Return to Article Details An Ostrowski type inequality for double integrals in terms of $L_p$-norms and applications in numerical integration

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION

Rev. Anal. Numér. Théor. Approx., vol. 32 (2003) no. 2, pp. 161-169
ictp.acad.ro/jnaat

AN OSTROWSKI TYPE INEQUALITY
FOR DOUBLE INTEGRALS IN TERMS OF $L_{p}$ -NORMS
AND APPLICATIONS IN NUMERICAL INTEGRATION

S. S. DRAGOMIR, N. S. BARNETT and P. CERONE*

Abstract

An inequality of the Ostrowski type for double integrals and applications in Numerical Analysis in connection with cubature formulae are given. MSC 2000. Primary: 26D15; Secondary: 41A55.

Keywords. Ostrowski's inequality, cubature formulae.

1. INTRODUCTION

In 1938, A. Ostrowski proved the following integral inequality [5, p. 468].
Theorem 1. Let

f : [a, b] \to R

be continuous on

[a, b]

and differentiable on

(a, b)

whose derivative

f^{'} : (a, b) \to R

is bounded on

(a, b)

, i.e.,

{‖ f^{'} ‖}_{\infty} := sup_{t \in (a, b)} | f^{'} (t) | < \infty

Then we have the inequality

| f (x) - \frac{1}{b - a} \int_{a}^{b} f (t) d t | \leq [\frac{1}{4} + \frac{{(x - \frac{a + b}{2})}^{2}}{(b - a)^{2}}] (b - a) {‖ f^{'} ‖}_{\infty}, \forall x \in [a, b]

The constant

\frac{1}{4}

is the best possible.
For some generalizations see the book [5, pp. 468-484] by Mitrinović, Pečarić and Fink.

Some applications of the above results in Numerical Integration and for special means have been given in 3 by S. S. Dragomir and S. Wang.

In [4] Dragomir and Wang established the following Ostrowski type inequality for differentiable mappings whose derivatives belong to

L_{p}

-spaces.

Theorem 2. Let

f : I \subseteq R \to R

be a differentiable mapping on

I^{\circ}

and

a, b \in I^{\circ}

with

a < b

. If

f^{'} \in L_{p} (a, b), p > 1, \frac{1}{p} + \frac{1}{q} = 1

, then we have the inequality:

| f (x) - \frac{1}{b - a} \int_{a}^{b} f (t) d t | \leq \frac{1}{b - a} {[\frac{(x - a)^{q + 1} + (b - x)^{q + 1}}{q + 1}]}^{\frac{1}{q}} {‖ f^{'} ‖}_{p}, \forall x \in [a, b]

where

{‖ f^{'} ‖}_{p} := {(\int_{a}^{b} {| f^{'} (t) |}^{p} d t)}^{\frac{1}{p}}

is the

L_{p} (a, b)

-norm.
Note that the above inequality can also be obtained from Theorem 1 [5], p. 471] due to A. M. Fink.

For other Ostrowski type inequalities, see the papers [1], [2] and [4].
In 1975, G. N. Milovanović generalized Theorem 1, where

f

is a function of several variables [5, p. 468].

Theorem 3. Let

f : R^{m} \to R

be a differentiable function defined on

D = {(x_{1}, \dots, x_{m}) : a_{i} \leq x_{i} \leq b_{i}, i = 1, \dots, m}

and let

| \frac{\partial f}{\partial x_{i}} | \leq M_{i}, M_{i} > 0

i = 1, \dots, m

, in

D

. Furthermore, let function

x ⟼ p (x)

be integrable and

p (x) > 0

, for every

x \in D

. Then for every

x \in D

, we have the inequality:

| f (x) - \frac{\int_{D} p (y) f (y) d y}{\int_{D} p (y) d y} | \leq \frac{\sum_{i = 1}^{m} M_{i} \int_{D} p (y) | x_{i} - y_{i} | d y}{\int_{D} p (y) d y}

In the present paper we point out an Ostrowski type inequality for double integrals in terms of

L_{p}

-norms and apply it in Numerical Integration obtaining a general cubature formula.

2. THE RESULTS

The following inequality of Ostrowski's type for mappings of two variables holds:

Theorem 4. Let

f : [a, b] \times [c, d] \to R

be a continuous mapping on

[a, b] \times [c, d], f_{x, y}^{''} = \frac{\partial^{2} f}{\partial x \partial y}

exists on

(a, b) \times (c, d)

and is in

L_{p} ((a, b) \times (c, d))

, i.e.,

{‖ f_{s, t}^{''} ‖}_{p} := {(\int_{a}^{b} \int_{c}^{d} {| \frac{\partial^{2} f (x, y)}{\partial x \partial y} |}^{p} d x d y)}^{\frac{1}{p}} < \infty, p > 1

then we have the inequality:

\begin{aligned} (1) & ∣ \int_{a}^{b} \int_{c}^{d} f (s, t) d s d t - [(b - a) \int_{c}^{d} f (x, t) d t + (d - c) \int_{a}^{b} f (s, y) d s \\ - (d - c) (b - a) f (x, y)] ∣\leq \end{aligned}

\leq {[\frac{(x - a)^{q + 1} + (b - x)^{q + 1}}{q + 1}]}^{\frac{1}{q}} {[\frac{(y - c)^{q + 1} + (d - y)^{q + 1}}{q + 1}]}^{\frac{1}{q}} {‖ f_{s, t}^{''} ‖}_{p}

for all

(x, y) \in [a, b] \times [c, d]

, where

\frac{1}{p} + \frac{1}{q} = 1, p > 1

.
Proof. Integrating by parts successively, we have the equality:

\begin{aligned} (2) & \int_{a}^{x} \int_{c}^{y} (s - a) (t - c) f_{s, t}^{''} (s, t) d t d s = \\ = (y - c) (x - a) f (x, y) - (y - c) \int_{a}^{x} f (s, y) d s - (x - a) \int_{c}^{y} f (x, t) d t \\ + \int_{a}^{x} \int_{c}^{y} f (s, t) d s d t \end{aligned}

By similar computations, we have

\begin{aligned} (3) & \int_{a}^{x} \int_{y}^{d} (s - a) (t - d) f_{s, t}^{''} (s, t) d s d t \\ = (x - a) (d - y) f (x, y) - (d - y) \int_{a}^{x} f (s, y) d s \\ - (x - a) \int_{y}^{d} f (x, t) d t + \int_{a}^{x} \int_{c}^{y} f (s, t) d s d t \end{aligned}

Now,

\begin{aligned} (4) & \int_{x}^{b} \int_{y}^{d} (s - b) (t - d) f_{s, t}^{''} (s, t) d s d t \\ = (d - y) (b - x) f (x, y) - (d - y) \int_{x}^{b} f (s, y) d s \\ - (b - x) \int_{y}^{d} f (x, t) d t + \int_{x}^{b} \int_{y}^{d} f (s, t) d s d t \end{aligned}

and finally

\begin{aligned} (5) & \int_{x}^{b} \int_{c}^{y} (s - b) (t - c) f_{s, t}^{''} (s, t) d s d t \\ = (y - c) (b - x) f (x, y) - (y - c) \int_{x}^{b} f (s, y) d s \\ - (b - x) \int_{c}^{y} f (x, t) d t + \int_{x}^{b} \int_{c}^{y} f (s, t) d s d t \end{aligned}

If we add the equalities (2) - (5) we get, in the right hand side:

[(y - c) (x - a) + (x - a) (d - y) + (d - y) (b - x) + (y - c) (b - x)] f (x, y) -

\begin{aligned} - (d - c) \int_{a}^{x} f (s, y) d s - (d - c) \int_{x}^{b} f (s, y) d s - (b - a) \int_{c}^{y} f (x, t) d t \\ - (b - a) \int_{y}^{d} f (x, t) d t + \int_{a}^{x} \int_{c}^{y} f (s, t) d s d t + \int_{a}^{x} \int_{y}^{d} f (s, t) d s d t \\ + \int_{x}^{b} \int_{y}^{d} f (s, t) d s d t + \int_{x}^{b} \int_{c}^{y} f (s, t) d s d t \\ = & (d - c) (b - a) f (x, y) - (d - c) \int_{a}^{b} f (s, y) d s - (b - a) \int_{c}^{b} f (x, t) d t \\ + \int_{a}^{b} \int_{c}^{d} f (s, t) d s d t \end{aligned}

For the first part, let us define the kernels:

p : [a, b]^{2} \to R, q : [c, d]^{2} \to R

given by:

p (x, s) := {\begin{cases} s - a, & if s \in [a, x] \\ s - b, & if s \in (x, b] \end{cases}

and

q (y, t) := {\begin{cases} t - c, & if t \in [c, y] \\ t - d, & if t \in (y, d] \end{cases}

Now, we deduce that the left part can be represented as:

\int_{a}^{b} \int_{c}^{d} p (x, s) q (y, t) f_{s, t}^{''} (s, t) d s d t

Consequently, we get the identity

\begin{aligned} (6) & \int_{a}^{b} \int_{c}^{d} p (x, s) q (y, t) f_{s, t}^{''} (s, t) d s d t = \\ = (d - c) (b - a) f (x, y) - (d - c) \int_{a}^{b} f (s, y) d s \\ - (b - a) \int_{c}^{d} f (x, t) d t + \int_{a}^{b} \int_{c}^{d} f (s, t) d s d t \end{aligned}

for all

(x, y) \in [a, b] \times [c, d]

.
Now, using the identity (6), we get

\begin{matrix} ∣ \int_{a}^{b} \int_{c}^{d} f (s, t) d s d t - [(b - a) \int_{c}^{d} f (x, t) d t + (d - c) \int_{a}^{b} f (s, y) d s \\ - (d - c) (b - a) f (x, y)] ∣\leq \\ \leq \int_{a}^{b} \int_{c}^{d} | p (x, s) q (y, t) | \cdot | f_{s, t}^{''} (s, t) | d s d t \end{matrix}

Using Hölder's integral inequality for double integrals, we get

\begin{aligned} \int_{a}^{b} \int_{c}^{d} | p (x, s) q (y, t) | | f_{s, t}^{''} (s, t) | d s d t \leq \\ \leq {(\int_{a}^{b} \int_{c}^{d} | p (x, s) q (y, t) |^{q} d s d t)}^{\frac{1}{q}} {(\int_{a}^{b} \int_{c}^{d} {| f_{s, t}^{''} (s, t) |}^{p} d s d t)}^{\frac{1}{p}} \\ = {(\int_{a}^{b} | p (x, s) |^{q} d s)}^{\frac{1}{q}} {(\int_{c}^{d} | q (y, t) |^{q} d t)}^{\frac{1}{q}} {‖ f_{s, t}^{''} ‖}_{p} \\ = {[\frac{(x - a)^{q + 1} + (b - x)^{q + 1}}{q + 1}]}^{\frac{1}{q}} {[\frac{(y - c)^{q + 1} + (d - y)^{q + 1}}{q + 1}]}^{\frac{1}{q}} {‖ f_{s, t}^{''} ‖}_{p} \end{aligned}

and the theorem is proved.
Corollary 5. Under the above assumptions, we have the inequality:

\begin{aligned} (7) & | \int_{a}^{b} \int_{c}^{d} f (s, t) d s d t - [(b - a) \int_{c}^{d} f (\frac{a + b}{2}, t) d t \\ + (d - c) \int_{a}^{b} f (s, \frac{c + d}{2}) d s - (d - c) (b - a) f (\frac{a + b}{2}, \frac{c + d}{2})] ∣\leq \\ \leq \frac{(b - a)^{1 + \frac{1}{q}} (d - c)^{1 + \frac{1}{q}}}{4 (q + 1)^{\frac{2}{q}}} {‖ f_{s, t}^{''} ‖}_{p} \end{aligned}

Remark 1. Consider the mapping

g : [α, β] \to R, g (t) = (t - α)^{m} + (β - t)^{m}

m \geq 1

. Taking into account the fact that one has the properties

inf_{t \in [α, β]} g (t) = g (\frac{α + β}{2}) = \frac{(β - α)^{m}}{2^{m - 1}}

and

sup_{t \in [α, β]} g (t) = g (α) = g (β) = (β - α)^{m}

then, the above inequality (7) is the best that can be obtained from (1).
Remark 2. Now, if we assume that

f (s, t) = h (s) h (t), h : [a, b] \to R

is continuous on

[a, b]

and suppose that

{‖ h^{'} ‖}_{p} < \infty

, then from (1) we get, for

x = y

\begin{matrix} ∣ \int_{a}^{b} h (s) d s \int_{a}^{b} h (s) d s - h (x) (b - a) \int_{a}^{b} h (s) d s \\ - h (x) (b - a) \int_{a}^{b} h (s) d s + (b - a)^{2} h^{2} (x) ∣\leq \\ \leq {[\frac{(x - a)^{q + 1} + (b - x)^{q + 1}}{q + 1}]}^{\frac{2}{q}} {‖ h^{'} ‖}_{p}^{2} \end{matrix}

i.e.,

{[\int_{a}^{b} h (s) d s - h (x) (b - a)]}^{2} \leq {[\frac{(x - a)^{q + 1} + (b - x)^{q + 1}}{q + 1}]}^{\frac{2}{q}} {‖ h^{'} ‖}_{p}^{2}

which is clearly equivalent to Ostrowski's inequality. Consequently (1) can be also regarded as a generalization for double integrals of the result embodied in Theorem 2.

3. APPLICATIONS FOR CUBATURE FORMULAE

Let us consider the arbitrary divisions

I_{n} : a = x_{0} < x_{1} < \dots < x_{n - 1} < x_{n} = b, J_{m} : c = y_{0} < y_{1} < \dots < y_{m - 1} < y_{m} = b

and

ξ_{i} \in [x_{i}, x_{i + 1}], i = 0, \dots, n - 1

η_{j} \in [y_{j}, y_{j + 1}], j = 0, \dots, m - 1

, be intermediate points. Consider the sum

\begin{aligned} C (f, I_{n}, J_{m}, ξ, η) := & \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} h_{i} \int_{y_{j}}^{y_{j + 1}} f (ξ_{i}, t) d t + \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} l_{j} \int_{x_{i}}^{x_{i + 1}} f (s, η_{j}) d s \\ - \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} h_{i} l_{j} f (ξ_{i}, η_{j}) \end{aligned}

for which we assume that the involved integrals can more easily be computed than the original double integral

D := \int_{a}^{b} \int_{c}^{d} f (s, t) d s d t

and

h_{i} := x_{i + 1} - x_{i}, i = 0, \dots, n - 1, l_{j} := y_{j + 1} - y_{j}, j = 0, \dots, m - 1 .

With this assumption, we can state the following cubature formula:
Theorem 6. Let

f : [a, b] \times [c, d] \to R

be as in Theorem 4 and

I_{n}, J_{m}, ξ

and

η

be as above. Then we have the cubature formula:

\int_{a}^{b} \int_{c}^{d} f (s, t) d s d t = C (f, I_{n}, J_{m}, ξ, η) + R (f, I_{n}, J_{m}, ξ, η)

where the remainder term

R (f, I_{n}, J_{m}, ξ, η)

satisfies the estimation:

\begin{aligned} (8) & | R (f, I_{n}, J_{m}, ξ, η) | \leq \\ \leq {‖ f_{s, t}^{''} ‖}_{p} {[\sum_{i = 0}^{n - 1} \frac{{(x_{i + 1} - ξ_{i})}^{q + 1} + {(ξ_{i} - x_{i})}^{q + 1}}{q + 1}]}^{\frac{1}{q}} {[\sum_{j = 0}^{m - 1} \frac{{(y_{j + 1} - η_{j})}^{q + 1} + {(η_{j} - y_{j})}^{q + 1}}{q + 1}]}^{\frac{1}{q}} \\ \leq \frac{{‖ f_{s, t}^{''} ‖}_{p}}{(q + 1)^{\frac{2}{q}}} \sum_{i = 0}^{n - 1} h_{i}^{1 + \frac{1}{q}} \sum_{j = 0}^{m - 1} l_{j}^{1 + \frac{1}{q}} \end{aligned}

for all

ξ

and

η

as above.

Proof. Apply Theorem 4 on the interval

[x_{i}, x_{i + 1}] \times [y_{j}, y_{j + 1}], i = 0, \dots, n - 1; j = 0, \dots, m - 1

, to get:

\begin{aligned} | \int_{x_{i}}^{x_{i + 1}} \int_{y_{j}}^{y_{j + 1}} f (s, t) d s d t - [h_{i} \int_{y_{j}}^{y_{j + 1}} f (ξ_{i}, t) d t + l_{j} \int_{x_{i}}^{x_{i + 1}} f (s, η_{j}) d s - h_{i} l_{j} f (ξ_{i}, η_{j})] | \\ \leq {[\frac{{(x_{i + 1} - ξ_{i})}^{q + 1} + {(ξ_{i} - x_{i})}^{q + 1}}{q + 1} \cdot \frac{{(y_{j + 1} - η_{j})}^{q + 1} + {(η_{j} - y_{j})}^{q + 1}}{q + 1}]}^{\frac{1}{q}} {[\int_{x_{i}}^{x_{i + 1}} \int_{y_{j}}^{y_{j + 1}} | f (s, t) |^{p} d s d t]}^{\frac{1}{p}} \end{aligned}

for all

i = 0, \dots, n - 1; j = 0, \dots, m - 1

.
Summing over

i

from 0 to

n - 1

and over

j

from 0 to

m - 1

and using the generalized triangle inequality and Hölder's discrete inequality for double sums, we deduce

\begin{aligned} | R (f, I_{n}, J_{m}, ξ, η) | \leq \\ \leq \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} {[\frac{{(x_{i + 1} - ξ_{i})}^{q + 1} + {(ξ_{i} - x_{i})}^{q + 1}}{q + 1} \cdot \frac{{(y_{j + 1} - η_{j})}^{q + 1} + {(η_{j} - y_{j})}^{q + 1}}{q + 1}]}^{\frac{1}{q}} \\ \times {[\int_{x_{i}}^{x_{i + 1}} \int_{y_{j}}^{y_{j + 1}} | f (s, t) |^{p} d s d t]}^{\frac{1}{p}} \\ \leq {[\sum_{i = 0}^{n - 1} (\frac{{(x_{i + 1} - ξ_{i})}^{q + 1} + {(ξ_{i} - x_{i})}^{q + 1}}{q + 1})]}^{\frac{1}{q}} {[\sum_{j = 0}^{m - 1} (\frac{{(y_{j + 1} - η_{j})}^{q + 1} + {(η_{j} - y_{j})}^{q + 1}}{q + 1})]}^{\frac{1}{q}} \\ \times {[\sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} \int_{x_{i}}^{x_{i + 1}} \int_{y_{j}}^{y_{j + 1}} | f (s, t) |^{p} d s d t]}^{\frac{1}{p}} \\ = {[\sum_{i = 0}^{n - 1} \frac{{(x_{i + 1} - ξ_{i})}^{q + 1} + {(ξ_{i} - x_{i})}^{q + 1}}{q + 1} \times \sum_{j = 0}^{m - 1} \frac{{(y_{j + 1} - η_{j})}^{q + 1} + {(η_{j} - y_{j})}^{q + 1}}{q + 1}]}^{\frac{1}{q}} \times {‖ f_{s, t}^{''} ‖}_{p} . \end{aligned}

To prove the second part, we observe that

{(x_{i + 1} - ξ_{i})}^{q + 1} + {(ξ_{i} - x_{i})}^{q + 1} \leq {(x_{i + 1} - x_{i})}^{q + 1}

and

{(y_{j + 1} - η_{j})}^{q + 1} + {(η_{j} - y_{j})}^{q + 1} \leq {(y_{j + 1} - y_{j})}^{q + 1}

for all

i, j

as above and the intermediate points

ξ_{i}

and

η_{j}

.
We omit the details.
Remark 3. As

\sum_{i = 0}^{n - 1} h_{i}^{1 + \frac{1}{q}} \leq [ν (h)]^{\frac{1}{q}} \sum_{i = 0}^{n - 1} h_{i} = (b - a) [ν (h)]^{\frac{1}{q}}

and

\sum_{j = 0}^{m - 1} l_{j}^{1 + \frac{1}{q}} \leq [μ (l)]^{\frac{1}{q}} \sum_{j = 0}^{m - 1} l_{j} = (d - c) [μ (l)]^{\frac{1}{q}},

where

ν (h) = max {h_{i} : i = 0, \dots, n - 1}

and

μ (l) = max {l_{j} : j = 0, \dots, m - 1},

the right hand side of (8) can be bounded by

\frac{1}{(q + 1)^{2 / q}} {‖ f_{s, t}^{''} ‖}_{p} (b - a) (d - c) [ν (h) μ (l)]^{\frac{1}{q}}

Now, define the sum

\begin{aligned} C_{M} (f, I_{n}, J_{m}) := & \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} h_{i} \int_{y_{j}}^{y_{j + 1}} f (\frac{x_{i} + x_{i + 1}}{2}, t) d t \\ + \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} l_{j} \int_{x_{i}}^{x_{i + 1}} f (s, \frac{y_{j} + y_{j + 1}}{2}) d s \\ - \sum_{i = 0}^{n - 1} \sum_{j = 0}^{m - 1} h_{i} l_{j} f (\frac{x_{i} + x_{i + 1}}{2}, \frac{y_{j} + y_{j + 1}}{2}) \end{aligned}

Then we have the best cubature formula we can get from Theorem 6.
Corollary 7. Under the above assumptions we have

\int_{a}^{b} \int_{c}^{d} f (s, t) d s d t = C_{M} (f, I_{n}, J_{m}) + R (f, I_{n}, J_{m})

where the remainder

R (f, I_{n}, J_{m})

satisfies the estimation:

| R (f, I_{n}, J_{m}) | \leq \frac{1}{4 (q + 1)^{2 / q}} {‖ f_{s, t}^{''} ‖}_{p} \sum_{i = 0}^{n - 1} h_{i}^{1 + \frac{1}{q}} \sum_{j = 0}^{m - 1} l_{j}^{1 + \frac{1}{q}}

REFERENCES

[1] Dragomir, S. S. and Wang, S., A new inequality of Ostrowski's type in

L_{1}

-norm and applications to some special means and to some numerical quadrature rules, Tamkang J. of Math., 28, pp. 239-244, 1997.
[2] Dragomir, S. S. and Wang, S., 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, Computers Math. Applic., 33, pp. 15-20, 1997.
[3] Dragomir, S. S. and Wang, S., Applications of Ostrowski's inequality to the estimation of error bounds for some special means and some numerical quadrature rules, Appl. Math. Lett., 11, pp. 105-109, 1998.
[4] Dragomir, S. S. and Wang, S., A new inequality of Ostrowski's type in

L_{p}

-norm, Indian J. Math., 40, no. 3, pp. 299-304, 1998.
[5] Mitrinović, D. S., Pec̆arić, J. E. and Fink, A. M., Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.

Received by the editors: September 29, 1998.

*School of Computer Science & Mathematics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia, e-mail:
{sever, neil, pc}@matilda.vu.edu.au.