SOME NEW INEQUALITIES OF OSTROWSKI TYPE

S.S. DRAGOMIR

Abstract. By the use of the Cauchy mean value theorem, some new inequalities of Ostrowski type 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} + \begin{matrix}(x − \frac{a+b}{2}) & 2 \\ (b − a) & 2 \end{matrix} ] (b − a) M

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

A simple proof of this fact can be done by using the identity:

(1.2)  f (x) = \frac{1}{b − a} ∫_{a}^{b} f (t) dt + \frac{1}{b − a} ∫_{a}^{b} p (x, t) f′ (t) dt, x ∈ [a, b] ,

where

p (x, t) := \{ \begin{matrix}t − a & if & a ≤ t ≤ x \\ t − b & if & x < t ≤ b \end{matrix}

which also holds for absolutely continuous functions f : [a, b] → ℝ.

The following Ostrowski type result for absolutely continuous functions holds (see [2], [3] and [4]).

1991 Mathematics Subject Classification. Primary 26D10, 26D15; Secondary 26D99.

Key words and phrases. Cauchy mean value theorem, Ostrowski Inequality.

Theorem 2. Let f : [a, b] → ℝ be absolutely continuous on [a, b]. Then, for all x ∈ [a, b], we have:

|f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| (1.3)

≤ \{ \begin{matrix}[\frac{1}{4} + (\frac{x− \frac{a+b}{2}}{b−a} )^{2}] (b − a) ∥f′∥_{∞} & if & f′ ∈ L_{∞} [a, b] ; \\ _{\frac{1}{(p+1)^{\frac{1}{p}}}} [(\frac{x−a}{b−a})^{p+1} + (\frac{b−x}{b−a})^{p+1}]^{\frac{1}{p}} (b − a)^{\frac{1}{p}} ∥f′∥_{q} & if & f′ ∈ L_{q} [a, b] , \\ & & \frac{1}{p} + \frac{1}{q} = 1, p > 1; \\ [\frac{1}{2} + |\frac{x− \frac{a+b}{2}}{b−a} |] ∥f′∥_{1} ; \end{matrix}

where ∥·∥_{r} (r ∈ [1, ∞]) are the usual Lebesgue norms on L_{r} [a, b], i.e.,

∥g∥_{∞} := ess sup_{t∈[a,b]} |g (t)|

and

∥g∥_{r} := (∫_{a}^{b} |g (t)|^{r} dt)^{\frac{1}{r}} , r ∈ [1, ∞).

The constants \frac{1}{4}, _{\frac{1}{(p+1)^{\frac{1}{p}}}} and \frac{1}{2} respectively are sharp in the sense presented in

Theorem 1.

The above inequalities can also be obtained from the Fink result in [5] on choosing n = 1 and performing some appropriate computations.

If one drops the condition of absolute continuity and assumes that f is Hölder continuous, then one may state the result (see [6]): Theorem 3. Let f : [a, b] → ℝ be of r − H−Hölder type, i.e.,

(1.4)  |f (x) − f (y)| ≤ H |x − y|^{r} , for all x, y ∈ [a, b] ,

where r ∈ (0, 1] and H > 0 are fixed. Then for all x ∈ [a, b] we have the inequality:

|f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| (1.5)

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

The constant \frac{1}{r+1} is also sharp in the above sense.

Note that if r = 1, i.e., f is Lipschitz continuous, then we get the following version of Ostrowski’s inequality for Lipschitzian functions (with L instead of H) (see [7])

(1.6)  |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) L.

Here the constant \frac{1}{4} is also best.

Moreover, if one drops the condition of the continuity of the function, and assumes that it is of bounded variation, then the following result may be stated (see [8]).

Theorem 4. Assume that f : [a, b] → ℝ is of bounded variation and denote by ⋁_{a}^{b}

its total variation. Then

(1.7)  |f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ [\frac{1}{2} + |\frac{x − \frac{a+b}{2}}{b − a} |] ⋁_{a}^{b} (f)

for all x ∈ [a, b]. The constant \frac{1}{2} is the best possible.

If we assume more about f, i.e., f is monotonically increasing, then the inequality (1.7) may be improved in the following manner [9] (see also [10]). Theorem 5. Let f : [a, b] → ℝ be monotonic nondecreasing. Then for all x ∈ [a, b], we have the inequality:

|f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| (1.8)

≤ \frac{1}{b − a} \{[2x − (a + b)] f (x) + ∫_{a}^{b} sgn (t − x) f (t) dt\}
≤ \frac{1}{b − a} \{(x − a) [f (x) − f (a)] + (b − x) [f (b) − f (x)]\}
≤ [\frac{1}{2} + |\frac{x − \frac{a+b}{2}}{b − a} |] [f (b) − f (a)] .

All the inequalities in (1.8) are sharp and the constant \frac{1}{2} is the best possible.

In this paper we point out different Ostrowski type inequalities assuming some special properties for the derivative of the function f around a given point x ∈ (a, b).

2. The Results

The following theorem holds. Theorem 6. 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

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

Then we have the inequality (2.2)

|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) .

Proof. Let x ∈ (a, b) and define the mapping g_{1,x} : (a, x) → ℝ, g_{1,x} (t) = (x − t)^{p}.

Applying the Cauchy mean value theorem, for any t ∈ (a, x) there exists a η ∈ (t, x) such that

[f (t) − f (x)] g′_{1,x} (η) = [g_{1,x} (t) − g_{1,x} (x)] f′ (η)

i.e.,

(−p) (f (t) − f (x)) (x − η)^{p−1} = (x − t)^{p} f′ (η)

from where we obtain

(2.3)  |f (t) − f (x)| = \frac{(x − t)^{p} |f′ (η)|}{p (x − η)^{p−1}} ≤ \frac{(x − t)^{p}}{p} M_{p} (x) , t ∈ (a, x) .

We define the mapping g_{2,x} : (x, b) → ℝ, g_{2,x} (t) = (t − x)^{p}. Applying the Cauchy mean value theorem, we can find a ξ ∈ (x, t) such that

[f (t) − f (x)] p (ξ − x)^{p−1} = (t − x)^{p} f′ (ξ)

from where we get

(2.4)  |f (t) − f (x)| = \frac{(t − x)^{p} |f′ (ξ)|}{p (ξ − x)^{p−1}} ≤ \frac{(t − x)^{p}}{p} M_{p} (x) , t ∈ (x, b) .

In conclusion, by (2.3) and (2.4) we may write

(2.5)  |f (t) − f (x)| ≤ \frac{1}{p}M_{p} (x) |t − x|^{p} for all t ∈ (a, b) .

Integrating (2.5) over t on [a, b], we get

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

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

= \frac{1}{p}M_{p} (x) \frac{1}{b − a} [∫_{a}^{x} (x − t)^{p} dt + ∫_{x}^{b} (t − x)^{p} dt]

= \frac{1}{p}M_{p} (x) \frac{(x − a)^{p+1} + (b − x)^{p+1}}{(p + 1) (b − a)} ,

and the inequality (2.2) is proved.

Remark 1. For p = 1, we obtain

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

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

where ∥f′∥_{∞} := sup_{t∈(a,b)} |f′ (t)| < ∞, which is Ostrowski’s inequality (1.1). It is

obvious that for p > 1, the accuracy order provided by (2.2) is higher than 1, as provided by the classical Ostrowski’s inequality. Remark 2. If p ∈ (0, 1) and f′ ∈ L_{∞} [a, b], then obviously

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

for all x ∈ [a, b].

The following mid-point formula holds. Corollary 1. Let f and p be as in Theorem 6. Assume that

M_{p} (\frac{a + b}{2}) := sup_{u∈(a,b)} \{|\frac{a + b}{2} − u|^{1−p} |f′ (u)|\} < ∞.

Then we have the midpoint inequality

(2.6)  |f (\frac{a + b}{2}) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{(b − a)^{p}}{p (p + 1) 2^{p}} M_{p} (\frac{a + b}{2}) .

Before we continue our presentation, we recall the following special means:

(a) The arithmetic mean

A = A (a, b) := \frac{a + b}{2}, a, b ≥ 0;

(b) The geometric mean G = G (a, b) := \sqrt{ab}, a, b ≥ 0;

(c) The harmonic mean

H = H (a, b) := \frac{2ab}{a + b}, a, b > 0;

(d) The logarithmic mean

L = L (a, b) := \{ \begin{matrix}a & if & a = b, \\ \frac{b−a}{ln b−ln a} & if & a ≠ b, \end{matrix} a, b > 0;

(e) The identric mean

I = I (a, b) := \{ \begin{matrix}a & if & a = b, \\ \frac{1}{e} (\frac{b^{b}}{a^{a}} )^{\frac{1}{b−a}} & if & a ≠ b, \end{matrix} a, b > 0;

(f) The p−logarithmic mean

L_{p} = L_{p} (a, b) := \{ \begin{matrix}[\frac{b^{p+1}−a^{p+1}}{(p+1)(b−a)}]^{\frac{1}{p}} & if & a ≠ b, \\ a & if & a = b, \end{matrix} a, b > 0;

where p ∈ ℝ\ {−1, 0} .

The following result also holds. Theorem 7. 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

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

Then we have the inequality:

(2.8)  |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} (a, x) − (b − x) L^{−1} (b, x) − \frac{2}{x} (x − A) & if & p = −1 \end{matrix}

for all x ∈ [a, b]. Proof. Consider the mapping g : [a, b] → ℝ, g (x) = x^{p}. Applying the Cauchy mean value theorem, then for any x and t ∈ [a, b], there exists a η between x and t such that

[f (t) − f (x)] g′ (η) = [g (t) − g (x)] f′ (η)

i.e.,

(f (t) − f (x)) pη^{p−1} = (t^{p} − x^{p}) f′ (η)

from where we obtain:

|f (t) − f (x)| = \begin{matrix}|f′ (η)| & |t^{p} − x^{p}| \\ |p| & η & p−1 \end{matrix} ≤ \frac{K_{p} (f′)}{|p|} |t^{p} − x^{p}| .

In conclusion, for any t, x ∈ [a, b], we have the inequality

(2.9)  |f (t) − f (x)| ≤ \frac{K_{p} (f′)}{|p|} |t^{p} − x^{p}| .

Integrating (2.9) over t on [a, b], we get

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

For p > 0, we have

∫_{a}^{b} |t^{p} − x^{p}| dt = ∫_{a}^{x} (x^{p} − t^{p}) dt + ∫_{x}^{b} (t^{p} − x^{p}) dt
= 2x^{p} (x − A) + (b − x) L_{p}^{p} (b, x) − (x − a) L_{p}^{p} (x, a) .

For p ∈ (−∞, −1) ∪ (−1, 0), we have

∫_{a}^{b} |x^{p} − t^{p}| dt = ∫_{a}^{x} (t^{p} − x^{p}) dt + ∫_{x}^{b} (x^{p} − t^{p}) dt
= (x − a) L_{p}^{p} (x, a) − (b − x) L_{p}^{p} (b, x) − 2x^{p} (x − A)

and, finally, for p = −1, we have

∫_{a}^{b} |\frac{1}{x} − \frac{1}{t}| dt = ∫_{a}^{x} (\frac{1}{t} − \frac{1}{x}) dt + ∫_{x}^{b} (\frac{1}{x} − \frac{1}{t}) dt
= (x − a) L^{−1} (a, x) − (b − x) L^{−1} (b, x) − \frac{2}{x} (x − A)

and the theorem is proved.

The following corollary is natural.

Corollary 2. With the assumptions in Theorem 7, we have the midpoint inequality

|f (A) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{K_{p} (f′)}{|p|} (2.10)

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

The following theorem also holds.

Theorem 8. Let f : [a, b] → ℝ be continuous on [a, b] (with a > 0) and differentiable on (a, b). If

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

then we have the inequality

|f (x) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| (2.12)

≤ \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 all x ∈ [a, b].

Proof. Consider the mapping g : [a, b] → ℝ, g (t) = ln t. Applying the Cauchy mean value theorem for any x and t ∈ [a, b] there exists a η between x and t such that

(f (t) − f (x)) g′ (η) = (g (t) − g (x)) f′ (η)

i.e.,

(f (t) − f (x)) \frac{1}{η} = (ln t − ln x) f′ (η)

from where we get

|f (t) − f (x)| = |ηf′ (η)| |ln t − ln x| ≤ P (f′) |ln t − ln x| .

In conclusion, for any t, x ∈ [a, b], we have the inequality

(2.13)  |f (t) − f (x)| ≤ P (f′) |ln t − ln x| .

Integrating (2.13) over t on [a, b], we get

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

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

= P (f′) \frac{1}{b − a} [∫_{a}^{x} (ln x − ln t) dt + ∫_{x}^{b} (ln t − ln x) dt]

= P (f′) \frac{1}{b − a} [(x − a) ln x − (x − a) ln I (a, x) + (b − x) ln I (b, x) − (b − x) ln x]
= \frac{1}{b − a} [2 (x − A) ln x + (b − x) ln I (x, b) − (x − a) ln I (a, x)] P′ (f)

and the theorem is proved.

The following corollary is natural. Corollary 3. With the assumptions of Theorem 8, we have the inequality

(2.14)  |f (A) − \frac{1}{b − a} ∫_{a}^{b} f (t) dt| ≤ \frac{1}{2}P (f′) ln [\frac{I (A, b)}{I (a, A)}] ,

where A = A (a, b) = \frac{a+b}{2}.

References

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[10] P. CERONE and S.S. DRAGOMIR, Midpoint type rules from an inequalities point of view, in Analytic-Computational Methods in Applied Mathematics, G.A. Anastassiou (Ed), CRC Press, New York, 2000, 135-200.

School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia.

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