NORM ESTIMATES FOR THE DIFFERENCE BETWEEN BOCHNER’S INTEGRAL AND THE CONVEX COMBINATION

OF FUNCTION’S VALUES

P. CERONE, Y.J. CHO, S.S. DRAGOMIR, J.K. KIM, AND S.S. KIM

Abstract. Norm estimates between the Bochner integral of a vector-valued function in Banach spaces having the Radon-Nikodym property and the convex combination of function’s values taken on a division of the interval [a, b] , are given.

1. Introduction

A Banach space X with the property that every absolutely continuous X−valued function is almost everywhere differentiable is said to be a Radon-Nikodym space [6, pp. 217–219] or [2, 12] (see also [3]). For example, every reflexive Banach space (in particular, every Hilbert space) is a Radon-Nikodym space, but the space L_{∞} [0, 1] of all 𝕂−valued, essentially bounded functions defined on the interval [0, 1], endowed with the norm

∥g∥_{∞} := ess sup_{t∈[0,1]} |g (t)| ,

is a Banach space which is not a Radon-Nikodym space.

A function f : [a, b] → X is measurable if there exists a sequence of simple functions (f_{n}) (with f_{n} : [a, b] → X) which converges punctually a.e. on [a, b].

It is well-known that a measurable function f : [a, b] → X is Bochner integrable if and only if its norm, i.e., the function t ⟼ ∥f∥ (t) := ∥f (t)∥ : [a, b] → X is Lebesgue integrable on [a, b], (see for example [11]).

Further, we use the integration by parts formula. This holds under the following general conditions:

Let −∞ < a < b < ∞ and f, g be two mappings defined on [a, b] such that f is ℂ-valued and g is X-valued, where X is a real or complex Banach space. If f, g are differentiable on [a, b] and their derivatives are Bochner integrable on [a, b], then

(B) ∫_{a}^{b} f′g = f (b) g (b) − f (a) g (b) − (B) ∫_{a}^{b} fg′.

For some results on the Ostrowski inequality for real-valued functions, see [1], [5], [9] and [10], and the references therein.

The following theorem concerning a version of Ostrowski’s inequality for vectorvalued functions has been obtained in [3].

Date: July 16, 2003.

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

Key words and phrases. Bochner’s Integral, Ostrowski Inequality, Quadrature Formulae.

Theorem 1. Let (X; ∥·∥) be a Banach space with the Radon-Nikodym property and f : [a, b] → X an absolutely continuous function on [a, b] with the property that f′ ∈ L_{∞} ([a, b] ; X), i.e.,

|∥f′∥|_{[a,b],∞} := ess sup_{t∈[a,b]} ∥f′ (t)∥ < ∞.

Then we have the inequalities:

∥f (s) − \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt∥ (1.1)

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

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

≤ \frac{1}{2} (b − a) |∥f′∥|_{[a,b],∞};

for any s ∈ [a, b], where (B) ∫_{a}^{b} f (t) dt is the Bochner integral of f.

Bounds involving the p−norms, p ∈ [1, ∞), of the derivative f′, are embodied in the following theorem [3].

Theorem 2. Let (X, ∥·∥) be a Banach space with the Radon-Nikodym property and f : [a, b] → X be an absolutely continuous function on [a, b] with the property that f′ ∈ L_{p} ([a, b] ; X), p ∈ [1, ∞), i.e.,

(1.2)  |∥f′∥|_{[a,b],p} := (∫_{a}^{b} ∥f′ (t)∥^{p} dt)^{\frac{1}{p}} < ∞.

Then we have the inequalities

∥f (s) − \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt∥ (1.3)

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

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

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

It is the main aim of this paper to point out estimates between the Bochner integral of a vector-valued function with values in Banach spaces having the Radon-Nikodym property and the convex combination of values taken on a given division of the interval [a, b] . The obtained results naturally extend Ostrowski’s type inequalities mentioned above. Some particular cases for two and three points rules are also given.

2. The Results

Let a ≤ b and c ∈ ℝ. Define the mapping

μ_{p} (a, c, b) := ∫_{a}^{b} |t − c|^{p} dt if p ∈ [1, ∞);
μ_{∞} (a, c, b) := max_{t∈[a,b]} |t − c| if p = ∞.

We observe that (1) If c < a, then

μ_{p} (a, c, b) = ∫_{a}^{b} (t − c)^{p} dt

= \frac{1}{p + 1} [(b − c)^{p+1} − (a − c)^{p+1}] , if p ∈ [1, ∞)

and

μ_{∞} (a, c, b) = b − c.

(2) If c ∈ [a, b] , then
μ_{p} (a, c, b) = ∫_{a}^{c} (c − t)^{p} dt + ∫_{c}^{b} (t − c)^{p} dt

= \frac{1}{p + 1} [(c − a)^{p+1} + (b − c)^{p+1}]

if p ∈ [1, ∞) and

μ_{∞} (a, c, b) = max (c − a, b − c) = \frac{1}{2} (b − a) + |c − \frac{a + b}{2}| .

(3) If b < c, then
μ_{p} (a, c, b) = ∫_{a}^{b} (c − t)^{p} dt

= \frac{1}{p + 1} [(c − a)^{p+1} − (c − b)^{p+1}] , if p ∈ [1, ∞)

and

μ_{∞} (a, c, b) = c − a.

Consequently, we may conclude that

μ_{p} (a, c, b) = \{ \begin{matrix}\frac{1}{p+1} [(b − c)^{p+1} − (a − c)^{p+1}] & if c < a; \\ \frac{1}{p+1} [(c − a)^{p+1} + (b − c)^{p+1}] & if c ∈ [a, b] ; \\ \frac{1}{p+1} [(c − a)^{p+1} − (c − b)^{p+1}] & if b < c; \end{matrix}

if p ∈ [1, ∞) and

μ_{∞} (a, c, b) = \{ \begin{matrix}b − c & if c < a; \\ \frac{1}{2} (b − a) + |c − \frac{a+b}{2} | & if c ∈ [a, b] ; \\ c − a & if b < c. \end{matrix}

The following integral identity is interesting in itself as well.

Lemma 1. Let f : [a, b] → X be an absolutely continuous function on the Banach space X, X is with the property of Radon-Nikodym, a ≤ x_{1} ≤ ⋯ ≤ x_{n−1} ≤ x_{n} ≤ b and p_{i} > 0 (i = 1, …, n) with ∑_{i=1}^{n} p_{i} = 1. Then we have the equality:

(2.1) ∑_{i=1}^{n} p_{i}f (x_{i}) − \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt = \frac{1}{b − a} (B) ∫_{a}^{x_{1}} (t − a) f′ (t) dt

+ \frac{1}{b − a} ∑_{i=1}^{n−1} (B) ∫_{x_{i}}^{x_{i+1}} [t − (P_{i}b + P̄_{i}a)] f′ (t) dt

+ \frac{1}{b − a} (B) ∫_{x_{n}}^{b} (t − b) f′ (t) dt,

where (B) ∫_{a}^{b} f (t) dt is the Bochner integral, P_{i} := ∑_{k=1}^{i} p_{k} and P̄_{i} = 1 − P_{i}. The sum in the middle is assumed to be zero when n = 1.

Proof. We know that, on utilizing the integration by parts formula, for any x ∈ [a, b] , we have the representation (see for example [3])

(2.2)  f (x) = \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt + \frac{1}{b − a} (B) ∫_{a}^{b} k (x, t) f′ (t) dt,

where

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

BOCHNER’S INTEGRAL 5

Putting in (2.2) x = x_{i} (i = 1, …, n) , multiplying by p_{i} ≥ 0 and summing over i from 1 to n, we deduce

(2.3) ∑_{i=1}^{n} p_{i}f (x_{i}) = \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt + \frac{1}{b − a} (B) ∫_{a}^{b} [∑_{i=1}^{n} p_{i}k (x_{i}, t)] f′ (t) dt.

However,

k (x_{1}, t) = \{ \begin{matrix}t − a & if a ≤ t ≤ x_{1} ≤ b, \\ t − b & if a ≤ x_{1} < t ≤ b, \end{matrix}

· · · · · · · · · · · · · · · · · · · · · · · · · · ·

k (x_{n}, t) = \{ \begin{matrix}t − a & if a ≤ t ≤ x_{n} ≤ b, \\ t − b & if a ≤ x_{n} < t ≤ b, \end{matrix}

then

(2.4)  S (x̄, p̄, t) := ∑_{i=1}^{n} p_{i}k (x_{i}, t)

= \{ \begin{matrix}p_{1} (t − a) + p_{2} (t − a) + ⋯ + p_{n−1} (t − a) + p_{n} (t − a) , & a ≤ t ≤ x_{1} ≤ b, \\ p_{1} (t − b) + p_{2} (t − a) + ⋯ + p_{n−1} (t − a) + p_{n} (t − a) , & a ≤ x_{1} < t ≤ x_{2} ≤ b, \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ p_{1} (t − b) + p_{2} (t − b) + ⋯ + p_{n−1} (t − b) + p_{n} (t − a) , & a ≤ x_{n−1} ≤ t ≤ x_{n} ≤ b, \\ p_{1} (t − b) + p_{2} (t − b) + ⋯ + p_{n−1} (t − b) + p_{n} (t − b) , & a ≤ x_{n} < t ≤ b, \end{matrix}

= \{ \begin{matrix}t − a, & a ≤ t ≤ x_{1} ≤ b, \\ p_{1} (t − b) + (p_{2} + ⋯ + p_{n}) (t − a) , & a ≤ x_{1} < t ≤ x_{2} ≤ b, \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ (p_{1} + ⋯ + p_{n−1}) (t − b) + p_{n} (t − a) , & a ≤ x_{n−1} ≤ t ≤ x_{n} ≤ b, \\ t − b, & a ≤ x_{n} < t ≤ b, \end{matrix}

= \{ \begin{matrix}t − a, & & & a ≤ t ≤ x_{1} ≤ b, \\ t − [p_{1}b + (p_{2} + ⋯ + p_{n}) a] , & & & a ≤ x_{1} < t ≤ x_{2} ≤ b, \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ t − [(p_{1} + ⋯ + p_{n−1}) b + p_{n}a] , & & & a ≤ x_{n−1} ≤ t ≤ x_{n} ≤ b, \\ t − b, & & & a ≤ x_{n} < t ≤ b, \end{matrix}

= \{ \begin{matrix}t − a, & & a ≤ t ≤ x_{1} ≤ b, \\ t − (P_{1}b + P̄_{1}a) , & & a ≤ x_{1} < t ≤ x_{2} ≤ b, \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ t − (P_{i}b + P̄_{i}a) & & a ≤ x_{i} ≤ t ≤ x_{i+1} ≤ b, \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ t − (P_{n−1}b + P̄_{n−1}a) , & & a ≤ x_{n−1} ≤ t ≤ x_{n} ≤ b, \\ t − b, & & a ≤ x_{n} < t ≤ b. \end{matrix}

6 P. CERONE, Y.J. CHO, S.S. DRAGOMIR, J.K. KIM, AND S.S. KIM

Consequently, by (2.3), we have

∑_{i=1}^{n} p_{i}f (x_{i}) (2.5)

= \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt + \frac{1}{b − a} (B) ∫_{a}^{b} S (x̄, p̄, t) f′ (t) dt (by (2.4))

= \frac{1}{b − a} (B) ∫_{a}^{b} f (t) dt + \frac{1}{b − a} (B) ∫_{a}^{x_{1}} (t − a) f′ (t) dt

+ \frac{1}{b − a} ∑_{i=1}^{n−1} (B) ∫_{a}^{b} [t − (P_{i}b + P̄_{i}a)] f′ (t) dt

+ \frac{1}{b − a} (B) ∫_{x_{n}}^{b} (t − b) f′ (t) dt,

and the representation (2.1) is proved.

The following result in approximating the Bochner integral (B) ∫_{a}^{b} f (t) dt in terms of the convex combination of (f (x_{i}))_{i=\overline{1,n}} with the weights (p_{i})_{i=\overline{1,n}} holds. Theorem 3. Assume that f : [a, b] → X, (x_{i})_{i=\overline{1,n}} and (p_{i})_{i=\overline{1,n}} are as in Lemma 1. Then we have the inequality:

(2.6) ∥(B) ∫_{a}^{b} f (t) dt − (b − a) ∑_{i=1}^{n} p_{i}f (x_{i})∥

≤ ∫_{a}^{x_{1}} (t − a) ∥f′ (t)∥ dt + ∑_{i=1}^{n−1} ∫_{x_{i}}^{x_{i+1}} |t − (P_{i}b + P̄_{i}a)| ∥f′ (t)∥ dt

+ ∫_{x_{n}}^{b} (b − t) ∥f′ (t)∥ dt

≤ \{ \begin{matrix}(x_{1} − a) ∥|f′|∥_{[a,x_{1}],1} + ∑_{i=1}^{n−1} μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) ∥|f′|∥_{[x_{i},x_{i+1}],1} \\ + (b − x_{n}) ∥|f′|∥_{[x_{n},b],1} \\ \begin{matrix}(x_{1}−a) & ^{1+ \frac{1}{q}} \\ (q+1) & \frac{1}{q} \end{matrix} ∥|f′|∥_{[a,x_{1}],p} + ∑_{i=1}^{n−1} [μ_{q} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})]^{\frac{1}{q}} ∥|f′|∥_{[x_{i},x_{i+1}],p} \\ +\begin{matrix}(b−x_{n}) & ^{1+ \frac{1}{q}} \\ (q+1) & \frac{1}{q} \end{matrix} ∥|f′|∥_{[x_{n},b],p} where p > 1, \frac{1}{p} + \frac{1}{q} = 1 \\ and if f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{(x_{1}−a)^{2}}{2} ∥|f′|∥_{[a,x_{1}],∞} + ∑_{i=1}^{n−1} μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) ∥|f′|∥_{[x_{i},x_{i+1}],∞} \\ +\frac{(b−x_{n})^{2}}{2} ∥|f′|∥_{[x_{n},b],∞} if f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}max & (x_{1} − a, max_{i=\overline{1,n−1}} \{μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})\} , b − x_{n}) & ∥|f′|∥ & [a,b],1 \\ if f′ ∈ L_{1} ([a, b] ; X) ; \\ [\frac{(x_{1}−a)^{q+1}}{q+1} + ∑_{i=1}^{n−1} μ_{q} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) + \frac{(b−x_{n})^{q+1}}{q+1} ] & \frac{1}{q} & ∥|f′|∥ & [a,b],p \\ where p > 1, \frac{1}{p} + \frac{1}{q} = 1 and if f′ ∈ L_{p} ([a, b] ; X) ; \\ [\frac{(x_{1}−a)^{2}}{2} + ∑_{i=1}^{n−1} μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) + \frac{(b−x_{n})^{2}}{2} ] & ∥|f′|∥ & [a,b],∞ \\ if f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

where L_{p} ([a, b] ; X) , p ∈ [1, ∞] are the usual vector-valued Lebesgue spaces and

∥|h|∥_{[α,β],∞} := ess sup_{t∈[α,β]} ∥h (t)∥ ,
∥|h|∥_{[α,β],p} := (∫_{α}^{β} ∥h (t)∥^{p} dt)^{\frac{1}{p}} , p ≥ 1,

and the functions μ_{q} (·, ·, ·) , q ∈ [1, ∞] were defined above. Proof. Using the properties of the norm, we have, by (2.1), that

∥(B) ∫_{a}^{b} f (t) dt − (b − a) ∑_{i=1}^{n} p_{i}f (x_{i})∥

≤ ∫_{a}^{x_{1}} (t − a) ∥f′ (t)∥ dt + ∑_{i=1}^{n−1} ∫_{x_{i}}^{x_{i+1}} |t − (P_{i}b + P̄_{i}a)| ∥f′ (t)∥ dt

+ ∫_{x_{n}}^{b} (b − t) ∥f′ (t)∥ dt

and the first inequality in (2.6) is proved.

Now, observe that

∫_{a}^{x_{1}} (t − a) ∥f′ (t)∥ dt

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

8 P. CERONE, Y.J. CHO, S.S. DRAGOMIR, J.K. KIM, AND S.S. KIM

and

∑_{i=1}^{n−1} ∫_{x_{i}}^{x_{i+1}} |t − (P_{i}b + P̄_{i}a)| ∥f′ (t)∥ dt

≤ \{ \begin{matrix}sup_{t∈[x_{i},x_{i+1}]} |t − (P_{i}b + P̄_{i}a)| ∥|f′|∥_{[x_{i},x_{i+1}],1} ; \\ (∫_{x_{i}}^{x_{i+1}} |t − (P_{i}b + P̄_{i}a)|^{q} dt) & \frac{1}{q} & ∥|f′|∥ & [x_{i},x_{i+1}],p \\ if p > 1, \frac{1}{p} + \frac{1}{q} = 1, and f′ ∈ L_{p} ([a, b] ; X) ; \\ ∫_{x_{i}}^{x_{i+1}} |t − (P_{i}b + P̄_{i}a)| dt ∥|f′|∥_{[x_{i},x_{i+1}],∞} if f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

= \{ \begin{matrix}μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) ∥|f′|∥_{[x_{i},x_{i+1}],1} ; \\ [μ_{q} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})] & \frac{1}{q} & ∥|f′|∥ & [x_{i},x_{i+1}],p \\ if p > 1, \frac{1}{p} + \frac{1}{q} = 1, and f′ ∈ L_{p} ([a, b] ; X) ; \\ μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) ∥|f′|∥_{[x_{i},x_{i+1}],∞} if f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

and ∫_{x_{n}}^{b} (b − t) ∥f′ (t)∥ dt

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

giving the second inequality in (2.6).

Finally, observe that

(x_{1} − a) ∥|f′|∥_{[a,x_{1}],1} + ∑_{i=1}^{n−1} μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) ∥|f′|∥_{[x_{i},x_{i+1}],1}

+ (b − x_{n}) ∥|f′|∥_{[x_{n},b],1}

≤ max \{x_{1} − a, max_{i=\overline{1,n−1}} \{μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})\} , b − x_{n}\}

× [∥|f′|∥_{[a,x_{1}],1} + ∑_{i=1}^{n−1} ∥|f′|∥_{[x_{i},x_{i+1}],1} + ∥|f′|∥_{[x_{n},b],1}]

≤ max \{x_{1} − a, max_{i=\overline{1,n−1}} \{μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})\} , b − x_{n}\} ∥|f′|∥_{[a,b],1} ,

by the discrete Hölder’s inequality we have that

\begin{matrix}(x_{1} − a) & 1+ \frac{1}{q} \\ (q + 1) & \frac{1}{q} \end{matrix} ∥|f′|∥_{[a,x_{1}],p} + ∑_{i=1}^{n−1} [μ_{q} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})]^{\frac{1}{q}} ∥|f′|∥_{[x_{i},x_{i+1}],p}

+\begin{matrix}(b − x_{n}) & 1+ \frac{1}{q} \\ (q + 1) & \frac{1}{q} \end{matrix} ∥|f′|∥_{[x_{n},b],p}

≤ \{[\begin{matrix}(x_{1} − a) & 1+ \frac{1}{q} \\ (q + 1) & \frac{1}{q} \end{matrix} ]^{q} + ∑_{i=1}^{n−1} ([μ_{q} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1})]^{\frac{1}{q}} )^{q}

+ [\begin{matrix}(b − x_{n}) & 1+ \frac{1}{q} \\ (q + 1) & \frac{1}{q} \end{matrix} ]^{q}\}^{\frac{1}{q}} × [∥|f′|∥_{[x_{n},b],p}^{p} + ∥|f′|∥_{[a,x_{1}],p}^{p} + ∑_{i=1}^{n−1} ∥|f′|∥_{[x_{i},x_{i+1}],p}^{p}]^{\frac{1}{p}}

= [\frac{(x_{1} − a)^{q+1}}{q + 1} + ∑_{i=1}^{n−1} μ_{q} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) + \frac{(b − x_{n})^{q+1}}{q + 1} ]^{\frac{1}{q}} ∥|f′|∥_{[a,b],p}

and

\frac{(x_{1} − a)^{2}}{2} ∥|f′|∥_{[a,x_{1}],∞} + ∑_{i=1}^{n−1} μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) ∥|f′|∥_{[x_{i},x_{i+1}],∞}

+\frac{(b − x_{n})^{2}}{2} ∥|f′|∥_{[x_{n},b],∞}

≤ [\frac{(x_{1} − a)^{2}}{2} + ∑_{i=1}^{n−1} μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) + \frac{(b − x_{n})^{2}}{2} ]

× max \{∥|f′|∥_{[a,x_{1}],∞} , max_{i=\overline{1,n−1}} ∥|f′|∥_{[x_{i},x_{i+1}],∞} , ∥|f′|∥_{[x_{n},b],∞}\}

= [\frac{(x_{1} − a)^{2}}{2} + ∑_{i=1}^{n−1} μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) + \frac{(b − x_{n})^{2}}{2} ] ∥|f′|∥_{[a,b],∞}

and the theorem is completely proved.

It is a natural assumption to consider the weights p_{i} > 0 (i = 1, …, n) for which ξ_{i} := P_{i}b+P̄_{i}a (∈ [a, b]) will be in the interval [x_{i}, x_{i+1}] (i = 1, …, n) . In this case we have:

μ_{∞} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) = \frac{1}{2}h_{i} + |P_{i}b + P̄_{i}a − \frac{x_{i} + x_{i+1}}{2} | ,

where h_{i} := x_{i+1} − x_{i}, and for p ∈ [1, ∞)

μ_{p} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) = \frac{1}{p + 1} [(P_{i}b + P̄_{i}a − x_{i})^{p+1} + (x_{i+1} − P_{i}b − P̄_{i}a)^{p+1}] .

Note that for p = 1, we have

μ_{1} (x_{i}, P_{i}b + P̄_{i}a, x_{i+1}) = \frac{1}{4}h_{i}^{2} + (P_{i}b + P̄_{i}a − \frac{x_{i} + x_{i+1}}{2} )^{2} .

The following corollary is important for applications.

Corollary 1. With the assumptions of Lemma 1 and if x_{i} ≤ P_{i}b + P̄_{i}a ≤ x_{i+1} for each i = 1, …, n − 1, then we have the inequalities:

(2.7)  ∥(B) ∫_{a}^{b} f (t) dt − (b − a) ∑_{i=1}^{n} p_{i}f (x_{i})∥

≤ \{ \begin{matrix}(x_{1} − a) ∥|f′|∥_{[a,x_{1}],1} + ∑_{i=1}^{n−1} [\frac{1}{2}h_{i} + |P_{i}b + P̄_{i}a − \frac{x_{i}+x_{i+1}}{2} |] ∥|f′|∥_{[x_{i},x_{i+1}],1} \\ + (b − x_{n}) ∥|f′|∥_{[x_{n},b],1} ; \\ \begin{matrix}(x_{1}−a) & ^{1+ \frac{1}{q}} \\ (q+1) & \frac{1}{q} \end{matrix} & ∥|f′|∥ & [a,x_{1}],p \\ +_{\frac{1}{(q+1)^{\frac{1}{q}}}} ∑_{i=1}^{n−1} [(P_{i}b + P̄_{i}a − x_{i})^{q+1} + (x_{i+1} − P_{i}b − P̄_{i}a)^{q+1}]^{\frac{1}{q}} ∥|f′|∥_{[x_{i},x_{i+1}],p} \\ +\begin{matrix}(b−x_{n}) & ^{1+ \frac{1}{q}} \\ (q+1) & \frac{1}{q} \end{matrix} ∥|f′|∥_{[x_{n},b],p} where p > 1, \frac{1}{p} + \frac{1}{q} = 1 \\ and if f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{(x_{1}−a)^{2}}{2} ∥|f′|∥_{[a,x_{1}],∞} + ∑_{i=1}^{n−1} [\frac{1}{4}h_{i}^{2} + (P_{i}b + P̄_{i}a − \frac{x_{i}+x_{i+1}}{2} )^{2}] ∥|f′|∥_{[x_{i},x_{i+1}],∞} \\ +\frac{(b−x_{n})^{2}}{2} ∥|f′|∥_{[x_{n},b],∞} if f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}max & (x_{1} − a, \frac{1}{2} max_{i=\overline{1,n−1}} h_{i} + max_{i=\overline{1,n−1}} |P_{i}b + P̄_{i}a − \frac{x_{i}+x_{i+1}}{2} | , b − x_{n}) & ∥|f′|∥ & [a,b],1 \\ if f′ ∈ L_{1} ([a, b] ; X) ; \\ \frac{1}{(q+1)^{\frac{1}{q}}} & [(x_{1} − a)^{q+1} + ∑_{i=1}^{n−1} & [(P_{i}b + P̄_{i}a − x_{i})^{q+1} \\ + (x_{i+1} − P_{i}b − P̄_{i}a)^{q+1}] + (b − x_{n})^{q+1} ]^{\frac{1}{q}} ∥|f′|∥_{[a,b],p} \\ if p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ [\frac{(x_{1}−a)^{2}}{2} + ∑_{i=1}^{n−1} [\frac{1}{4}h_{i}^{2} + (P_{i}b + P̄_{i}a − \frac{x_{i}+x_{i+1}}{2} )^{2}] + \frac{(b−x_{n})^{2}}{2} ] & ∥|f′|∥ & [a,b],∞ \\ if f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

Remark 1. For n = 1, we recapture from (2.7) the Ostrowski type inequalities incorporated in Theorems 1 and 2.

3. The Case of Two Points

The following proposition is a particular case of Corollary 1 for n = 2 and will be considered with some details that are important for applications.

Proposition 1. Let (X, ∥·∥) be a Banach space with the Radon-Nikodym property and f : [a, b] → X be an absolutely continuous function on [a, b] . If a ≤ x_{1} ≤ x_{2} ≤ b (b > a) and t ∈ [0, 1] satisfies the condition

(0 ≤) \frac{x_{1} − a}{b − a} ≤ t ≤ \frac{x_{2} − a}{b − a} (≤ 1) ,

then we have the inequalities

(3.1)  ∥(B) ∫_{a}^{b} f (t) dt − (b − a) [tf (x_{2}) + (1 − t) f (x_{1})]∥

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

≤ \{ \begin{matrix}max \{x_{1} − a, \frac{1}{2} (x_{2} − x_{1}) + |tb + (1 − t) a − \frac{x_{1}+x_{2}}{2} | , b − x_{2}\} ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{(q+1)^{1/q}} & \{(x_{1} − a)^{q+1} + (tb + (1 − t) a − x_{1})^{q+1} + (x_{2} − tb − (1 − t) a)^{q+1} \\ + (b − x_{2})^{q+1}\}^{1/q} ∥|f′|∥_{[a,b],p}, p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ [\frac{(x_{1}−a)^{2}}{2} + \frac{1}{4} (x_{2} − x_{1})^{2} + [tb + (1 − t) a − \frac{x_{1}+x_{2}}{2} ]^{2} + \frac{(b−x_{2})^{2}}{2} ] ∥|f′|∥_{[a,b],∞} , \\ f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

The following particular inequalities are of interest.

1. If x_{1} = a, x_{2} = b, then for any t ∈ [0, 1] , we have the inequalities

(3.2)  ∥(B) ∫_{a}^{b} f (t) dt − (b − a) [tf (b) + (1 − t) f (a)]∥

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

12 P. CERONE, Y.J. CHO, S.S. DRAGOMIR, J.K. KIM, AND S.S. KIM

The best inequality one can get from (3.2) is for t = \frac{1}{2}, obtaining the trapezoidal rule

∥(B) ∫_{a}^{b} f (t) dt − (b − a) .\frac{f (b) + f (a)}{2}∥ (3.3)

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

2. If x_{1} = \frac{3a+b}{4}, x_{2} = \frac{a+3b}{4}, then for any t ∈ [\frac{1}{4}, \frac{3}{4}] we have the inequalities

(3.4)  ∥(B) ∫_{a}^{b} f (t) dt − (b − a) [tf (\frac{3a + b}{4}) + (1 − t) f (\frac{a + 3b}{4})]∥

≤ \{ \begin{matrix}\frac{b−a}{4} ∥|f′|∥_{[a, \frac{3a+b}{4}],1} + [\frac{1}{4} (b − a) + |tb + (1 − t) a − \frac{a+b}{2} |] ∥|f′|∥_{[\frac{3a+b}{4} , \frac{a+3b}{4}],1} \\ +\frac{b−a}{4} ∥|f′|∥_{[\frac{a+3b}{4} ,b],1} ; \\ \frac{1}{4^{q+1}(q+1)^{1/q}} & (b − a) & 1+1/q & ∥|f′|∥ & [a, \frac{3a+b}{4}],p \\ +\frac{1}{(q+1)^{1/q}} [(tb + (1 − t) a − \frac{3a+b}{4})^{q+1} \\ + (\frac{a+3b}{4} − tb − (1 − t) a)^{q+1}] & 1/q & ∥|f′|∥ & [\frac{3a+b}{4} , \frac{a+3b}{4}],p \\ +\frac{1}{4^{q+1}(q+1)^{1/q}} (b − a)^{1+1/q} ∥|f′|∥_{[\frac{a+3b}{4} ,b],p}, p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{(b−a)^{2}}{8} ∥|f′|∥_{[a, \frac{3a+b}{4}],∞} + [\frac{1}{16} (b − a)^{2} + [tb + (1 − t) a − \frac{a+b}{2}]^{2}] ∥|f′|∥_{[\frac{3a+b}{4} , \frac{a+3b}{4}],∞} \\ +\frac{(b−a)^{2}}{8} ∥|f′|∥_{[\frac{a+3b}{4} ,b],∞}, f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}[\frac{1}{4} (b − a) + |tb + (1 − t) a − \frac{a+b}{2} |] ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{(q+1)^{1/q}} & [\frac{2(b−a)^{q+1}}{4^{q+1}} + (tb + (1 − t) a − \frac{3a+b}{4})^{q+1} + (\frac{a+3b}{4} − tb − (1 − t) a)^{q+1}] & 1/q \\ × (b − a)^{1+1/q} ∥|f′|∥_{[a,b],p} , p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ [\frac{1}{8} (b − a)^{2} + (tb + (1 − t) a − \frac{a+b}{2})^{2}] ∥|f′|∥_{[a,b],∞} , f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

The best inequality one can get from (3.4) is for t = \frac{1}{2}, obtaining

∥(B) ∫_{a}^{b} f (t) dt − (b − a) .\frac{f (\frac{a+3b}{4}) + f (\frac{3a+b}{4})}{2}∥ (3.5)

≤ \{ \begin{matrix}\frac{1}{4} (b − a) ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{4(q+1)^{1/q}} (b − a)^{1+1/q} ∥|f′|∥_{[a,b],p} , p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{1}{8} (b − a)^{2} ∥|f′|∥_{[a,b],∞} , f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

Remark 2. One may realize that, instead of using the trapezoidal rule in approximating the Bochner integral (B) ∫_{a}^{b} f (t) dt, that one should use the rule

(3.6)  QT (f; a, b) := (b − a) .\frac{f (\frac{a+3b}{4}) + f (\frac{3a+b}{4})}{2},

that provides twice better theoretical accuracy.

4. The Case of Three Points

The case of three points is important for applications since it contains amongst other Simpson’s quadrature rule.

The following proposition holds.

Proposition 2. Let (X, ∥·∥) be a Banach space with the Radon-Nikodym property and f : [a, b] → X be an absolutely continuous function on [a, b] . If a ≤ x_{1} ≤ x_{2} ≤ x_{3} ≤ b (b > a) and α, β, γ ∈ [0, 1] with α + β + γ = 1 satisfies the condition

(4.1)  (0 ≤) \frac{x_{1} − a}{b − a} ≤ α ≤ \frac{x_{2} − a}{b − a} ≤ α + β ≤ \frac{x_{3} − a}{b − a} (≤ 1) ,

then we have the inequalities

(4.2)  ∥(B) ∫_{a}^{b} f (t) dt − (b − a) [αf (x_{1}) + βf (x_{2}) + (1 − α − β) f (x_{3})]∥

≤ \{ \begin{matrix}(x_{1} − a) ∥|f′|∥_{[a,x_{1}],1} + [\frac{1}{2} (x_{2} − x_{1}) + |αb + (1 − α) a − \frac{x_{1}+x_{2}}{2} |] ∥|f′|∥_{[x_{1},x_{2}],1} \\ + [\frac{1}{2} (x_{3} − x_{2}) + |(α + β) b + (1 − α − β) a − \frac{x_{2}+x_{3}}{2} |] ∥|f′|∥_{[x_{2},x_{3}],1} \\ + (b − x_{2}) ∥|f′|∥_{[x_{2},b],1} ; \\ \frac{1}{(q+1)^{1/q}} & \{(x_{1} − a)^{1+1/q} ∥|f′|∥_{[a,x_{1}],p} \\ + [(αb + (1 − α) a − x_{1})^{q+1} + (x_{2} − αb − (1 − α) a)^{q+1}]^{1/q} ∥|f′|∥_{[x_{1},x_{2}],p} \\ + [((α + β) b + (1 − α − β) a − x_{2})^{q+1} + (x_{3} − (α + β) b − (1 − α − β) a)^{q+1}]^{1/q} ∥|f′|∥_{[x_{2},x_{3}],p} \\ + (b − x_{3})^{1+1/q} ∥|f′|∥_{[x_{3},b],p}\}, p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{(x_{1}−a)^{2}}{2} ∥|f′|∥_{[a,x_{1}],∞} + [\frac{1}{4} (x_{2} − x_{1})^{2} + [αb + (1 − α) a − \frac{x_{1}+x_{2}}{2} ]^{2}] ∥|f′|∥_{[x_{1},x_{2}],∞} \\ [\frac{1}{4} (x_{3} − x_{2})^{2} + [(α + β) b + (1 − α − β) a − \frac{x_{2}+x_{3}}{2} ]^{2}] & ∥|f′|∥ & [x_{2},x_{3}],∞ \\ +\frac{(b−x_{3})^{2}}{2} ∥|f′|∥_{[x_{3},b],∞}, f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}max & \{x_{1} − a, \frac{1}{2} (x_{2} − x_{1}) + |αb + (1 − α) a − \frac{x_{1}+x_{2}}{2} | , \\ \frac{1}{2} (x_{2} − x_{1}) + |(α + β) b + (1 − α − β) a − \frac{x_{2}+x_{3}}{2} | , b − x_{2}\} ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{(q+1)^{1/q}} & \{(x_{1} − a)^{q+1} + (αb + (1 − α) a − x_{1})^{q+1} + (x_{2} − αb − (1 − α) a)^{q+1} \\ + ((α + β) b + (1 − α − β) a − x_{2})^{q+1} + (x_{3} − (α + β) b − (1 − α − β) a)^{q+1} \\ + (b − x_{3})^{q+1}\} ∥|f′|∥_{[a,b],p}, p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ [\frac{(x_{1}−a)^{2}}{2} + \frac{1}{4} (x_{2} − x_{1})^{2} + [αb + (1 − α) a − \frac{x_{1}+x_{2}}{2} ]^{2} \\ +\frac{1}{4} (x_{3} − x_{2})^{2} + [(α + β) b + (1 − α − β) a − \frac{x_{2}+x_{3}}{2} ]^{2} + \frac{(b−x_{3})^{2}}{2} ] ∥|f′|∥_{[a,b],∞} , \\ f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

The following particular inequalities are of interest. 1.Assume that x_{1} = a, x_{2} = \frac{a+b}{2}, x_{3} = b and α, β ∈ [0, 1] so that 0 ≤ α ≤ \frac{1}{2} ≤

α + β ≤ 1, then we have the inequalities

(4.3) ∥(B) ∫_{a}^{b} f (t) dt − (b − a) [αf (a) + βf (\frac{a + b}{2}) + (1 − α − β) f (b)]∥

≤ \{ \begin{matrix}[\frac{1}{4} (b − a) + |αb + (1 − α) a − \frac{3a+b}{4} |] & ∥|f′|∥ & [a, \frac{a+b}{2}],1 \\ + [\frac{1}{4} (b − a) + |(α + β) b + (1 − α − β) a − \frac{a+3b}{4} |] ∥|f′|∥_{[\frac{a+b}{2} ,b],1} ; \\ \frac{1}{(q+1)^{1/q}} & \{[α^{q+1} (b − a)^{q+1} + (\frac{a+b}{2} − αb − (1 − α) a)^{q+1}]^{1/q} ∥|f′|∥_{[a, \frac{a+b}{2}],p} \\ + [((α + β) b + (1 − α − β) a − \frac{a+b}{2})^{q+1} + (1 − α − β)^{q+1} (b − a)^{q+1}]^{1/q}\} ∥|f′|∥_{[\frac{a+b}{2} ,b],p} \\ p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ [\frac{1}{16} (b − a)^{2} + [αb + (1 − α) a − \frac{3a+b}{4}]^{2}] & ∥|f′|∥ & [a, \frac{a+b}{2}],∞ \\ [\frac{1}{16} (b − a)^{2} + [(α + β) b + (1 − α − β) a − \frac{a+3b}{4}]^{2}] & ∥|f′|∥ & [\frac{a+b}{2} ,b],∞ \\ f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}max & \{[\frac{1}{4} (b − a) + |αb + (1 − α) a − \frac{3a+b}{4} |] , \\ [\frac{1}{4} (b − a) + |(α + β) b + (1 − α − β) a − \frac{a+3b}{4} |]\} & ∥|f′|∥ & [a,b],1 \\ \frac{1}{(q+1)^{1/q}} & \{α^{q+1} (b − a)^{q+1} + (\frac{a+b}{2} − αb − (1 − α) a)^{q+1} \\ + ((α + β) b + (1 − α − β) a − \frac{a+b}{2})^{q+1} + (1 − α − β)^{q+1} (b − a)^{q+1}\}^{1/q} ∥|f′|∥_{[a,b],p} \\ p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \{\frac{1}{8} (b − a)^{2} + [αb + (1 − α) a − \frac{3a+b}{4}]^{2} \\ + [(α + β) b + (1 − α − β) a − \frac{a+3b}{4}]^{2}\} & ∥|f′|∥ & [a,b],∞ \\ f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

It is easy to see that, the best inequality one can derive from (4.3) is the one for α = \frac{1}{4} and β = \frac{3}{4}, getting

(4.4)  ∥(B) ∫_{a}^{b} f (t) dt − (b − a) [\frac{f (a) + f (b)}{4} + \frac{1}{2}f (\frac{a + b}{2})]∥

≤ \{ \begin{matrix}\frac{1}{4} (b − a) ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{2^{2+1/q}(q+1)^{1/q}} & (b − a) & 1+1/q & \{∥|f′|∥_{[a, \frac{a+b}{2}],p} + ∥|f′|∥_{[\frac{a+b}{2} ,b],p}\} \\ p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{1}{16} (b − a)^{2} [∥|f′|∥_{[a, \frac{a+b}{2}],∞} + ∥|f′|∥_{[\frac{a+b}{2} ,b],∞}] , f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}\frac{1}{4} (b − a) ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{2^{2+1/q}(q+1)^{1/q}} (b − a)^{1+1/q} ∥|f′|∥_{[a,b],p} , p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{1}{8} (b − a)^{2} ∥|f′|∥_{[a,b],∞} f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

The inequality (4.3) incorporates Simpson’s rule as well. Indeed, if we choose α = \frac{1}{6}, β = \frac{4}{6}, then we get from (4.3) the following result

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

≤ \{ \begin{matrix}\frac{1}{3} (b − a) ∥|f′|∥_{[a,b],1} ; \\ \frac{1}{(q+1)^{1/q}} & \frac{(2^{q+1}+1)^{1/q}}{6^{1+1/q}} & (b − a) & 1+1/q & \{∥|f′|∥_{[a, \frac{a+b}{2}],p} + ∥|f′|∥_{[\frac{a+b}{2} ,b],p}\} \\ p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{5}{72} (b − a)^{2} [∥|f′|∥_{[a, \frac{a+b}{2}],∞} + ∥|f′|∥_{[\frac{a+b}{2} ,b],∞}] , f′ ∈ L_{∞} ([a, b] ; X) ; \end{matrix}

≤ \{ \begin{matrix}\frac{1}{3} (b − a) ∥|f′|∥_{[a,b],1} ; \\ \frac{(2^{q+1}+1)^{1/q}}{2·3^{1+1/q}(q+1)^{1/q}} (b − a)^{1+1/q} ∥|f′|∥_{[a,b],p} , p > 1, \frac{1}{p} + \frac{1}{q} = 1, f′ ∈ L_{p} ([a, b] ; X) ; \\ \frac{5}{36} (b − a)^{2} ∥|f′|∥_{[a,b],∞} f′ ∈ L_{∞} ([a, b] ; X) . \end{matrix}

Remark 3. It is obvious that, if the values in a, \frac{a+b}{2} and b of the function f : [a, b] → X are available, then one should choose the rule

QS (f; a, b) := (b − a) [\frac{f (a) + f (b)}{4} + \frac{1}{2}f (\frac{a + b}{2})]

that provides a better approximation for the Bochner integral (B) ∫_{a}^{b} f (t) dt than the classical Simpson’s rule.

2. If one chooses x_{1} = \frac{3a+b}{4}, x_{2} = \frac{a+b}{2}, x_{3} = \frac{a+3b}{4}, then by the use of inequality (4.2) , that one can derive estimates for the norm of difference

(B) ∫_{a}^{b} f (t) dt − (b − a) [αf (\frac{3a + b}{4}) + βf (\frac{a + b}{2}) + (1 − α − β) f (\frac{a + 3b}{4})]

in terms of the Lebesgue norms of the derivative f′.

We omit the details.

Acknowledgement: S. S. Dragomir and Y. J. Cho greatly acknowledge the financial support from the Brain Pool Program (2002) of the Korean Federation of Science and Technology Societies. The research was performed under the ”Memorandum of Understanding” between Victoria University and Gyeongsang National University.

References

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

[2] W. ARENDT, C.J.K. BATTY, M. HIEBER and F. NEUBRANDER, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag Basel, 2001.

[3] N.S. BARNETT, C. BUŞE, P. CERONE and S.S. DRAGOMIR, Ostrowski’s inequality for vector-valued functions and applications, Comp. & Math. Appl., 44(2002), 559-572.

[4] S. BOCHNER, Integration von Funktionen, deren Werte die Elemente eines Vektorraumes, Fund. Math., 20 (1933), 262-276.

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

[6] J. DIETSEL and J.J. UHL, Jr., Vector Measures, Amer. Math. Society, Mathematical Surveys, 15, Providence, Rhode Island, (1977).

[7] S.S. DRAGOMIR and S. WANG, 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 (1997), 239-244.

[8] S.S. DRAGOMIR and S. WANG, A new inequality of Ostrowski’s type in L_{p}−norm, Indian J. of Math., 40(3) (1998), 295-304.

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

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

[11] E. HILLE and R.S. PHILIPS, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ., Vol. 31, Providence, R.I. (1957).

[12] J. MIKUSIŃSKI, The Bochner Integral, Birkhäuser Verlag Basel, 1978.

[13] R. NAGEL (Ed.), One-Parameter Semigroups of Positive Operators, Springer Lect. Notes in Math., 1184 (1986).

[14] A. PAZY, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, (1983).

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

E-mail address: pc@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/cerone/index.html

Department of Mathematics Education, College of Education, Gyeongsang National University, Chinju 660-701, KOREA

E-mail address: yjcho@nongae.gsnu.ac.kr

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

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

Department of Mathematics, Kyungnam University, Masan,, Kyungnam 631-701, Korea

E-mail address: jongkyuk@kyungnam.ac.kr

Department of Mathematics, Dongeui University, Pusan 614-714, Korea

E-mail address: sskim@dongeui.ac.kr