AN OSTROWSKI LIKE INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS

S.S. DRAGOMIR

Abstract. In this paper we point out an Ostrowski type inequality for convex functions which complement in a sense the recent results for functions of bounded variation and absolutely continuous functions. Applications in connection with the Hermite-Hadamard inequality are also considered.

1. Introduction

In 1938, A. Ostrowski [9] proved the following integral inequality

(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) ∥f′∥_{∞}

provided f is differentiable and ∥f′∥_{∞} = sup_{t∈(a,b)} |f′ (t)| < ∞.

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

In the last 5 years, many authors have concentrated their efforts in generalising (1.1) and have applied the obtained results in different fields, including Numerical Integration, Probability Theory and Statistics, Information Theory, etc. For a comprehensive approach in the field, see the recent book [5] where many other references may be found.

One direction of generalising (1.1) was pointed out by the author in [2] – [4]. Let us recall here a couple of the main results obtained in the above papers. Theorem 1. Let I_{k} : a = x_{0} < x_{1} < ⋯ < x_{k−1} < x_{k} = b be a division of the interval [a, b] and α_{i} (i = 0, …, k + 1) be k + 2 points such that α_{0} = a, α_{i} ∈ [x_{i−1}, x_{i}] (i = 1, …, k) and α_{k+1} = b. If f : [a, b] → ℝ is of bounded variation on [a, b] , then we have the inequality:

|∫_{a}^{b} f (x) dx − ∑_{i=0}^{k} (α_{i+1} − α_{i}) f (x_{i})| (1.2)

≤ [\frac{1}{2}ν (h) + max \{|α_{i+1} − \frac{x_{i} + x_{i+1}}{2} | , i = 0, …, k − 1\}] ⋁_{a}^{b} (f) ,

where ν (h) := max {h_{i}|i = 0, …, k − 1} , h_{i} := x_{i+1} − x_{i} (i = 0, …, k − 1) and ⋁_{a}^{b} (f) is the total variation of f on [a, b] .

Date: December 12, 2001.

1991 Mathematics Subject Classification. Primary 26D15, 26D10.

Key words and phrases. Ostrowski type inequalities, Convex functions, Hermite-Hadamard type inequalities.

The constant \frac{1}{2} is sharp in the sense that it cannot be replaced by a smaller constant.

If one would assume more for the function f, for example, absolute continuity, then the following result holds. Theorem 2. Under the assumptions of Theorem 1 for I_{k} and α_{i} (i = 0, …, k + 1) and if f : [a, b] → ℝ is absolutely continuous on [a, b] , then

(1.3)  |∫_{a}^{b} f (x) dx − ∑_{i=0}^{k} (α_{i+1} − α_{i}) f (x_{i})|

≤ \{ \begin{matrix}[\frac{1}{4} ∑_{i=0}^{k−1} h_{i}^{2} + ∑_{i=0}^{k−1} (α_{i+1} − \frac{x_{i}+x_{i+1}}{2} )^{2}] ∥f′∥_{∞} & if f′ ∈ L_{∞} [a, b] ; \\ _{\frac{1}{(q+1)^{\frac{1}{q}}}} [ ∑_{i=0}^{k−1} [(α_{i+1} − x_{i})^{q+1} + (x_{i+1} − α_{i+1})^{q+1}]]^{\frac{1}{q}} ∥f′∥_{p} & if f′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ [\frac{1}{2}ν (h) + max \{|α_{i+1} − \frac{x_{i}+x_{i+1}}{2} | , i = 0, …, k − 1\}] ∥f′∥_{1} , \end{matrix}

where ∥·∥_{p} (p ∈ [1, ∞]) are the Lebesgue norms, i.e.,

∥h∥_{∞} : = ess sup_{t∈[a,b]} |h (t)| ,

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

The constants \frac{1}{4}, _{\frac{1}{(q+1)^{\frac{1}{q}}}} and \frac{1}{2} are best in the sense mentioned above.

In this paper, the case of convex functions f : [a, b] → ℝ is examined. Some particular cases in connection with the well known Hermite-Hadamard inequality for convex functions are also considered.

2. The Results

The following result holds. Theorem 3. Let I_{k} : a = x_{0} < x_{1} < ⋯ < x_{k−1} < x_{k} = b be a division of the interval [a, b] and α_{i} (i = 0, …, k + 1) be k + 2 points such that α_{0} = a, α_{i} ∈ [x_{i−1}, x_{i}] (i = 1, …, k) and α_{k+1} = b. If f : [a, b] → ℝ is a convex function on [a, b] , then we have the inequality:

\frac{1}{2} ∑_{i=0}^{k−1} [(x_{i+1} − α_{i+1})^{2} f′_{+} (α_{i+1}) − (α_{i+1} − x_{i})^{2} f′_{−} (α_{i+1})] (2.1)

≤ ∑_{i=0}^{k} (α_{i+1} − α_{i}) f (x_{i+1}) − ∫_{a}^{b} f (t) dt

≤ \frac{1}{2} ∑_{i=0}^{k−1} [(x_{i+1} − α_{i+1})^{2} f′_{−} (x_{i+1}) − (α_{i+1} − x_{i})^{2} f′_{+} (x_{i})] .

The constant \frac{1}{2} is sharp in both inequalities.

Proof. Using the integration by parts formula, we may prove the equality (see for example [3]):

(2.2)  ∑_{i=0}^{k} (α_{i+1} − α_{i}) f (x_{i+1}) − ∫_{a}^{b} f (t) dt = ∑_{i=0}^{k−1} ∫_{x_{i}}^{x_{i+1}} (t − α_{i+1}) f′ (t) dt

for any locally absolutely continuous function f : (a, b) → ℝ.

Since f is convex, then it is locally Lipschitzian on (a, b) and thus the above equality holds. Also, we have

(2.3)  f′_{+} (x_{i}) ≤ f′ (t) ≤ f′_{−} (α_{i+1}) for a.e. t ∈ [x_{i}, α_{i+1}]

and

(2.4)  f′_{+} (α_{i+1}) ≤ f′ (t) ≤ f′_{−} (x_{i+1}) for a.e. t ∈ [α_{i+1}, x_{i+1}] .

Using (2.3) and (2.4), we may write that

(2.5)  f′_{−} (α_{i+1}) ∫_{x_{i}}^{α_{i+1}} (t − α_{i+1}) dt ≤ ∫_{x_{i}}^{α_{i+1}} f′ (t) (t − α_{i+1}) dt
≤ f′_{+} (x_{i}) ∫_{x_{i}}^{α_{i+1}} (t − α_{i+1}) dt

and

(2.6) f′_{+} (α_{i+1}) ∫_{α_{i+1}}^{x_{i+1}} (t − α_{i+1}) dt ≤ ∫_{α_{i+1}}^{x_{i+1}} f′ (t) (t − α_{i+1}) dt
≤ f′_{−} (x_{i+1}) ∫_{α_{i+1}}^{x_{i+1}} (t − α_{i+1}) dt.

Adding (2.5) and (2.6) and taking into account that

∫_{x_{i}}^{α_{i+1}} (t − α_{i+1}) dt = −\frac{1}{2} (α_{i+1} − x_{i})^{2}

and

∫_{α_{i+1}}^{x_{i+1}} (t − α_{i+1}) dt = \frac{1}{2} (x_{i+1} − α_{i+1})^{2} ,

we get

\frac{1}{2} [(x_{i+1} − α_{i+1})^{2} f′_{+} (α_{i+1}) − (α_{i+1} − x_{i})^{2} f′_{−} (α_{i+1})] (2.7)

≤ ∫_{x_{i}}^{x_{i+1}} (t − α_{i+1}) f′ (t) dt
≤ \frac{1}{2} [(x_{i+1} − α_{i+1})^{2} f′_{−} (x_{i+1}) − (α_{i+1} − x_{i})^{2} f′_{+} (x_{i})]

for any i = 0, …, k − 1.

If we sum (2.7) over i from 0 to k − 1 and use the identity (2.2), we deduce the desired result (2.1).

The sharpness will be proved in what follows for a particular case.

It is natural to consider the following particular case.

Corollary 1. Let L_{k} and f be as in the above theorem. Then we have the inequality

(2.8) 0 ≤ \frac{1}{8} ∑_{i=0}^{k−1} [f′_{+} (\frac{x_{i} + x_{i+1}}{2} ) − f′_{−} (\frac{x_{i} + x_{i+1}}{2} )] (x_{i+1} − x_{i})^{2}

≤ \frac{1}{2} [(x_{1} − a) f (a) + ∑_{i=1}^{k−1} (x_{i+1} − x_{i−1}) f (x_{i}) + (b − x_{k−1}) f (b)]

− ∫_{a}^{b} f (t) dt

≤ \frac{1}{8} ∑_{i=0}^{k−1} [f′_{−} (x_{i+1}) − f′_{+} (x_{i})] (x_{i+1} − x_{i})^{2} .

The constant \frac{1}{8} in both inequalities is sharp.

The proof follows by the above theorem choosing α_{i} = \frac{x_{i−1}+x_{i}}{2} , i = 1, …, k and taking into account that (see also [2])

∑_{i=0}^{k} (α_{i+1} − α_{i}) f (x_{i}) (2.9)

= \frac{1}{2} [(x_{1} − a) f (a) + ∑_{i=1}^{k−1} (x_{i+1} − x_{i−1}) f (x_{i}) + (b − x_{k−1}) f (b)] .

The following corollary for equidistant partitioning also holds.

Corollary 2. Let

I_{k} : x_{i} := a + (b − a) · \frac{i}{k} (i = 0, …, k)

be an equidistant partitioning of [a, b] . If f : [a, b] → ℝ is convex on [a, b] , then we have the inequalities

(2.10) 0 ≤ \frac{(b − a)^{2}}{8n^{2}} ∑_{i=0}^{k−1} \{f′_{+} [a + (i + \frac{1}{2}) \frac{b − a}{n}]

− f′_{−} [a + (i + \frac{1}{2}) \frac{b − a}{n}]\}

≤ \frac{1}{k} · \frac{f (a) + f (b)}{2} (b − a)

+\frac{b − a}{k} ∑_{i=1}^{k−1} f [\frac{(k − i) a + ib}{k}] − ∫_{a}^{b} f (t) dt

≤ \frac{(b − a)^{2}}{8n^{2}} ∑_{i=0}^{k−1} \{f′_{−} [a + (i + 1) · \frac{b − a}{n}] − f′_{+} [a + i · \frac{b − a}{n}]\} .

The following particular cases which hold when we assume differentiability conditions may be stated.

Corollary 3. If α_{i} ∈ (a, b) for i = 1, …, k are points of differentiability for f, then we have the inequality

∑_{i=0}^{k−1} (x_{i+1} − x_{i}) (\frac{x_{i} + x_{i+1}}{2} − α_{i+1}) f′ (α_{i+1}) (2.11)

≤ ∑_{i=0}^{k} (α_{i+1} − α_{i}) f (x_{i+1}) − ∫_{a}^{b} f (t) dt.

If we denote by ν (I_{n}) := max {x_{i+1} − x_{i}|i = 0, …, k − 1} , then the following corollary also holds.

Corollary 4. If x_{i} (i = 1, …, k − 1) are points of differentiability for f then

\frac{1}{2} [(x_{1} − a) f (a) + ∑_{i=0}^{k−1} (x_{i+1} − x_{i−1}) f (x_{i}) + (b − x_{k−1}) f (b)] − ∫_{a}^{b} f (t) dt (2.12)

≤ \frac{1}{8} [ν (I_{n})]^{2} [f′_{−} (b) − f′_{+} (a)] .

3. Some Particular Inequalities

(1) If we choose x_{0} = a, x_{1} = b, α_{0} = a, α_{1} = x ∈ (a, b) , α_{2} = b, then from (2.1) we deduce (see also [6])

\frac{1}{2} [(b − x)^{2} f′_{+} (x) − (x − a)^{2} f′_{−} (x)] (3.1)

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

The constant \frac{1}{2} is sharp in both inequalities (see for example [6]). If x = \frac{a+b}{2}, then by (3.1) one deduces (see also [6])

(3.2)  0 ≤ \frac{1}{8} (b − a)^{2} [f′_{+} (\frac{a + b}{2}) − f′_{−} (\frac{a + b}{2})]
≤ \frac{f (a) + f (b)}{2} · (b − a) − ∫_{a}^{b} f (t) dt
≤ \frac{1}{8} (b − a)^{2} [f′_{−} (b) − f′_{+} (a)]

and the constant \frac{1}{8} in both inequalities is sharp (see for example [6]). If one would assume that x ∈ (a, b) is a point of differentiability, then

(3.3) (b − a) (\frac{a + b}{2} − x) f′ (x) ≤ (x − a) f (a) + (b − x) f (b) − ∫_{a}^{b} f (t) dt.

(2) If we choose a = x_{0} < x < x_{2} = b and the numbers α_{0} = a, α ∈ (a, x], β ∈ [x, b) and α_{3} = b, then by Theorem 3, we deduce

\frac{1}{2} [(x − α)^{2} f′_{+} (α) − (α − a)^{2} f′_{−} (α) + (b − β)^{2} f′_{+} (β) − (β − x)^{2} f′_{−} (β)] (3.4)

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

The constant \frac{1}{2} is sharp in both inequalities.

(a) Note that if we let α → a+ and β → b−, then from (3.4), by taking into account firstly that (x − α)^{2} f′_{+} (a) ≤ (x − α)^{2} f′_{+} (α) and − (β − x)^{2} f′_{−} (b) ≤ − (β − x)^{2} f′_{−} (β) , we may deduce the inequality obtained in [7]:

\frac{1}{2} [(b − x)^{2} f′_{+} (x) − (x − a)^{2} f′_{−} (x)] (3.5)

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

The constant \frac{1}{2} is sharp in both inequalities (see for example [7]). If in (3.5) we choose x = \frac{a+b}{2}, then (see also [7])

(3.6)  0 ≤ \frac{1}{8} (b − a)^{2} [f′_{+} (\frac{a + b}{2}) − f′_{−} (\frac{a + b}{2})]
≤ ∫_{a}^{b} f (t) dt − (b − a) f (\frac{a + b}{2})
≤ \frac{1}{8} (b − a)^{2} [f′_{−} (b) − f′_{+} (a)]

and the constant \frac{1}{8} is sharp in both inequalities. We may state now the following result for convex functions improving Hermite-Hadamard integral inequalities. Proposition 1. Let f : [a, b] → ℝ be a convex function on [a, b] . Then

(3.7)  0 ≤ \frac{1}{8} (b − a) [f′_{+} (\frac{a + b}{2}) − f′_{−} (\frac{a + b}{2})]
≤ \frac{1}{b − a} ∫_{a}^{b} f (t) dt − f (\frac{a + b}{2})
≤ \frac{f (a) + f (b)}{2} − \frac{1}{b − a} ∫_{a}^{b} f (t) dt
≤ \frac{1}{8} (b − a) [f′_{−} (b) − f′_{+} (a)] .

The constant \frac{1}{8} is sharp in both parts.

If one would assume that x ∈ (a, b) is a differentiability point for f, then we have the inequality [7]

(3.8)  (b − a) (\frac{a + b}{2} − x) f′ (x) ≤ ∫_{a}^{b} f (t) dt − (b − a) f (x) .

(b) If we choose α = \frac{a+x}{2} and β = \frac{x+b}{2}, then by (3.4) we have the three point inequality:

(3.9) 0 ≤ \frac{1}{8} \{(x − a)^{2} [f′_{+} (\frac{a + x}{2}) − f′_{−} (\frac{a + x}{2})]

+ (b − x)^{2} [f′_{+} (\frac{x + b}{2}) − f′_{−} (\frac{x + b}{2})]\}

≤ \frac{1}{2} [(x − a) f (a) + f (x) (b − a) + (b − x) f (b)] − ∫_{a}^{b} f (t) dt
≤ \frac{1}{8} \{(x − a)^{2} [f′_{+} (x) − f′_{−} (a)] + (b − x)^{2} [f′_{−} (b) − f′_{+} (x)]\}

for any x ∈ (a, b) . The constant \frac{1}{8} is sharp in both parts. If in (3.9) we choose x = \frac{a+b}{2}, then we get

(3.10) 0 ≤ \frac{1}{32} (b − a)^{2} [f′_{+} (\frac{3a + b}{4}) − f′_{−} (\frac{3a + b}{4})

+ f′_{+} (\frac{a + 3b}{4}) − f′_{−} (\frac{a + 3b}{4})]

≤ \frac{1}{2} · [\frac{f (a) + f (b)}{2} + f (\frac{a + b}{2})] (b − a) − ∫_{a}^{b} f (t) dt
≤ \frac{1}{32} (b − a)^{2} [f′_{−} (b) − f′_{+} (\frac{a + b}{2}) + f′_{−} (\frac{a + b}{2}) − f′_{+} (a)]

If one would assume that f is differentiable in \frac{a+b}{2}, then we get the following reverse of Bullen’s inequality

(3.11)  0 ≤ \frac{1}{2} · [\frac{f (a) + f (b)}{2} + f (\frac{a + b}{2})] (b − a) − ∫_{a}^{b} f (t) dt
≤ \frac{1}{32} (b − a)^{2} [f′_{−} (b) − f′_{+} (a)] .

The constant \frac{1}{32} is sharp.

(c) Now, if we choose α = \frac{5a+b}{6}, β = \frac{a+5b}{6} and x ∈ [\frac{5a+b}{6}, \frac{a+5b}{6}] in (3.4), then we have the inequalities

\frac{1}{2} [(x − \frac{5a + b}{6})^{2} f′_{+} (\frac{5a + b}{6}) − \frac{(b − a)^{2}}{36} f′_{−} (\frac{5a + b}{6}) (3.12)

+ \frac{(b − a)^{2}}{36} f′_{+} (\frac{a + 5b}{6}) − (\frac{a + 5b}{6} − x)^{2} f′_{−} (\frac{a + 5b}{6})]

≤ \frac{b − a}{3} [\frac{f (a) + f (b)}{2} + 2f (x)] − ∫_{a}^{b} f (t) dt

≤ \frac{1}{2} [(x − \frac{5a + b}{6})^{2} f′_{−} (x) − \frac{(b − a)^{2}}{36} f′_{+} (a)

+ \frac{(b − a)^{2}}{36} f′_{−} (b) − (\frac{a + 5b}{6} − x)^{2} f′_{+} (x)] .

If in (3.12) we choose x = \frac{a+b}{2}, then we get the Simpson’s inequality

\frac{1}{18} (b − a)^{2} [f′_{+} (\frac{5a + b}{6}) − \frac{1}{4}f′_{−} (\frac{5a + b}{6}) (3.13)

+ \frac{1}{4}f′_{+} (\frac{a + 5b}{6}) − f′_{−} (\frac{a + 5b}{6})]

≤ \frac{b − a}{3} [\frac{f (a) + f (b)}{2} + 2f (\frac{a + b}{2})] − ∫_{a}^{b} f (t) dt

≤ \frac{1}{18} (b − a)^{2} [f′_{−} (\frac{a + b}{2}) − \frac{1}{4}f′_{+} (a) + \frac{1}{4}f′_{−} (b) − f′_{+} (\frac{a + b}{2})] .

If the function is differentiable on (a, b) , then we get

−\frac{1}{24} (b − a)^{2} [f′ (\frac{a + 5b}{6}) − f′ (\frac{5a + b}{6})] (3.14)

≤ \frac{b − a}{3} [\frac{f (a) + f (b)}{2} + 2f (\frac{a + b}{2})] − ∫_{a}^{b} f (t) dt

≤ \frac{1}{72} (b − a)^{2} [f′_{−} (b) − f′_{+} (a)] .

References

[1] G. ANASTASSIOU, Ostrowski type inequalities, Proc. Amer. Math. Soc., 123(12) (1995), 3775-3781.

[2] S.S. DRAGOMIR, The Ostrowski integral inequality for mappings of bounded variation, Bull. Austral. Math. Soc., 60 (1999), 495-508.

[3] S.S. DRAGOMIR, A generalisation of Ostrowski integral inequality for mappings whose derivatives belong to L_{p} [a, b] , 1 < p < ∞ and applications in numerical integration, J. Math. Anal. Appl., 225 (2001), 605-626.

[4] S.S. DRAGOMIR, A generalisation of Ostrowski integral inequality for mappings whose derivatives belong to L_{1} [a, b] , and applications in numerical integration, J. Computational Analysis and Appl., 3(4) (2001), 343-360.

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

[6] S.S. DRAGOMIR, A generalised trapezoid type inequality for convex functions, (Preprint) RGMIA Res. Rep. Coll., 5(1) (2002), Article 9. [ONLINE]

   http://rgmia.vu.edu.au/v5n1.html

[7] S.S. DRAGOMIR, An Ostrowski type inequality for convex functions, (Preprint) RGMIA Res. Rep. Coll., 5(1) (2002), Article 5. [ONLINE] http://rgmia.vu.edu.au/v5n1.html

[8] A.M. FINK, Bounds on the derivation of a function from its averages, Czech. Math. J., 42 (1992), 289-310.

[9] A. OSTROWSKI, Über die Absolutabweichung einer differentiienbaren Funcktion von ihren Integralwittelwert, Comment. Math. Helv., 10 (1938), 220-227.

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