AN INEQUALITY OF OSTROWSKI TYPE VIA POMPEIU’S MEAN VALUE THEOREM

S.S. DRAGOMIR

Abstract. An inequality providing some bounds for the integral mean via Pompeiu’s mean value theorem and applications for quadrature rules and special means are given.

1. Introduction

The following result is known in the literature as Ostrowski’s inequality [1].

Theorem 1. Let f : [a, b] → ℝ be a differentiable mapping on (a, b) with the property that |f′ (t)| ≤ M for all t ∈ (a, b) . Then

(1.1)  |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ [\frac{1}{4} + (\frac{x − \frac{a+b}{2}}{b − a} )^{2}] (b − a) M,

for all x ∈ [a, b] . The constant \frac{1}{4} is best possible in the sense that it cannot be replaced by a smaller constant.

In [2], the author has proved the following Ostrowski type inequality.

Theorem 2. Let f : [a, b] → ℝ be continuous on [a, b] with a > 0 and differentiable on (a, b) . Let p ∈ ℝ\ {0} and assume that

K_{p} (f′) := sup_{u∈(a,b)} {u^{1−p} |f′ (u)|} < ∞.

Then we have the inequality

(1.2) |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{K_{p} (f′)}{|p| (b − a)}

×\{ \begin{matrix}2x^{p} (x − A) + (b − x) L_{p}^{p} (b, x) − (x − a) L_{p}^{p} (x, a) & if p ∈ (0, ∞) ; \\ (x − a) L_{p}^{p} (x, a) − (b − x) L_{p}^{p} (b, x) − 2x^{p} (x − A) & if p ∈ (−∞, −1) ∪ (−1, 0) \\ (x − a) L^{−1} (x, a) − (b − x) L^{−1} (b, x) − \frac{2}{x} (x − A) & if p = −1, \end{matrix}

for any x ∈ (a, b) , where for a ≠ b,

A = A (a, b) := \frac{a + b}{2}, is the arithmetic mean,

Date: February 27, 2003.

1991 Mathematics Subject Classification. Primary 26D15, 26D10; Secondary 41A55.

Key words and phrases. Ostrowski’s inequality, Pompeiu mean value theorem, quadrature rules, Special means.

L_{p} = L_{p} (a, b) = [\frac{b^{p+1} − a^{p+1}}{(p + 1) (b − a)}]^{\frac{1}{p}} , is the p − logarithmic mean p ∈ ℝ\ \{−1, 0\} ,

and

L = L (a, b) := \frac{b − a}{ln b − ln a} is the logarithmic mean.

Another result of this type obtained in the same paper is:

Theorem 3. Let f : [a, b] → ℝ be continuous on [a, b] (with a > 0) and differentiable on (a, b) . If

P (f′) := sup_{u∈(a,b)} |uf′ (x)| < ∞,

then we have the inequality

(1.3) |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{P (f′)}{b − a} [ln [\begin{matrix}[I (x, b)] & b−x \\ [I (a, x)] & x−a \end{matrix} ] + 2 (x − A) ln x]

for any x ∈ (a, b) , where for a ≠ b

I = I (a, b) := \frac{1}{e} (\frac{b^{b}}{a^{a}} )^{\frac{1}{b−a}} , is the identric mean.

If some local information around the point x ∈ (a, b) is available, then we may state the following result as well [2].

Theorem 4. Let f : [a, b] → ℝ be continuous on [a, b] and differentiable on (a, b) . Let p ∈ (0, ∞) and assume, for a given x ∈ (a, b) , we have that

M_{p} (x) := sup_{u∈(a,b)} {|x − u|^{1−p} |f′ (u)|} < ∞.

Then we have the inequality

(1.4) |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt|

≤ \frac{1}{p (p + 1) (b − a)} [(x − a)^{p+1} + (b − x)^{p+1}] M_{p} (x) .

For recent results in connection to Ostrowski’s inequality see the papers [3],[4] and the monograph [5].

The main aim of this paper is to provide some complementary results, where instead of using Cauchy mean value theorem, we use Pompeiu mean Value Theorem to evaluate the integral mean of an absolutely continuous function. Applications for quadrature rules and particular instances of functions are given as well.

2. Pompeiu’s Mean Value Theorem

In 1946, Pompeiu [6] derived a variant of Lagrange’s mean value theorem, now known as Pompeiu’s mean value theorem (see also [7, p. 83]).

Theorem 5. For every real valued function f differentiable on an interval [a, b] not containing 0 and for all pairs x_{1} ≠ x_{2} in [a, b] , there exists a point ξ in (x_{1}, x_{2}) such that

(2.1)  \frac{x_{1}f (x_{2}) − x_{2}f (x_{1})}{x_{1} − x_{2}} = f (ξ) − ξf′ (ξ) .

Proof. Define a real valued function F on the interval [\frac{1}{b}, \frac{1}{a}] by

(2.2)  F (t) = tf (\frac{1}{t}) .

Since f is differentiable on (\frac{1}{b}, \frac{1}{a}) and

(2.3)  F′ (t) = f (\frac{1}{t}) − \frac{1}{t}f′ (\frac{1}{t}) ,

then applying the mean value theorem to F on the interval [x, y] ⊂ [\frac{1}{b}, \frac{1}{a}] we get

(2.4)  \frac{F (x) − F (y)}{x − y} = F′ (η)

for some η ∈ (x, y) .

Let x_{2} = \frac{1}{x}, x_{1} = \frac{1}{y} and ξ = \frac{1}{η} . Then, since η ∈ (x, y) , we have

x_{1} < ξ < x_{2}.

Now, using (2.2) and (2.3) on (2.4), we have

\frac{xf (\frac{1}{x}) − yf (\frac{1}{y} )}{x − y} = f (\frac{1}{η} ) − \frac{1}{η} f′ (\frac{1}{η} ) ,

that is

\frac{x_{1}f (x_{2}) − x_{2}f (x_{1})}{x_{1} − x_{2}} = f (ξ) − ξf′ (ξ) .

This completes the proof of the theorem.

Remark 1. Following [7, p. 84 – 85], we will mention here a geometrical interpretation of Pompeiu’s theorem. The equation of the secant line joining the points (x_{1}, f (x_{1})) and (x_{2}, f (x_{2})) is given by

y = f (x_{1}) + \frac{f (x_{2}) − f (x_{1})}{x_{2} − x_{1}} (x − x_{1}) .

This line intersects the y−axis at the point (0, y) , where y is

y = f (x_{1}) + \frac{f (x_{2}) − f (x_{1})}{x_{2} − x_{1}} (0 − x_{1})

= \frac{x_{1}f (x_{2}) − x_{2}f (x_{1})}{x_{1} − x_{2}}.

The equation of the tangent line at the point (ξ, f (ξ)) is

y = (x − ξ) f′ (ξ) + f (ξ) .

The tangent line intersects the y−axis at the point (0, y) , where

y = −ξf′ (ξ) + f (ξ) .

Hence, the geometric meaning of Pompeiu’s mean value theorem is that the tangent of the point (ξ, f (ξ)) intersects on the y−axis at the same point as the secant line connecting the points (x_{1}, f (x_{1})) and (x_{2}, f (x_{2})) .

3. Evaluating the Integral Mean

The following result holds.

Theorem 6. Let f : [a, b] → ℝ be continuous on [a, b] and differentiable on (a, b) with [a, b] not containing 0. Then for any x ∈ [a, b] , we have the inequality

(3.1)  |\frac{a + b}{2} · \frac{f (x)}{x} − \frac{1}{b − a} ∫_{a}^{b} f (t) dt|

≤ \frac{b − a}{|x|} [\frac{1}{4} + (\frac{x − \frac{a+b}{2}}{b − a} )^{2}] ∥f − ℓf′∥_{∞} ,

where ℓ (t) = t, t ∈ [a, b] .

The constant \frac{1}{4} is sharp in the sense that it cannot be replaced by a smaller constant.

Proof. Applying Pompeiu’s mean value theorem, for any x, t ∈ [a, b] , there is a ξ between x and t such that

tf (x) − xf (t) = [f (ξ) − ξf′ (ξ)] (t − x)

giving

(3.2)  |tf (x) − xf (t)| ≤ sup_{ξ∈[a,b]} |f (ξ) − ξf′ (ξ)| |x − t|
= ∥f − ℓf′∥_{∞} |x − t|

for any t, x ∈ [a, b] .

Integrating over t ∈ [a, b] , we get

(3.3) |f (x) ∫_{a}^{b} tdt − x ∫_{a}^{b} f (t) dt| ≤ ∥f − ℓf′∥_{∞} ∫_{a}^{b} |x − t| dt

= ∥f − ℓf′∥_{∞} [\frac{(x − a)^{2} + (b − x)^{2}}{2} ]

= ∥f − ℓf′∥_{∞} [\frac{1}{4} (b − a)^{2} + (x − \frac{a + b}{2})^{2}]

and since ∫_{a}^{b} tdt = \frac{b^{2}−a^{2}}{2} , we deduce from (3.3) the desired result (3.1). Now, assume that (3.2) holds with a constant k > 0, i.e.,

(3.4)  |\frac{a + b}{2} · \frac{f (x)}{x} − \frac{1}{b − a} ∫_{a}^{b} f (t) dt|

≤ \frac{b − a}{|x|} [k + (\frac{x − \frac{a+b}{2}}{b − a} )^{2}] ∥f − ℓf′∥_{∞} ,

for any x ∈ [a, b] .

Consider f : [a, b] → ℝ, f (t) = αt + β; α, β ≠ 0. Then

∥f − ℓf′∥_{∞} = |β| ,
\frac{1}{b − a} ∫_{a}^{b} f (t) dt = \frac{a + b}{2} · α + β,

and by (3.4) we deduce

|\frac{a + b}{2} (α + \frac{β}{x} ) − (\frac{a + b}{2}α + β)| ≤ \frac{b − a}{|x|} [k + (\frac{x − \frac{a+b}{2}}{b − a} )^{2}] |β|

giving

(3.5)  |\frac{a + b}{2} − x| ≤ (b − a) k + (\frac{x − \frac{a+b}{2}}{b − a} )^{2}

for any x ∈ [a, b] .

If in (3.5) we let x = a or x = b, we deduce k ≥ \frac{1}{4}, and the sharpness of the constant is proved.

The following interesting particular case holds.

Corollary 1. With the assumptions in Theorem 6, we have

(3.6)  |f (\frac{a + b}{2}) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{(b − a)}{2 |a + b|} ∥f − ℓf′∥_{∞} .

4. The Case of Weighted Integrals

We will consider now the weighted integral case.

Theorem 7. Let f : [a, b] → ℝ be continuous on [a, b] and differentiable on (a, b) with [a, b] not containing 0. If w : [a, b] → ℝ is nonnegative integrable on [a, b] , then for each x ∈ [a, b] , we have the inequality:

(4.1) |∫_{a}^{b} f (t) w (t) dt − \frac{f (x)}{x} ∫_{a}^{b} tw (t) dt|

≤ ∥f − ℓf′∥_{∞} [sgn (x) (∫_{a}^{x} w (t) dt − ∫_{x}^{b} w (t) dt)

+ \frac{1}{|x|} (∫_{x}^{b} tw (t) dt − ∫_{a}^{x} tw (t) dt)]

Proof. Using the inequality (3.2), we have

|f (x) ∫_{a}^{b} tw (t) dt − x ∫_{a}^{b} f (t) w (t) dt| (4.2)

≤ ∥f − ℓf′∥_{∞} ∫_{a}^{b} w (t) |x − t| dt

= ∥f − ℓf′∥_{∞} [∫_{a}^{x} w (t) (x − t) dt + ∫_{x}^{b} w (t) (t − x) dt]

= ∥f − ℓf′∥_{∞} [x ∫_{a}^{x} w (t) dt − ∫_{a}^{x} tw (t) dt + ∫_{x}^{b} tw (t) dt − x ∫_{x}^{b} w (t) dt]

= ∥f − ℓf′∥_{∞} [x (∫_{a}^{x} w (t) dt − ∫_{x}^{b} w (t) dt) + ∫_{x}^{b} tw (t) dt − ∫_{a}^{x} tw (t) dt]

from where we get the desired inequality (4.1).

Now, if we assume that 0 < a < b, then

(4.3)  a ≤ \frac{∫_{a}^{b} tw (t) dt}{∫_{a}^{b} w (t) dt} ≤ b

provided ∫_{a}^{b} w (t) dt > 0.

With this extra hypothesis, we may state the following corollary.

Corollary 2. With the above assumptions, we have

(4.4) |f (\frac{∫_{a}^{b} tw (t) dt}{∫_{a}^{b} w (t) dt}) − \frac{1}{∫_{a}^{b} w (t) dt} ∫_{a}^{b} f (t) w (t) dt|

≤ ∥f − ℓf′∥_{∞} [\frac{∫_{a}^{x} w (t) dt − ∫_{x}^{b} w (t) dt}{∫_{a}^{b} w (t) dt} + \frac{∫_{x}^{b} w (t) tdt − ∫_{a}^{x} tw (t) dt}{∫_{a}^{b} tw (t) dt}]

5. A Quadrature Formula

We assume in the following that 0 < a < b.

Consider the division of the interval [a, b] given by

I_{n} : a = x_{0} < x_{1} < ⋯ < x_{n−1} < x_{n} = b,

and ξ_{i} ∈ [x_{i}, x_{i+1}], i = 0, …, n − 1 a sequence of intermediate points. Define the quadrature

(5.1)  S_{n} (f, I_{n}, ξ) := ∑_{i=0}^{n−1} \frac{f (ξ_{i})}{ξ_{i}} · \frac{x_{i+1}^{2} − x_{i}^{2}}{2}

= ∑_{i=0}^{n−1} \frac{f (ξ_{i})}{ξ_{i}} · \frac{x_{i} + x_{i+1}}{2} · h_{i},

where h_{i} := x_{i+1} − x_{i}, i = 0, …, n − 1.

The following result concerning the estimate of the remainder in approximating the integral ∫_{a}^{b} f (t) dt by the use of S_{n} (f, I_{n}, ξ) holds.

Theorem 8. Assume that f : [a, b] → ℝ is continuous on [a, b] and differentiable on (a, b) . Then we have the representation

(5.2)  ∫_{a}^{b} f (t) dt = S_{n} (f, I_{n}, ξ) + R_{n} (f, I_{n}, ξ)

where S_{n} (f, I_{n}, ξ) is as defined in (5.1), and the remainder R_{n} (f, I_{n}, ξ) satisfies the estimate

(5.3)  |R_{n} (f, I_{n}, ξ)| ≤ ∥f − ℓf′∥_{∞} ∑_{i=0}^{n−1} \frac{h_{i}^{2}}{ξ_{i}} [\frac{1}{4} + |\frac{ξ_{i} − \frac{x_{i}+x_{i+1}}{2}}{h_{i}} |^{2}]

≤ \frac{1}{2} ∥f − ℓf′∥_{∞} ∑_{i=0}^{n−1} \frac{h_{i}^{2}}{ξ_{i}} ≤ \frac{∥f − ℓf′∥_{∞}}{2a} ∑_{i=0}^{n−1} h_{i}^{2}.

Proof. Apply Theorem 6 on the interval [x_{i}, x_{i+1}] for the intermediate points ξ_{i} to obtain

|∫_{x_{i}}^{x_{i+1}} f (t) dt − \frac{f (ξ_{i})}{ξ_{i}} · \frac{x_{i} + x_{i+1}}{2} · h_{i}| (5.4)

≤ \frac{1}{ξ_{i}} h_{i}^{2} [\frac{1}{4} + (\frac{ξ_{i} − \frac{x_{i}+x_{i+1}}{2}}{h_{i}} )^{2}] ∥f − ℓf′∥_{∞}

≤ \frac{1}{2ξ_{i}} h_{i}^{2} ∥f − ℓf′∥_{∞} ≤ \frac{1}{2a}h_{i}^{2} ∥f − ℓf′∥_{∞}

for each i ∈ {0, …, n − 1} .

Summing over i from 1 to n − 1 and using the generalised triangle inequality, we deduce the desired estimate (5.3).

Now, if we consider the mid-point rule (i.e., we choose ξ_{i} = \frac{x_{i}+x_{i+1}}{2} above, i ∈ {0, …, n − 1})

M_{n} (f, I_{n}) := ∑_{i=0}^{n−1} f (\frac{x_{i} + x_{i+1}}{2} ) h_{i},

then, by Corollary 1, we may state the following result as well.

Corollary 3. With the assumptions of Theorem 8, we have

(5.5)  ∫_{a}^{b} f (t) dt = M_{n} (f, I_{n}) + R_{n} (f, I_{n}) ,

where the remainder satisfies the estimate:

(5.6)  |R_{n} (f, I_{n})| ≤ \frac{∥f − ℓf′∥_{∞}}{2} ∑_{i=0}^{n−1} \frac{h_{i}^{2}}{x_{i} + x_{i+1}}

≤ \frac{∥f − ℓf′∥_{∞}}{4a} ∑_{i=0}^{n−1} h_{i}^{2}.

6. Applications for Special Means

For 0 < a < b, let us consider the means

A = A (a, b) := \frac{a + b}{2},
G = G (a, b) := \sqrt{a · b},
H = H (a, b) := \frac{2}{\frac{1}{a} + \frac{1}{b}} ,
L = L (a, b) := \frac{b − a}{ln b − ln a} (logarithmic mean),

I = I (a, b) := \frac{1}{e} (\frac{b^{b}}{a^{a}} )^{\frac{1}{b−a}} (identric mean)

and the p−logarithmic mean

L_{p} = L_{p} (a, b) = [\frac{b^{p+1} − a^{p+1}}{(p + 1) (b − a)}]^{\frac{1}{p}} , p ∈ ℝ\ \{−1, 0\} .

It is well known that

H ≤ G ≤ L ≤ I ≤ A,

and, denoting L_{0} := I and L_{−1} = L, the function ℝ ∋ p ↦ L_{p} ∈ ℝ is monotonic increasing.

In the following we will use the following inequality obtained in Corollary 1,

(6.1)  |f (\frac{a + b}{2}) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{(b − a)}{2 (a + b)} ∥f − ℓf′∥_{∞} ,

provided 0 < a < b.

(1) Consider the function f : [a, b] ⊂ (0, ∞) → ℝ, f (t) = t^{p}, p ∈ ℝ\ {−1, 0} . Then

f (\frac{a + b}{2}) = [A (a, b)]^{p} ,
\frac{1}{b − a} ∫_{a}^{b} f (t) dt = L_{p}^{p} (a, b) ,

∥f − ℓf′∥_{[a,b],∞} = \{ \begin{matrix}(1 − p) a^{p} & if p ∈ (−∞, 0) \ \{−1\} , \\ |1 − p| b^{p} & if p ∈ (0, 1) ∪ (1, ∞) . \end{matrix}

Consequently, by (6.1) we deduce

(6.2) |A^{p} (a, b) − L_{p}^{p} (a, b)|

≤ \frac{1}{4} × \{ \begin{matrix}\frac{(1 − p) a^{p} (b − a)}{A (a, b)} & if p ∈ (−∞, 0) \ \{−1\} , \\ \frac{|1 − p| b^{p} (b − a)}{A (a, b)} & if p ∈ (0, 1) ∪ (1, ∞) . \end{matrix}

(2) Consider the function f : [a, b] ⊂ (0, ∞) → ℝ, f (t) = \frac{1}{t}. Then

f (\frac{a + b}{2}) = \frac{1}{A (a, b)},

\frac{1}{b − a} ∫_{a}^{b} f (t) dt = \frac{1}{L (a, b)},

∥f − ℓf′∥_{[a,b],∞} = \frac{2}{a}.

Consequently, by (6.1) we deduce

(6.3)  0 ≤ A (a, b) − L_{p} (a, b) ≤ \frac{b − a}{2a}L (a, b) .

(3) Consider the function f : [a, b] ⊂ (0, ∞) → ℝ, f (t) = ln t. Then

f (\frac{a + b}{2}) = ln [A (a, b)] ,

\frac{1}{b − a} ∫_{a}^{b} f (t) dt = ln [I (a, b)] ,

∥f − ℓf′∥_{[a,b],∞} = max \{|ln (\frac{a}{e})| , |ln (\frac{b}{e})|\} .

Consequently, by (6.1) we deduce

(6.4)  1 ≤ \frac{A (a, b)}{I (a, b)} ≤ exp \{\frac{b − a}{4A (a, b)} max \{|ln (\frac{a}{e})| , |ln (\frac{b}{e})|\}\} .

References

[1] A. OSTROWSKI, Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.

[2] S.S. DRAGOMIR, Some new inequalities of Ostrowski type, RGMIA Res. Rep. Coll., 5(2002),

   Supplement, Article 11, [ON LINE:http://rgmia.vu.edu.au/v5(E).html]to apper in Gazette,

Austral. Math. Soc.

[3] G. A. ANASTASSIOU, Multidimensional Ostrowski inequalities, revisited. Acta Math. Hungar., 97 (2002), no. 4, 339–353.

[4] G. A. ANASTASSIOU, Univariate Ostrowski inequalities, revisited. Monatsh. Math., 135 (2002), no. 3, 175–189.

[5] S.S. DRAGOMIR and T.M. RASSIAS (Eds), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht/Boston/London, 2002.

[6] D. POMPEIU, Sur une proposition analogue au théorème des accroissements finis, Mathematica (Cluj, Romania), 22(1946), 143-146.

[7] P.K. SAHOO and T. RIEDEL, Mean Value Theorems and Functional Equations, World Scientific, Singapore, New Jersey, London, Hong Kong, 2000.

School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, MCMC 8001, VIC, Australia.

   E-mail address: sever@matilda.vu.edu.au
   URL: http://rgmia.vu.edu.au/SSDragomirWeb.html