THE GENERALISED INTEGRATION BY PARTS FORMULA FOR APPELL SEQUENCES AND RELATED RESULTS

S.S. DRAGOMIR

Abstract. A generalised integration by parts formula for sequences of absolutely continuous functions that satisfy the w−Appell condition and different estimates for the remainder are provided. Applications for particular instances of such sequences are pointed out as well.

1. Introduction

In [6], Matić et. al introduced the concept of harmonic sequences of polynomials by assuming that the polynomial {P_{n}}_{n∈ℕ} satisfies the condition

(1.1)  P_{0} = 1, P′_{n} (t) = P_{n−1} (t) for all t ∈ ℝ and n ∈ ℕ.

With this assumption, they proved the following generalised Taylor’s formula: Theorem 1. Let I ⊂ ℝ be a closed interval and a ∈ I. If f : I → ℝ is any function such that, for some n ∈ ℕ, f^{(n)} is absolutely continuous, then for any x ∈ I

(1.2) f (x) = f (a) + ∑_{k=1}^{n} (−1)^{k+1} [P_{k} (x) f^{(k)} (x) − P_{k} (a) f^{(k)} (a)] + R_{n} (f; a, x) ,

where

(1.3)  R_{n} (f; a, x) = (−1)^{n} ∫_{a}^{x} P_{n} (t) f^{(n+1)} (t) dt

and {P_{n}}_{n∈ℕ} is a harmonic sequence of polynomials.

As examples of such polynomials, they mentioned the following

P_{n} (t) := \frac{1}{n!} (t − x)^{n} , t ∈ ℝ;

or

P_{n} (t) := \frac{1}{n!} (t − \frac{a + x}{2})^{n} , t ∈ ℝ;

or

P_{n} (t) := \frac{(x − a)^{n}}{n!} B_{n} (\frac{t − a}{x − a}) , n ≥ 1, P_{0} (t) = 1,

where B_{n} (·) are the Bernoulli’s polynomials, or

P_{n} (t) := \frac{(x − a)^{n}}{n!} E_{n} (\frac{t − a}{x − a}) , n ≥ 1, P_{0} (t) = 1,

where E_{n} (·) are the Euler’s polynomials.

Date: January 25, 2001.

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

Key words and phrases. Integration by parts formula, Appell polynomials, Korkine identity, Analytic inequalities.

Amongst others, they proved the following general estimation result for the remainder R_{n} (f; a, x). Corollary 1. Under the assumptions of Theorem 1 and if x ≥ a, then

(1.4)  |R_{n} (f; a, x)| ≤ \{ \begin{matrix}max_{t∈[a,x]} |P_{n} (t)| ∫_{a}^{x} |f^{(n+1)} (s)| ds; \\ max_{t∈[a,x]} |f^{(n+1)} (t)| ∫_{a}^{x} |P_{n} (s)| ds; \\ (∫_{a}^{x} |P_{n} (s)|^{q} ds)^{\frac{1}{q}} (∫_{a}^{x} |f^{(n+1)} (s)|^{p} ds)^{\frac{1}{p}} , \\ where \frac{1}{p} + \frac{1}{q} = 1, p > 1. \end{matrix}

Now, if one would choose in (1.2) f (x) = ∫_{a}^{b} g (t) dt and put x = b, we could then state the following generalised integration by parts formula

(1.5) ∫_{a}^{b} g (t) dt = ∑_{k=1}^{n} (−1)^{k+1} [P_{k} (b) g^{(k−1)} (b) − P_{k} (a) g^{(k−1)} (a)]
+S_{n} (f; a, b) ,

where

(1.6)  S_{n} (f; a, b) = (−1)^{n} ∫_{a}^{b} P_{n} (t) g (t) dt.

Using the classical notation for the Lebesgue norms,

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

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

the remainder (1.6) may be bounded in the following manner

(1.7)  |S_{n} (g; a, b)| ≤ \{ \begin{matrix}∥P_{n}∥_{∞} ∥g^{(n)}∥_{1} , \\ ∥P_{n}∥_{1} ∥g^{(n)}∥_{∞} , \\ ∥P_{n}∥_{q} ∥g^{(n)}∥_{p}, where \frac{1}{p} + \frac{1}{q} = 1, p > 1. \end{matrix}

For other results based on the integration by parts formula (1.5), see [1] – [5] and [7] – [9].

2. The Generalised Integration by Parts Formula

We shall start with the following definition. Definition 1. Let w : [a, b] → ℝ be an absolutely continuous function on [a, b]. The sequence of absolutely continuous functions \{w_{k}\}_{k=\overline{0,n}} (n ≥ 1) defined on [a, b] are said to be of w−Appell type if

(2.1)  w_{0} = w a.e. on [a, b] ;

(2.2)  w′_{k} = w_{k−1} a.e. on [a, b] , for all k = 1, …, n.

Remark 1. It is obvious that any sequence of harmonic polynomials is a sequence of w−Appell type with w = 1.

Remark 2. Having given an absolutely continuous function w : [a, b] → ℝ we may construct a sequence of w−Appell type in the following canonical fashion:

w_{1} (t) = ∫_{a}^{t} w (s) ds + c_{1}, t ∈ [a, b] , c_{1} ∈ ℝ;
w_{2} (t) = ∫_{a}^{t} w_{1} (s) ds + c_{2}, t ∈ [a, b] , c_{2} ∈ ℝ;
…
w_{n} (t) = ∫_{a}^{t} w_{n−1} (s) ds + c_{n}, t ∈ [a, b] , c_{n} ∈ ℝ.

The following generalised integration by parts formula associated with the sequence \{w_{k}\}_{k=\overline{0,n}} naturally holds.

Lemma 1. Let w : [a, b] → ℝ be an absolutely continuous function on [a, b] and \{w_{k}\}_{k=\overline{0,n}} a sequence of w−Appell type. If g : [a, b] → ℝ is such that g^{(n−1)}is absolutely continuous on [a, b] and w_{n}g^{(n)} ∈ L_{1} [a, b], then we have the equality

(2.3)  ∫_{a}^{b} w (t) g (t) dt = A_{n} (w, g; a, b) + R_{n} (w, g; a, b) ,

where

(2.4)  A_{n} (w, g; a, b) = ∑_{k=1}^{n} (−1)^{k−1} [w_{k} (b) g^{(k−1)} (b) − w_{k} (a) g^{(k−1)} (a)]

and

(2.5)  R_{n} (w, g; a, b) = (−1)^{n} ∫_{a}^{b} w_{n} (t) g^{(n)} (t) dt.

Proof. We prove (2.3) by mathematical induction.

For n = 1, we have

∫_{a}^{b} w (t) g (t) dt = w_{1} (b) g (b) − w_{1} (a) g (a) − ∫_{a}^{b} w_{1} (t) g^{(1)} (t) dt

which follows on applying the integration by parts formula taking into account that w′_{1} = w a.e. on [a, b].

Assume that (2.3) holds for m ∈ {1, …, n − 1}, i.e.,

(2.6) ∫_{a}^{b} w (t) g (t) dt = ∑_{k=1}^{m} (−1)^{k−1} [w_{k} (b) g^{(k−1)} (b) − w_{k} (a) g^{(k−1)} (a)]

+ (−1)^{m} ∫_{a}^{b} w_{m} (t) g^{(m)} (t) dt.

As w_{m} and g^{(m)} are absolutely continuous and w′_{m+1} = w_{m} a.e. on [a, b], then, by the integration by parts formula we may write

(2.7)  ∫_{a}^{b} w_{m} (t) g^{(m)} (t) dt = ∫_{a}^{b} w′_{m+1} (t) g^{(m)} (t) dt
= w_{m+1} (b) g^{(m)} (b) − w_{m+1} (a) g^{(m)} (a)
− ∫_{a}^{b} w_{m+1} (t) g^{(m+1)} (t) dt.

Using (2.6), we get

∫_{a}^{b} w (t) g (t) dt = ∑_{k=1}^{m} (−1)^{k−1} [w_{k} (b) g^{(k−1)} (b) − w_{k} (a) g^{(k−1)} (a)]

+ (−1)^{m} [w_{m+1} (b) g^{(m)} (b) − w_{m+1} (a) g^{(m)} (a)

− ∫_{a}^{b} w_{m+1} (t) g^{(m+1)} (t) dt]

= ∑_{k=1}^{m+1} (−1)^{k−1} [w_{k} (b) g^{(k−1)} (b) − w_{k} (a) g^{(k−1)} (a)]

+ (−1)^{m+1} ∫_{a}^{b} w_{m+1} (t) g^{(m+1)} (t) dt

showing that (2.3) holds for n = m + 1 as well.

The lemma is thus proved.

Remark 3. If w_{n} = P_{n}, {P_{n}}_{n∈ℕ} is a sequence of harmonic polynomials, then by (2.3) we recapture the identity (1.6).

The following result concerning estimates of the remainder R_{n} (w, g; a, b) holds.

Theorem 2. With the assumptions of Lemma 1, we have the estimates:

(2.8) |R_{n} (g; a, b)| ≤ \{ \begin{matrix}∥w_{n}∥_{1} ∥g^{(n)}∥_{∞} & if & w_{n} ∈ L_{1} [a, b] , g^{(n)} ∈ L_{∞} [a, b] ; \\ ∥w_{n}∥_{q} ∥g^{(n)}∥_{p}, & if & w_{n} ∈ L_{q} [a, b] , g^{(n)} ∈ L_{p} [a, b] ; \\ & and & \frac{1}{p} + \frac{1}{q} = 1, p > 1; \\ ∥w_{n}∥_{∞} ∥g^{(n)}∥_{1} & if & w_{n} ∈ L_{∞} [a, b] . \end{matrix}

The proof follows by the representation (2.5) and Hölder’s inequalities for ∥·∥_{p} −norms (p ∈ [1, ∞]).

In what follows, we point out a number of useful examples.

1. Define w_{0}^{(1)} (t) := e^{αt}, α ∈ ℝ\ {0} and consider the sequence w_{k}^{(1)} (t) :=

\frac{1}{α^{k}} e^{αt}, k = 0, 1, . . .. Then

\frac{dw_{k}^{(1)} (t)}{dt} = \frac{1}{α^{k−1}} e^{αt} = w_{k−1}^{(1)} (t) , for k = 1, 2, …

showing that {w_{k}^{(1)}}_{k∈ℕ} is an w_{0}^{(1)}−Appell type sequence.

INTEGRATION BY PARTS FORMULA 5

If we use Lemma 1, we may state the identity:

(2.9)  ∫_{a}^{b} e^{αt}g (t) dt = ∑_{k=1}^{n} \begin{matrix}(−1) & k−1 \\ α & k \end{matrix} [e^{αb}g^{(k−1)} (b) − e^{αa}g^{(k−1)} (a)]

+\begin{matrix}(−1) & n \\ α & n \end{matrix} ∫_{a}^{b} e^{αt}g^{(n)} (t) dt.

As

∥w_{n}^{(1)}∥_{1} = ∫_{a}^{b} |\frac{e^{αt}}{α^{n}} | dt = \frac{1}{|α|^{n}} ∫_{a}^{b} e^{αt}dt = \frac{1}{|α|^{n}} · |\frac{e^{αb} − e^{αa}}{α} | ,

∥w_{n}^{(1)}∥_{q} = (∫_{a}^{b} |\frac{e^{αt}}{α^{n}} |^{q} dt)^{\frac{1}{q}} = \frac{1}{|α|^{n}} (∫_{a}^{b} e^{αqt}dt)^{\frac{1}{q}} = \frac{1}{|α|^{n}} |\frac{e^{αqb} − e^{αqa}}{αq} |^{\frac{1}{q}} ,

∥w_{n}^{(1)}∥_{∞} = \frac{1}{|α|^{n}} sup_{t∈[a,b]} |e^{αt}| = \frac{1}{|α|^{n}} max \{e^{αa}, e^{αb}\}
= \frac{1}{|α|^{n}} · \frac{e^{αa} + e^{αb} + |e^{αa} − e^{αb}|}{2}.

Then, by Theorem 2 we may state the inequality:

|∫_{a}^{b} e^{αt}g (t) dt − ∑_{k=1}^{n} \begin{matrix}(−1) & k−1 \\ α & k \end{matrix} [e^{αb}g^{(k−1)} (b) − e^{αa}g^{(k−1)} (a)]| (2.10)

≤ \{ \begin{matrix}\frac{|e^{αb} − e^{αa}|}{|α|^{n+1}} ∥g^{(n)}∥_{∞} & if & g^{(n)} ∈ L_{∞} [a, b] ; \\ \frac{1}{|α|^{n}} |\frac{e^{αqb} − e^{αqa}}{αq} |^{\frac{1}{q}} ∥g^{(n)}∥_{p}, & if & g^{(n)} ∈ L_{p} [a, b] ; \\ & and & \frac{1}{p} + \frac{1}{q} = 1, p > 1; \\ \frac{1}{|α|^{n}} · \frac{e^{αa} + e^{αb} + |e^{αa} − e^{αb}|}{2} ∥g^{(n)}∥_{1} , \end{matrix}

where α ∈ ℝ\ {0}.

2. Define w_{0}^{(2)} (t) = t^{α}, α > −1 and consider

w_{k}^{(2)} (t) := \frac{t^{α+k}}{(α + 1) (α + 2) ⋯ (α + k)}, k = 0, 1, ….

Then

\frac{dw_{k}^{(1)} (t)}{dt} = \frac{t^{α+k−1}}{(α + 1) (α + 2) ⋯ (α + k − 1)} = w_{k−1}^{(2)} (t) , for k = 1, 2, …,

which shows that {w_{k}^{(2)}}_{k=0,1,...} is a w_{0}^{(2)}−Appell type sequence.

If we use Lemma 1, we may state the identity:

(2.11) ∫_{a}^{b} t^{α}g (t) dt = ∑_{k=1}^{n} (−1)^{k−1} \frac{[b^{α+k}g^{(k−1)} (b) − a^{α+k}g^{(k−1)} (a)]}{(α + 1) (α + 2) ⋯ (α + k)}

+ \begin{matrix}(−1) & n \\ (α + 1) & ⋯ & (α + n) \end{matrix} ∫_{a}^{b} t^{α+n}g^{(n)} (t) dt, [a, b] ∈ (0, ∞) .

6 S.S. DRAGOMIR

As, for [a, b] ⊂ (0, ∞),

∥w_{n}^{(2)}∥_{1} = ∫_{a}^{b} |\frac{t^{α+n}}{(α + 1) ⋯ (α + n)}| dt = \frac{b^{α+n+1} − a^{α+n+1}}{(α + 1) ⋯ (α + n) (α + n + 1)}
= \frac{(b − a)}{(α + 1) ⋯ (α + n)}L_{α+n}^{α+n} (a, b) ,

where

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

is the m−logarithmic mean, m ∈ ℝ\ {−1, 0}; and for q > 1

∥w_{n}^{(2)}∥_{q} = (∫_{a}^{b} |\frac{t^{α+n}}{(α + 1) ⋯ (α + n)}|^{q} dt)^{\frac{1}{q}}
= \frac{1}{(α + 1) ⋯ (α + n)} [\frac{b^{(α+n)q+1} − a^{(α+n)q+1}}{((α + n) q + 1) (b − a)} · (b − a)]^{\frac{1}{q} · \frac{(α+n)q}{(α+n)q}}
= \begin{matrix}(b − a) & \frac{1}{q} \\ (α + 1) & ⋯ & (α + n) \end{matrix}L_{(α+n)q}^{α+n} (a, b) ,

and, finally

∥w_{n}^{(2)}∥_{∞} = sup_{t∈[a,b]} |\frac{t^{α+n}}{(α + 1) ⋯ (α + n)}| = \frac{1}{(α + 1) ⋯ (α + n)}b^{α+n}, n ≥ 1

then by Theorem 2, we may state the inequality |∫_{a}^{b} t^{α}g (t) dt (2.12)

− ∑_{k=1}^{n} \begin{matrix}(−1) & k−1 \\ (α + 1) & (α + 2) & ⋯ & (α + k) \end{matrix} [b^{α+k}g^{(k−1)} (b) − a^{α+k}g^{(k−1)} (a)]|

≤ \{ \begin{matrix}\frac{(b − a) L_{α+n}^{α+n} (a, b)}{(α + 1) ⋯ (α + n)} ∥g^{(n)}∥_{∞} & if & g^{(n)} ∈ L_{∞} [a, b] ; \\ \frac{(b − a)^{\frac{1}{q}} L_{(α+n)q}^{α+n} (a, b)}{(α + 1) ⋯ (α + n)} ∥g^{(n)}∥_{p}, & if & g^{(n)} ∈ L_{p} [a, b] ; \\ & and & \frac{1}{p} + \frac{1}{q} = 1, p > 1; \\ \frac{b^{α+n}}{(α + 1) ⋯ (α + n)} ∥g^{(n)}∥_{1} , & if & g^{(n)} ∈ L_{1} [a, b] , \end{matrix}

for all α > −1.

3. Define w_{0}^{(3)} (t) = sin (αt), α ∈ ℝ\ {0} and consider the sequence

w_{k}^{(3)} (t) = \begin{matrix}(−1) & k \\ α & k \end{matrix} sin (αt + k · \frac{π}{2} ) .

INTEGRATION BY PARTS FORMULA 7

Then

\frac{dw_{k}^{(3)} (t)}{dt} = \begin{matrix}(−1) & k \\ α & k \end{matrix} α cos [αt + k · \frac{π}{2} ] = \begin{matrix}(−1) & k \\ α & k−1 \end{matrix} sin [\frac{π}{2} − (αt + k \frac{π}{2} )]
= \begin{matrix}(−1) & k−1 \\ α & k−1 \end{matrix} sin (αt + k \frac{π}{2} − \frac{π}{2} ) = \begin{matrix}(−1) & k−1 \\ α & k−1 \end{matrix} sin (αt + (k − 1) \frac{π}{2} )
= w_{k−1}^{(3)} (t) , t ∈ ℝ,

which shows that {w_{k}^{(3)}}_{k=0,1,...} is an w_{0}^{(3)}−Appell type sequence. If we use Lemma 1, we may state the identity

(2.13) ∫_{a}^{b} sin (αt) · g (t) dt

= ∑_{k=1}^{n} \frac{1}{α^{k}} [sin (αa + k \frac{π}{2} ) · g^{(k−1)} (a) − sin (αb + k \frac{π}{2} ) · g^{(k−1)} (b)]

+ \frac{1}{α^{n}} ∫_{a}^{b} sin (αt + k \frac{π}{2} ) g^{(n)} (t) dt.

We compute

∥w_{n}^{(3)}∥_{2} := [\frac{1}{α^{2n}} ∫_{a}^{b} sin^{2} (αt + n\frac{π}{2} ) dt]^{\frac{1}{2}}

= \frac{1}{|α|^{n}} [∫_{a}^{b} [\frac{1 − cos (2αt + n · π)}{2}] dt]^{\frac{1}{2}}

= \frac{1}{|α|^{n}} [\frac{1}{2} (b − a) − \frac{1}{4α} [sin (2αb + nπ) − sin (2αa + nπ)]]^{\frac{1}{2}}

= \begin{matrix}(b − a) & \frac{1}{2} \\ \sqrt{2} & |α| & n \end{matrix} [1 − \frac{sin (2αb + nπ) − sin (2αa + nπ)}{2α (b − a)}]^{\frac{1}{2}} .

Consequently, using Theorem 2 for Hilbertian norms, we may state that

|∫_{a}^{b} sin (αt) g (t) dt (2.14)

− ∑_{k=1}^{n} \frac{1}{α^{k}} [sin (αa + k \frac{π}{2} ) · g^{(k−1)} (a) − sin (αb + k \frac{π}{2} ) · g^{(k−1)} (b)]|

≤ \begin{matrix}(b − a) & \frac{1}{2} \\ \sqrt{2} & |α| & n \end{matrix} [1 − \frac{sin (2bα + nπ) − sin (2aα + nπ)}{2α (b − a)}]^{\frac{1}{2}} ∥g^{(n)}∥_{2} ,

provided g^{(n)} ∈ L_{2} [a, b] .

4. Define w_{0}^{(4)} (t) = cos (αt), α ∈ ℝ\ {0} and consider the sequence w_{k}^{(4)} (t)

= \begin{matrix}(−1) & ^{k} \\ α & ^{k} \end{matrix} cos (αt + k · \frac{π}{2} ). Then

\frac{dw_{k}^{(4)} (t)}{dt} = \begin{matrix}(−1) & k \\ α & k \end{matrix} [−α sin [αt + k · \frac{π}{2} ]] = \begin{matrix}(−1) & k−1 \\ α & k−1 \end{matrix} sin (αt + k \frac{π}{2} )
= \begin{matrix}(−1) & k−1 \\ α & k−1 \end{matrix} cos [\frac{π}{2} − (αt + k \frac{π}{2} )] = \begin{matrix}(−1) & k−1 \\ α & k−1 \end{matrix} cos [αt + (k − 1) \frac{π}{2} ]
= w_{k−1}^{(4)} (t) , t ∈ ℝ,

which shows that {w_{k}^{(4)}}_{k=0,1,...} is an w_{0}^{(4)}−Appell type sequence. If we use Lemma 1, we may state the identity

∫_{a}^{b} cos (αt) g (t) dt (2.15)

= ∑_{k=1}^{n} \frac{1}{α^{k}} [cos (αa + k \frac{π}{2} ) · g^{(k−1)} (a) − cos (αb + k \frac{π}{2} ) · g^{(k−1)} (b)]

+\frac{1}{α^{n}} ∫_{a}^{b} cos (αt + k \frac{π}{2} ) g^{(n)} (t) dt.

We compute

∥w_{n}^{(4)}∥_{2} : = [\frac{1}{α^{2n}} ∫_{a}^{b} cos^{2} (αt + n\frac{π}{2} ) dt]^{\frac{1}{2}}
= \frac{1}{|α|^{n}} [∫_{a}^{b} [\frac{1 + cos (2αt + n · π)}{2}] dt]^{\frac{1}{2}}
= \frac{1}{|α|^{n}} [\frac{1}{2} (b − a) + \frac{1}{4α} [sin (2αb + nπ) − sin (2αa + nπ)]]^{\frac{1}{2}}
= \begin{matrix}(b − a) & \frac{1}{2} \\ \sqrt{2} & |α| & n \end{matrix} [1 + \frac{sin (2αb + nπ) − sin (2αa + nπ)}{2α (b − a)}]^{\frac{1}{2}} .

Consequently, using Theorem 2 for Hilbertian norms, we may state that

(2.16) |∫_{a}^{b} cos (αt) g (t) dt

− ∑_{k=1}^{n} \frac{1}{α^{k}} [cos (αa + k \frac{π}{2} ) · g^{(k−1)} (a) − cos (αb + k \frac{π}{2} ) · g^{(k−1)} (b)]|

≤ \begin{matrix}(b − a) & \frac{1}{2} \\ \sqrt{2} & |α| & n \end{matrix} [1 + \frac{sin (2bα + nπ) − sin (2aα + nπ)}{2α (b − a)}]^{\frac{1}{2}} ∥g^{(n)}∥_{2} ,

provided g^{(n)} ∈ L_{2} [a, b].

INTEGRATION BY PARTS FORMULA 9

3. A Perturbed Version via Korkine’s Identity

The following identity which can be easily proved by direct computation is known in the literature as Korkine’s identity:

(3.1) \frac{1}{b − a} ∫_{a}^{b} u (t) v (t) dt − \frac{1}{b − a} ∫_{a}^{b} u (t) dt · \frac{1}{b − a} ∫_{a}^{b} v (t) dt

= \frac{1}{2 (b − a)^{2}} ∫_{a}^{b} ∫_{a}^{b} (u (t) − u (s)) (v (t) − v (s)) dtds,

provided that u, v : [a, b] → ℝ are measurable and all the involved integrals exist.

The following representation of the weighted integral ∫_{a}^{b} w (t) g (t) dt holds. Lemma 2. Let w : [a, b] → ℝ be an absolutely continuous function on [a, b] and \{w_{k}\}_{k=\overline{0,n+1}} be a sequence of w−Appell type. If g : [a, b] → ℝ is such that g^{(n−1)} is absolutely continuous on [a, b] and w_{n}g^{(n)} ∈ L_{1} [a, b], then we have the equality

(3.2)  ∫_{a}^{b} w (t) g (t) dt = B_{n} (w, g; a, b) + S_{n} (w, g; a, b) ,

where

(3.3) B_{n} (w, g; a, b) = A_{n} (w, g; a, b) + (−1)^{n} [w_{n+1} (b) − w_{n+1} (a)] [g^{(n−1)}; a, b]

and the remainder S_{n} (w, g; a, b) can be represented by

(3.4) S_{n} (w, g; a, b) = \frac{(−1)^{n}}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} (w_{n} (t) − w_{n} (s)) (g^{(n)} (t) − g^{(n)} (s)) dtds

where [g^{(n−1)}; a, b] is the devided difference, i.e., we recall that

[g^{(n−1)}; a, b] := \frac{g^{(n−1)} (b) − g^{(n−1)} (a)}{b − a}.

Proof. Using Korkine’s identity, we may write

∫_{a}^{b} w_{n} (t) g^{(n)} (t) dt = \frac{1}{b − a} ∫_{a}^{b} w_{n} (t) dt · ∫_{a}^{b} g^{(n)} (t) dt

+\frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} (w_{n} (t) − w_{n} (s)) (g^{(n)} (t) − g^{(n)} (s)) dtds

= [w_{n+1} (b) − w_{n+1} (a)] · \frac{g^{(n−1)} (b) − g^{(n−1)} (a)}{b − a}

+\frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} (w_{n} (t) − w_{n} (s)) (g^{(n)} (t) − g^{(n)} (s)) dtds,

giving the following representation for the remainder R_{n} (w, g; a, b):

R_{n} (w, g; a, b) = (−1)^{n} [w_{n+1} (b) − w_{n+1} (a)] · \frac{g^{(k−1)} (b) − g^{(k−1)} (a)}{b − a}

+\frac{(−1)^{n}}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} (w_{n} (t) − w_{n} (s)) (g^{(n)} (t) − g^{(n)} (s)) dtds.

Using the identity (2.3) we deduce (3.2).

For an absolutely continuous function h : [α, β] → ℝ, we denote

∥h∥_{[α,β],1} := |∫_{α}^{β} |h (τ)| dτ| .

Using this notation, we may state the following result involving the estimation of the remainder S (w, g; a, b) in the perturbed formula (3.2). Theorem 3. With the assumption of Lemma 2, we have

(3.5) |S (w, g; a, b)| ≤ \frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} ∥w_{n+1}∥_{[s,t],1} ∥g^{(n+1)}∥_{[s,t],1} dtds
= : M_{n} (w_{n+1}, g^{(n+1)}) .

Proof. Using the representation (3.4), we may write

|S (w, g; a, b)| ≤ \frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} |w_{n} (t) − w_{n} (s)| |g^{(n)} (t) − g^{(n)} (s)| dtds
= \frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} |∫_{s}^{t} w_{n+1} (τ) dτ| |∫_{s}^{t} g^{(n+1)} (σ) dσ| dtds
≤ \frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} ∥w_{n+1}∥_{[s,t],1} ∥g^{(n+1)}∥_{[s,t],1} dtds

and the estimate (3.5) is obtained.

In practical applications some bounds of M_{n} (w_{n+1}, g^{(n+1)}) could be more useful. Now, it obvious that

M_{n} (w_{n+1}, g^{(n+1)})

≤ \frac{1}{2 (b − a)}ess sup_{(t,s)∈[a,b]} \{∥w_{n+1}∥_{[s,t],1}\} · ∫_{a}^{b} ∫_{a}^{b} ∥g^{(n+1)}∥_{[s,t],1} dtds

= \frac{1}{2 (b − a)} ∥∥w_{n+1}∥_{[·,·],1}∥_{∞} ∥∥g^{(n+1)}∥_{[·,·],1}∥_{1} .

Also, by Hölder’s integral inequality for double integrals, we may write that

M_{n} (w_{n+1}, g^{(n+1)}) ≤ \frac{1}{2 (b − a)} (∫_{a}^{b} ∫_{a}^{b} ∥w_{n+1}∥_{[s,t],1}^{q} dtds)^{\frac{1}{q}}

× (∫_{a}^{b} ∫_{a}^{b} ∥g^{(n+1)}∥_{[s,t],1}^{p} dtds)^{\frac{1}{p}}

= \frac{1}{2 (b − a)} ∥∥w_{n+1}∥_{[·,·],1}∥_{q} ∥∥g^{(n+1)}∥_{[·,·],1}∥_{p} ,

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

In a similar manner,

M_{n} (w_{n+1}, g^{(n+1)}) ≤ \frac{1}{2 (b − a)} ∥∥w_{n+1}∥_{[·,·],1}∥_{1} ∥∥g^{(n+1)}∥_{[·,·],1}∥_{∞} .

Consequently, we may state the following corollary.

Corollary 2. With the assumptions of Lemma 2, we have the following bounds for the remainder S_{n} (w, g; a, b): (3.6)

|S_{n} (w, g; a, b)| ≤ \{ \begin{matrix}\frac{1}{2(b−a)} ∥∥w_{n+1}∥_{[·,·],1}∥_{∞} ∥∥g^{(n+1)}∥_{[·,·],1}∥_{1} ; \\ \frac{1}{2(b−a)} ∥∥w_{n+1}∥_{[·,·],1}∥_{q} ∥∥g^{(n+1)}∥_{[·,·],1}∥_{p} , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{2(b−a)} ∥∥w_{n+1}∥_{[·,·],1}∥_{1} ∥∥g^{(n+1)}∥_{[·,·],1}∥_{∞} , \end{matrix}

where ∥·∥_{s} (s ∈ [1, ∞]) are the usual Lebesgue norms on L_{s} ([a, b] × [a, b])^{2} .

Now, if we use the natural notations

∥h∥_{[t,s],β} := |∫_{t}^{s} |h (τ)|^{β} dτ|^{\frac{1}{β}} , β > 1

and

∥h∥_{[t,s],∞} := ess sup {|h (τ)| , τ ∈ [t, s] ([s, t])} ,

then by Hölder’s integral inequality, we have

(3.7) ∥w_{n+1}∥_{[s,t],1} ≤ |s − t|^{\frac{1}{α}} ∥w_{n+1}∥_{[s,t],β} , β > 1, \frac{1}{β} + \frac{1}{α} = 1 or α = 1, β = ∞

and

∥g^{(n+1)}∥_{[s,t],1} (3.8)

≤ |s − t|^{\frac{1}{γ}} ∥g^{(n+1)}∥_{[s,t],δ} , γ > 1, \frac{1}{γ} + \frac{1}{δ} = 1 or γ = 1, δ = ∞.

Multiplying (3.7) and (3.8) and integrating over (t, s) ∈ [a, b]^{2}, we deduce

M_{n} (w_{n+1}, g^{(n+1)}) (3.9)

≤ \frac{1}{2 (b − a)} ∫_{a}^{b} ∫_{a}^{b} |s − t|^{\frac{1}{α} + \frac{1}{γ}} ∥w_{n+1}∥_{[s,t],β} ∥g^{(n+1)}∥_{[s,t],δ} dtds,

where α, β > 1, \frac{1}{β} + \frac{1}{α} = 1 or α = 1, β = ∞; γ,δ > 1, \frac{1}{γ} + \frac{1}{δ} = 1 or γ = 1 and δ = ∞, which is an inequality interesting in itself.

The terms M_{n} (w_{n+1}, g^{(n+1)}) can be bounded in a simpler form as follows.

Corollary 3. With the assumptions of Lemma 2 and if α, β, γ, δ are as above and r_{i} > 1, (i = \overline{1, 3}), \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = 1, then we have the inequality:

(3.10)  M_{n} (w_{n+1}, g^{(n+1)}) ≤ K_{r_{1}} ∥∥w_{n+1}∥_{[·,·],β}∥_{r_{2}} ∥∥g^{(n+1)}∥_{[·,·],δ}∥_{r_{3}} ,

where

K_{r_{1}} = \frac{2^{\frac{1}{r_{1}} −1} (b − a)^{\frac{1}{α} + \frac{1}{δ} + \frac{2}{r_{1}} −1}}{[(\frac{1}{α} + \frac{1}{δ}) r_{1} + 1]^{\frac{1}{r_{1}}} [(\frac{1}{α} + \frac{1}{δ}) r_{1} + 2]^{\frac{1}{r_{1}}}} .

12 S.S. DRAGOMIR

Proof. Using Hölder’s integral inequality for three terms with r_{i} > 1, \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = 1, we may write: ∫_{a}^{b} ∫_{a}^{b} |s − t|^{\frac{1}{α} + \frac{1}{γ}} ∥w_{n+1}∥_{[s,t],β} ∥g^{(n+1)}∥_{[s,t],δ} dtds (3.11)

≤ (∫_{a}^{b} ∫_{a}^{b} |s − t|^{(\frac{1}{α} + \frac{1}{γ})r_{1}} dtds)^{\frac{1}{r_{1}}} × (∫_{a}^{b} ∫_{a}^{b} ∥w_{n+1}∥_{[s,t],β}^{r_{2}} dtds)^{\frac{1}{r_{2}}}
× (∫_{a}^{b} ∫_{a}^{b} ∥g^{(n+1)}∥_{[s,t],δ}^{r_{3}} dtds)^{\frac{1}{r_{3}}}
= (∫_{a}^{b} ∫_{a}^{b} |s − t|^{(\frac{1}{α} + \frac{1}{γ})r_{1}} dtds)^{\frac{1}{r_{1}}} ∥∥w_{n+1}∥_{[·,·],β}∥_{r_{2}} ∥∥g^{(n+1)}∥_{[·,·],δ}∥_{r_{3}} .

We remark that, for a positive p, we have

∫_{a}^{b} ∫_{a}^{b} |x − y|^{p} dxdy = ∫_{a}^{b} (∫_{a}^{b} |y − x|^{p} dy) dx
= ∫_{a}^{b} (∫_{a}^{x} (x − y)^{p} dy + ∫_{x}^{b} (y − x)^{p} dy) dx
= ∫_{a}^{b} [\frac{(x − a)^{p+1} + (b − x)^{p+1}}{p + 1} ] dx = \frac{2 (b − a)^{p+2}}{(p + 1) (p + 2)}

and then

(∫_{a}^{b} ∫_{a}^{b} |s − t|^{(\frac{1}{α} + \frac{1}{γ})r_{1}} dtds)^{\frac{1}{r_{1}}} = [\frac{2 (b − a)^{(\frac{1}{α} + \frac{1}{γ})r_{1}+2}}{[(\frac{1}{α} + \frac{1}{γ} ) r_{1} + 1] [(\frac{1}{α} + \frac{1}{γ} ) r_{1} + 2]}]^{\frac{1}{r_{1}}}
= \frac{2^{\frac{1}{r_{1}}} (b − a)^{\frac{1}{α} + \frac{1}{δ} + \frac{2}{r_{1}}}}{[(\frac{1}{α} + \frac{1}{δ}) r_{1} + 1]^{\frac{1}{r_{1}}} [(\frac{1}{α} + \frac{1}{δ}) r_{1} + 2]^{\frac{1}{r_{1}}}} .

Using (3.9) and (3.11) we obtain (3.10). Remark 4. If one would use the examples 1-4 considered in Section 2, some particular inequalities may be stated. We omit the details.

4. Some Bounds Via a Grüss Type Inequality

In [4], Cheng and Sun have proved the following integral inequality of Grüss type for univariate real functions: Lemma 3. Let h, g : [a, b] → ℝ be two integrable functions such that

(4.1)  φ ≤ g (x) ≤ Φ for a.e. x ∈ [a, b] .

Then

|\frac{1}{b − a} ∫_{a}^{b} h (t) g (t) dt − \frac{1}{b − a} ∫_{a}^{b} h (t) dt · \frac{1}{b − a} ∫_{a}^{b} g (t) dt| (4.2)

≤ \frac{1}{2} (Φ − φ) \frac{1}{b − a} ∫_{a}^{b} |h (t) − \frac{1}{b − a} ∫_{a}^{b} h (s) ds| dt.

For a generalization of this result in the abstract settings of Lebesgue integrals and weighted means, see [3].

The following result also holds. Theorem 4. Let w : [a, b] → ℝ be an absolutely continuous function on [a, b] and \{w_{k}\}_{k=\overline{0,n+1}} be a sequence of w−Appell type. If g : [a, b] → ℝ is such that g^{(n−1)} is absolutely continuous on [a, b], w_{n}g^{(n)} ∈ L_{1} [a, b], and there exists the constants γ_{n}, Γ_{n} so that

−∞ < γ_{n} ≤ g^{(n)} ≤ Γ_{n} < ∞ a.e. on [a, b] ,

then we have the equality

∫_{a}^{b} w (t) g (t) dt = B_{n} (w, g; a, b) + S_{n} (w, g; a, b) ,

where

(4.3) B_{n} (w, g; a, b) = A_{n} (w, g; a, b) + (−1)^{n} [w_{n+1} (b) − w_{n+1} (a)] [g^{(n−1)}; a, b] ,

A_{n} (w, g; a, b) is given in equation (2.4) and the remainder S_{n} (w, g; a, b) satisfies the estimate

(4.4)  |S_{n} (w, g; a, b)| ≤ \frac{1}{2} (Γ_{n} − γ_{n}) ∫_{a}^{b} |w_{n} (t) − [w_{n+1}; a, b]| dt

where [w_{n+1}; a, b] is the devided difference.

Proof. From (3.4) we have

S_{n} (w, g; a, b)

= (b − a) (−1)^{n} |\frac{1}{b − a} ∫_{a}^{b} w_{n} (t) g^{(n)} (t) dt

−\frac{1}{b − a} ∫_{a}^{b} w_{n} (t) dt · \frac{1}{b − a} ∫_{a}^{b} g^{(n)} (t) dt| .

Thus, by the use of Lemma 3 we have

|S_{n} (w, g; a, b)|
≤ \frac{1}{2} (Γ_{n} − γ_{n}) ∫_{a}^{b} |w_{n} (t) − \frac{1}{b − a} ∫_{a}^{b} w_{n} (s) ds| dt

and the theorem is proved.

Remark 5. If one would use the examples 1-4 considered in Section 2, some particular inequalities may be stated. We omit the details.

References

[1] P. CERONE, Three point rules in numerical integration, Nonlinear Analysis, 47 (2001), 2341-2352.

[2] P. CERONE and S.S. DRAGOMIR, New bounds for a perturbed generalised Taylor’s formula, East Asian Math. J., 17(2) (2001), 197-215.

[3] P. CERONE and S.S. DRAGOMIR, A refinement of the Grüss inequality and applications, submitted.

[4] X.-L. CHENG and J. SUN, A note on the perturbed trapezoid inequality, J. Ineq. Pure & Appl. Math., 3(2002), No. 2, Article 29. [ON LINE: http://jipam.vu.edu.au ].

[5] S.S. DRAGOMIR, Better bounds in some Ostrowski-Grüss type inequality, Tamkang J. Math., 32(3) (2001), 211-216.

[6] M. MATIĆ, J.E. PEČARIĆ and N. UJEVIĆ, On new estimation of the remainder in generalised Taylor’s formula, Math. Ineq. & Appl., 2(3) (1999), 343-361.

[7] C.E.M. PEARCE and J.E. PEČARIĆ, On Anastassiou’s generalisation of the Ostrowski inequality and related results, J. Comput. Anal. and Applic., 2(3) (2000), 215-235.

[8] C.E.M. PEARCE, J.E. PEČARIĆ, N. UJEVIĆ and S. VAROŠANEC, Generalisations of some inequalities of Ostrowski-Grüss type, Math. Ineq. & Appl., 3(1) (2000), 25-34.

[9] J.E. PEČARIĆ and S. VAROŠANEC, Harmonic polynomials and generalisation of Ostrowski inequality with applications in numerical integration, Nonlinear Analysis, 47 (2001), 2365-2374.

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   E-mail address: sever@matilda.vu.edu.au
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