 be two time scales. For i = 1, 2 let σi , ρi and i denote the forward jump operator, the backward jump operator, and the delta differentiation operator, respectively, on Ti . Suppose a < b are points in T1 , c < d are points in T2 , [a, b) is the half-closed bounded interval in T1 , and [c, d) is the half-closed bounded interval in T2 . Let us introduce a “rectangle” in T1 × T2 by R = [a, b) × [c, d) = {(t1 , t2 ) : t1 ∈ [a, b), t2 ∈ [c, d)} . Let f be a real-valued function on T1 × T2 . At (t1 , t2 ) ∈ T1 × T2 we say that f has a “1 partial derivative” f 1 (t1 , t2 ) (with respect to t1 ) if for each ε > 0 there exists a
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neighborhood Ut1 (open in the relative topology of T1 ), of t1 such that f (σ1 (t1 ), t2 ) − f (s, t2 ) − f 1 (t1 , t2 )(σ1 (t1 ) − s) ≤ ε |σ1 (t1 ) − s| for all s ∈ Ut1 . At (t1 , t2 ) ∈ T1 × T2 we say that f has a “2 partial derivative” f 2 (t1 , t2 ) (with respect to t2 ) if for each η > 0 there exists a neighborhood Ut2 , of t2 such that f (t1 , σ2 (t2 )) − f (t1 , l) − f 2 (t1 , t2 )(σ2 (t2 ) − l) ≤ η |σ2 (t2 ) − l| for all l ∈ Ut2 . Let f be a real-valued function on T1 × T2 . The function f is called rd-continuous in t2 if for every α1 ∈ T1 , the function f (α1 , t2 ) is rd-continuous on T2 . The function f is called rd-continuous in t1 if for every α2 ∈ T2 , the function f (t1 , α2 ) is rd-continuous on T1 . Let CCrd denote the set of functions f (t1 , t2 ) on T1 × T2 with the properties • f is rd-continuous in t1 , • f is rd-continuous in t2 , • if (x1 , x2 ) ∈ T1 × T2 with x1 right-dense or maximal and x2 right-dense or maximal, then f is continuous at (x1 , x2 ), • if x1 and x2 are both left-dense, then the limit of f (t1 , t2 ) exists as (t1 , t2 ) approaches (x1 , x2 ) along any path in the region RLL (x1 , x2 ) = {(t1 , t2 ) : t1 ∈ [a, x1 ] ∩ T1 , t2 ∈ [c, x2 ] ∩ T2 } . 1 Let CCrd be the set of all functions in CCrd for which both the 1 partial derivative and the 2 partial derivative exist and are in CCrd . We refer the reader to [5] for a comprehensive development of the calculus of the derivative and we refer the reader to [1–3] for an account of the calculus of the partial derivative and double integral.
2 Main Results Throughout this section, we suppose that (a) T1 is a time scale, a < b are points in T1 , (b) T2 is a time scale, c < d are points in T2 , (c) R is a rectangle in T1 × T2 defined by R = [a, b) × [c, d) = {(s, t) : s ∈ [a, b), t ∈ [c, d)} . Our main result is given in the following theorem. 1 Theorem 2.1 Let f, g ∈ CCrd ([a, b] × [c, d], R). Then b d 1 1 f (x, y)g(x, y) − g(x, y) f (σ1 (s), y)1 s + f (x, σ2 (t))2 t 2 b−a a d −c c b d 1 − f (σ1 (s), σ2 (t))2 t1 s (b − a)(d − c) a c b d f (x, y) 1 1 − g(σ1 (s), y)1 s + g(x, σ2 (t))2 t 2 b−a a d −c c
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b d 1 g(σ1 (s), σ2 (t))2 t1 s (b − a)(d − c) a c x y b d ∂ 2 f (α, β) 1 2 β1 α |g(x, y)| ≤ 2(b − a)(d − c) a c σ1 (s) σ2 (t) 2 β1 α x y 2 ∂ g(α, β) 2 β1 α 2 y1 x + |f (x, y)| σ1 (s) σ2 (t) 2 β1 α −
(2.1)
for any (x, y) ∈ [a, b] × [c, d]. Proof From the integration by parts formula for double integrals on time scales [1, p. 37], we have the following identities
x
y
f (x, y) − f (σ1 (s), y) − f (x, σ2 (t)) + f (σ1 (s), σ2 (t)) = σ1 (s)
σ2 (t)
∂ 2 f (α, β) 2 β1 α 2 β1 α (2.2)
and
x
y
∂ 2 g(α, β) 2 β1 α σ1 (s) σ2 (t) 2 β1 α (2.3) for (x, y), (s, t) ∈ [a, b] × [c, d]. Multiplying (2.2) by g(x, y), (2.3) by f (x, y), and adding the resulting identities, we have g(x, y) − g(σ1 (s), y) − g(x, σ2 (t)) + g(σ1 (s), σ2 (t)) =
2f (x, y)g(x, y) − g(x, y)[f (σ1 (s), y) + f (x, σ2 (t)) − f (σ1 (s), σ2 (t))] − f (x, y)[g(σ1 (s), y) + g(x, σ2 (t)) − g(σ1 (s), σ2 (t))] x y x y ∂ 2 f (α, β) ∂ 2 g(α, β) 2 β1 α + f (x, y) 2 β1 α. = g(x, y) σ1 (s) σ2 (t) 2 β1 α σ1 (s) σ2 (t) 2 β1 α (2.4) Integrating (2.4) on R, we get b d g(x, y) 1 1 f (σ1 (s), y)1 s + f (x, σ2 (t))2 t 2 b−a a d −c c b d 1 − f (σ1 (s), σ2 (t))2 t1 s (b − a)(d − c) a c b d f (x, y) 1 1 − g(σ1 (s), y)1 s + g(x, σ2 (t))2 t 2 b−a a d −c c b d 1 − g(σ1 (s), σ2 (t))2 t1 s (b − a)(d − c) a c b d x y 1 ∂ 2 f (α, β) = 2 β1 α g(x, y) 2(b − a)(d − c) a c σ1 (s) σ2 (t) 2 β1 α x y ∂ 2 g(α, β) (2.5) 2 β1 α 2 y1 x. + f (x, y) σ1 (s) σ2 (t) 2 β1 α
f (x, y)g(x, y) −
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From (2.5) and Theorem 2.3 in [3], it is easy to observe that the inequality (2.1) holds and the proof is complete. If we apply the Ostrowski type inequality for double integrals to different time scales, we will get some well-known and some new results. Corollary 2.1 Let T1 = T2 = R. We have b d 1 1 f (x, y)g(x, y) − g(x, y) f (s, y)ds + f (x, t)dt 2 b−a a d −c c b d 1 − f (s, t)dtds (b − a)(d − c) a c b d f (x, y) 1 1 − g(s, y)ds + g(x, t)dt 2 b−a a d −c c b d 1 − g(s, t)dtds (b − a)(d − c) a c x y 2 b d ∂ f (α, β) 1 dβdα |g(x, y)| ≤ 2(b − a)(d − c) a c ∂β∂α s t x y 2 ∂ g(α, β) dβdα dydx, + |f (x, y)| ∂β∂α s t which is exactly the Ostrowski type inequality for double integrals shown in [10, Theorem 1]. Corollary 2.2 Let T1 = T2 = Z and a = c = 0, b = k ∈ N and d = r ∈ N. Then k r k r 1 1 g(m, n) 1 f (s, n) + f (m, t) − f (s, t) f (m, n)g(m, n) − 2 k s=1 r t=1 kr s=1 t=1 k r k r 1 1 f (m, n) 1 g(s, n) + g(m, t) − g(s, t) − 2 k s=1 r t=1 kr s=1 t=1 n−1 n−1 k m−1 m−1 r 1 2 1 f (α, β) + |f (m, n)| 2 1 g(α, β) . |g(m, n)| ≤ 2kr α=s β=t
s=1 t=1
α=s β=t
This corresponds to the result obtained in [10, Theorem 2]. N
N
Corollary 2.3 Let T1 = q1 0 = {q1m : m ∈ N0 }, q1 > 1 and T2 = q2 0 = {q2n : n ∈ N0 }, q2 > 1 p and a = q1h , b = q1k and c = q2 , d = q2r . Then
r−1 t
k−1 s t+1 m s+1 n m n q f (q , q ) g(q , q ) t=p q2 f (q1 , q2 ) 1 2 1 1 2 s=h m n m n f (q , q )g(q , q ) − +
1 2 1 2 k−1 s r−1 t 2 s=h q1 t=p q2
k−1 r−1
−
s+1 t+1 s t t=p q1 q2 f (q1 , q2 )
k−1 s r−1 t ( s=h q1 )( t=p q2 )
s=h
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f (q1m , q2n ) − 2
r−1
k−1
k−1 r−1
s+1 s n s=h q1 g(q1 , q2 )
k−1 s s=h q1
+
t=p
q2t g(q1m , q2t+1 )
r−1 t t=p q2
s+1 t+1 s t t=p q1 q2 g(q1 , q2 )
k−1 s r−1 t ( s=h q1 )( t=p q2 ) m−2 n−2 r−1 k−1 1 β+1 β m n |g(q1 , q2 )| (f (q1α+1 , q2 ) − f (q1α+1 , q2 ) ≤ k−1 s r−1 t 2( s=h q1 )( t=p q2 ) s=h t=p α=s β=t β+1 β − f (q1α , q2 ) + f (q1α , q2 )) m−2 n−2 β+1 β β+1 β α+1 α+1 m n α α (g(q1 , q2 ) − g(q1 , q2 ) − g(q1 , q2 ) + g(q1 , q2 )) . + |f (q1 , q2 )|
−
s=h
α=s β=t
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