Three Point Identities and Inequalities for n-time Differentiable Functions P. Cerone and S.S. Dragomir Abstract. Identities and inequalities are obtained involving n-time differentiable functions in terms of evaluations at an interior and at the end points. It is shown how previous work is recaptured as particular instances of the current development. Generalised Taylor type series expansions are obtained and applications to numerical quadrature are demonstrated.
1. Introduction Recently, Cerone and Dragomir [2] obtained the following three point identity for f : [a, b] → R and α : [a, x] → R, β : (x, b] → R then Z b (1.1) f (t) dt − [(β (x) − α (x)) f (x) + (α (x) − a) f (a) + (b − β (x)) f (b)] a
=
Z
b
K (x, t) df (t) ,
a
where (1.2)
K (x, t) =
t − α (x) ,
t − β (x) ,
t ∈ [a, x] t ∈ (x, b].
They obtained a variety of inequalities for f satisfying different conditions such as bounded variation, Lipschitzian or monotonic. For f absolutely continuous then the above Riemann-Stieltjes integral would be equivalent to a Riemann integral and again a variety of bounds were obtained for f ∈ Lp [a, b], p ≥ 1. Inequalities of Gr¨ uss type and a number of premature variants were examined fully in the comprehensive article covering the situation in which f exhibits at most a first derivative. Applications to numerical quadrature were investigated covering rules of Newton-Cotes type containing the evaluation of the function at three possible points: the interior and extremities. The development included the midpoint, trapezoidal and Simpson type rules. However, unlike the classical rules, the results were not as restrictive in that the bounds are derived in terms of the behaviour of at most the first derivative and the Peano kernel (1.2). 1991 Mathematics Subject Classification. Primary 26D15 ; Secondary 41A55. Key words and phrases. Three Point Identities and Inequalities, Ostrowski and Gr¨ uss Inequalities, Newton-Cotes Quadrature. 1
2
P. CERONE AND S.S. DRAGOMIR
It is the aim of the current article to obtain inequalities for f (n) ∈ Lp [a, b], p ≥ 1 where f (n) are again evaluated at most at an interior point x and the end points. Results that involve the evaluation only at an interior point are termed Ostrowski type and those that involve only the boundary points will be referred to as trapezoidal type. In the numerical analysis literature these are also termed as Open and Closed Newton-Cotes rules (Atkinson [1]) respectively. In 1938, Ostrowski (see for example [32, p. 468]) proved the following integral inequality: Let f : I ⊆ R → R be a differentiable mapping on ˚ I (˚ I is the interior of I), 0 ˚ and let a, b ∈I with a < b. If f : (a, b) → R is bounded on (a, b), i.e., kf 0 k∞ := sup |f 0 (t)| < ∞, then we have the inequality: t∈(a,b)
(1.3)
" 2 # Z b x − a+b 1 1 2 f (t) dt ≤ + (b − a) kf 0 k∞ f (x) − 2 b−a a 4 (b − a)
for all x ∈ [a, b]. The constant 41 is sharp in the sense that it cannot be replaced by a smaller one. For applications of Ostrowski’s inequality to some special means and some numerical quadrature rules, we refer the reader to the recent paper [24] by S.S. Rb Dragomir and S. Wang who used integration by parts from a p (x, t) f 0 (t) dt to prove Ostrowski’s inequality (1.3) where p (x, t) is a peano kernel given by t − a, t ∈ [a, x] p (x, t) = (1.4) t − b, t ∈ (x, b]. Fink [27] used the integral remainder from a Taylor series expansion to show that for f (n−1) absolutely continuous on [a, b], then the identity ! Z Z b n−1 b X 1 (b − a) f (x) + Fk (x) = (1.5) KF (x, t) f (n) (t) dt f (t) dt − n a a k=1
is shown to hold where
n−1
(1.6)
KF (x, t) =
and Fk (x) =
p (x, t) (x − t) · with p (x, t) being given by (1.4) (n − 1)! n
i n−k h k k−1 k (x − a) f (k−1) (a) + (−1) (b − x) f (k−1) (b) . k!
Fink then proceeds to obtain a variety of bounds from (1.5), (1.6) for f (n) ∈ Lp [a, b]. Milovanovi´c and Peˇcari´c [31] earlier obtained a result for f (n) ∈ L∞ [a, b] although they did not use the integral form of the remainder. It may be noticed that (1.5) is again an identity that involves function evaluations at three points to approximate the integral from the resulting inequalities. See Mitrinovi´c, Peˇcari´c and Fink [33, Chapter XV] for further related results and papers [19], [20] and [21]. A number of other authors have obtained results in the literature that may be recaptured under the general formulation of the current paper. These will be highlighted throughout the article. The paper is structured as follows.
3
A variety of identities are obtained in Section 2 for f (n−1) absolutely continuous for a generalisation of the kernel (1.2). Specific forms are highlighted and a generalised Taylor-like expansion is obtained. Inequalities are developed in Section 3 and perturbed results through Gr¨ uss inequalities and premature variants are discussed in Section 4. Section 5 demonstrates the applicability of the inequalities to numerical integration. Concluding remarks are given in Section 6. 2. Some Integral Identities In this section identities are obtained involving n-time differentiable functions with evaluation at an interior point and at the end points. Theorem 1. Let f : [a, b] → R be a mapping such that f (n−1) is absolutely continuous on [a, b]. Further, let α : [a, x] → R and β : (x, b] → R. Then, for all x ∈ [a, b] the following identity holds, Z b n Kn (x, t) f (n) (t) dt (2.1) (−1) =
Z
a
b
a
n i X 1 h f (t) dt − Rk (x) f (k−1) (x) + Sk (x) , k! k=1
2
where the kernel Kn : [a, b] → R is given by (t−α(x))n , t ∈ [a, x] n! (2.2) Kn (x, t) := (t−β(x))n , t ∈ (x, b], n! ( k k−1 k Rk (x) = (β (x) − x) + (−1) (x − α (x)) (2.3) . k k−1 k and Sk (x) = (α (x) − a) f (k−1) (a) + (−1) (b − β (x)) f (k−1) (b) Proof. Let n
In (x) = (−1)
(2.4)
then from (2.3)
Z
b
n
Kn (x, t) f (n) (t) dt = (−1) Jn (a, x, b) a
Z
Jn (a, x, x) =
x
a
n
(t − α (x)) (n) f (t) dt n!
giving, upon using integration by parts n
(2.5)
Jn (a, x, x) =
(x − α (x)) (n−1) f (x) n! n (n−1) (a) n−1 (α (x) − a) f + (−1) − Jn−1 (a, x, x) . n!
Similarly, n
(β (x) − x) (n−1) f (x) n! n (b − β (x)) (n−1) f (b) − Jn−1 (x, x, b) + n! and so upon adding to (2.5) gives from (2.4) the recurrence relation Jn (x, x, b) =
(2.6)
n−1
(−1)
In (x) − In−1 (x) = −ω n (x) ,
4
P. CERONE AND S.S. DRAGOMIR
where
h i n! ω n (x) = Rn (x) f (n−1) (x) + Sn (x)
(2.7)
with Rn (x) and Sn (x) being given by (2.3). It may easily be shown that (2.8)
In (x) = −
n X
ω k (x) + I0 (x)
k=1
is a solution of (2.6) and so the theorem is proven since (2.8) is equivalent to (2.1) and In (x) is as given by (2.4). Remark 1. If we take n = 1 then an identity obtained by Cerone and Dragomir [2] results. In the same paper Riemann-Stieltjes integrals were also considered. Remark 2. If α (x) = a and β (x) = b then Sk (x) ≡ 0 and the Ostrowski type results for n-time differentiable functions of Cerone et al. [9] are recaptured. Merkle [30] also obtains Ostrowski type results. For α (x) = β (x) = x then R (x) ≡ 0 and the generalized trapezoidal type rules for n-time differentiable functions of Cerone et al. [10] are obtained. Qi [36] used a Taylor series whose remainder was not expressed in integral form so that only the supremum norm was possible. If the integral form of the remainder were used, then similar to Fink [27], the other Lp (a, b) norms for p ≥ 1 would be possible. However, this will not be pursued further here. For α (x) and β (x) at their respective midpoints, then the identity Z b n (2.9) (−1) Kn (x, t) f (n) (t) dt Z
=
a
a
b
f (t) dt − h
k
n X 2−k nh
k=1
k!
k
(b − x) + (−1) k−1
+ (x − a) f (k−1) (a) + (−1) results, where
(2.10)
Kn (x, t) =
n!
k
(x − a)
k
(b − x) f (k−1) (b)
n (t− a+x 2 ) , n! n (t− x+b 2 )
k−1
,
i
f (k−1) (x)
io
t ∈ [a, x] t ∈ (x, b].
As demonstrated in the above remarks, different choices of α (x) and β (x) give a variety of identities. The following corollary allows for α (x) and β (x) to be in the same relative position within their respective intervals. Corollary 1. Let f satisfy the conditions as stated in Theorem 1. Then the following identity holds for any γ ∈ [0, 1] and x ∈ [a, b]. Namely, Z b n (2.11) (−1) Cn (x, t) f (n) (t) dt =
Z
a
n h i X 1 n k k k−1 k (1 − γ) (b − x) + (−1) (x − a) f (k−1) (x) k! a k=1 h io k k−1 k (b − x) f (k−1) (b) , +γ k (x − a) f (k−1) (a) + (−1) b
f (t) dt −
5
where (2.12)
Cn (x, t) =
Proof. Let (2.13)
[t−(γx+(1−γ)a)]n , n!
[t−(γx+(1−γ)b)]n , n!
t ∈ [a, x] t ∈ (x, b].
α (x) = γx + (1 − γ) a and β (x) = γx + (1 − γ) b,
then
(2.14)
x − α (x) = (1 − γ) (x − a) , α (x) − a = γ (x − a) and β (x) − x = (1 − γ) (b − x) , b − β (x) = γ (b − x)
so that from (2.3) k
Rk (x) = (1 − γ) and
h
k
(b − x) + (−1)
k−1
k
(x − a)
i
h i k k−1 k Sk (x) = γ k (x − a) f (k−1) (a) + (−1) (b − x) f (k−1) (b) .
In addition, Cn (x, t) is the same as Kn (x, t) in (2.2) with α (x) and β (x) as given by (2.13) and hence the corollary is proven. The following Taylor-like formula with integral remainder also holds. Corollary 2. Let g : [a, y] → R be a mapping such that g (n) is absolutely continuous on [a, y]. Then for all x ∈ [a, y] we have the identity (2.15)
g (y) =
n i X 1 nh k k−1 k (β (x) − x) + (−1) (x − α (x)) g (k) (x) k! k=1 h io k k−1 k + (α (x) − a) g (k) (a) + (−1) (y − β (x)) g (k) (y) Z y n + (−1) τ n (x, t) g (n+1) (t) dt
g (a) +
a
where (2.16)
τ n (x, t) =
(t−α(x))n , n! (t−β(x))n , n!
t ∈ [a, x] t ∈ (x, y].
Proof. The proof is straight forward from Theorem 1 on taking f ≡ g 0 and b = y so that β (x) ∈ (x, y] and τ n (x, t) ≡ Kn (x, t) for t ∈ [a, y]. Remark 3. If α (x) = β (x) = x then we recapture the results of Cerone et al. [10], a trapezoidal type series expansion. That is, an expansion involving the end points. For α (x) = a, β (x) = b then a Taylor-like expansion of Cerone et al. [9] is reproduced as are the results of Merkle [30].
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P. CERONE AND S.S. DRAGOMIR
3. Integral Inequalities In this section we develop some inequalities from using the identities obtained in Section 2. Theorem 2. Let f : [a, b] → R be a mapping such that f (n−1) is absolutely continuous on [a, b] and, let α : [a, x] → R and β : (x, b] → R. Then the following inequalities hold for all x ∈ [a, b] Z n b i X 1 h (k−1) f (t) dt − (3.1) |Pn (x)| : = Rk (x) f (x) + Sk (x) a k! k=1 kf (n) k∞ Qn (1, x) if f (n) ∈ L∞ [a, b] , n! 1 kf (n) kp ≤ [Qn (q, x)] q if f (n) ∈ Lp [a, b] n! with p > 1, p1 + 1q = 1, (n) kf k1 n M (x) , if f (n) ∈ L1 [a, b] , n!
where (3.2)
Qn (q, x) =
M (x) =
(3.3)
1 h nq+1 nq+1 (α (x) − a) + (x − α (x)) nq + 1 i nq+1 nq+1 + (β (x) − x) + (b − β (x)) ,
b − a a + x x + b + α (x) − + β (x) − 2 2 2 a + x a + b x + b + α (x) − + x − + β (x) − , 2 2 2
1 2
Rk (x), Sk (x) are given by (2.3), and
(n)
f
∞
:= ess sup f (n) (t) < ∞ and f (n) := p
t∈[a,b]
Z
b
a
Proof. Taking the modulus of (2.1) then
(3.4)
(n) p f (t)
|Pn (x)| = |In (x)| ,
where Pn (x) is as defined by the left hand side of (3.1) and Z b (3.5) |In (x)| = Kn (x, t) f (n) (t) dt , a with Kn (x, t) given by (2.2). Now, observe that (3.6)
|In (x)|
≤ =
(n)
f
(n)
f
∞ ∞
kKn (x, ·)k1 Z b |Kn (x, t)| dt, a
! p1
, p ≥ 1.
7
where, from (2.2), Z b (3.7) |Kn (x, t)| dt
=
a
1 n! +
=
(Z
Z
α(x)
Z
n
|t − α (x)| dt +
a
β(x)
n
|t − β (x)| dt +
x
Z
x
n
α(x)
|t − α (x)| dt
b
)
n
β(x)
|t − β (x)| dt
h 1 n+1 n+1 (α (x) − a) + (x − α (x)) (n + 1)! i n+1 n+1 + (β (x) − x) + (b − β (x)) .
Thus, on combining (3.4), (3.6) and (3.7), the first inequality in (3.1) is obtained. Further, using H¨older’s integral inequality we have the result
1 1
(3.8) + = 1, with p > 1 |In (x)| ≤ f (n) kKn (x, ·)kq where p q p 1 ! q
Z b
q = f (n) |Kn (x, t)| dt . p
Now,
(3.9)
Z
b
a
q
|Kn (x, t)| dt =
1 n! +
a
(Z
α(x)
|t − α (x)|
a
Z
β(x)
x
nq
|t − β (x)|
nq
dt +
dt +
Z
Z
x α(x)
|t − α (x)|
b
β(x)
|t − β (x)|
nq
nq
dt
)
dt
1 Qn (q, x) , n! where Qn (q, x) is as given by (3.2). Combing (3.9) with (3.8) gives the second inequality in (3.1). Finally, let us observe that from (3.4) =
(3.10)
|In (x)|
≤ kKn (x, ·)k∞ f (n) = f (n) 1
=
= where (3.11)
(n)
f
1
n!
(n)
f
1
n!
sup |Kn (x, t)|
1 t∈[a,b]
n
n
n
n
max {|a − α (x)| , |x − α (x)| , |b − β (x)| , |x − β (x)| }
M n (x) ,
M (x) = max {M1 (x) , M2 (x)}
with M1 (x) = max {α (x) − a, x − α (x)} and M2 (x) = max {β (x) − x, b − β (x)} .
8
P. CERONE AND S.S. DRAGOMIR
The well known identity (3.12)
max {X, Y } =
may be used to give
X − Y X +Y + 2 2
+ α (x) −
M1 (x) =
x−a 2
and M2 (x) =
b−x 2
(3.13)
+ β (x) −
a+x , 2
x+b . 2
Using the identity (3.12) again gives, from (3.11), M1 (x) + M2 (x) M1 (x) − M2 (x) + M (x) = 2 2
which on substituting (3.13) gives (3.3) and so from (3.10) and (3.4) readily results in the third inequality in (3.1) and the theorem is completely proved. Remark 4. Various choices of α (·) and β (·) allow us to reproduce many of the earlier inequalities involving function and derivative evaluations at an interior point and/or boundary points. For other related results see Chapter XV of [32]. If α (x) = a and β (x) = b then Sk (x) ≡ 0 and Ostrowski type results for n-time differentiable functions of Cerone et al. [9] are reproduced (see also [35]). Further, taking n = 1 recaptures the results of Dragomir and Wang [22]-[25] and n = 2 gives the results of Cerone, Dragomir and Roumeliotis [5]-[8]. The n = 2 case is of importance since with x = a+b 2 the classic midpoint rule is obtained. However, here the bound is obtained for f 00 ∈ Lp [a, b] for p ≥ 1 rather than the traditional f 00 ∈ L∞ [a, b], see for example [16] and [17]. If α (x) = β (x) = x then R (x) ≡ 0 and inequalities are obtained for a generalised trapezoidal type rule in which functions are assumed to be n-time differentiable, recapturing the results in Cerone et al. [10]. Taking n = 2 the classic trapezoidal rule in which the bound involves the behaviour of the second derivative is recaptured as presented in Dragomir et al. [18]. Taking α (·) and β (·) to be other than at the extremities results in three point inequalities for n−time differentiable functions. Cerone and Dragomir [2] presented results for functions that at most admit a first derivative. Remark 5. It should be noted that the bounds in (3.1) may themselves be bounded since α (·), β (·) and x have not been explicitly specified. To demonstrate, consider the mappings, for t ∈ [A, B], (3.14)
θ
θ
h1 (t) = (t − A) + (B − t) , θ > 1 and h2 (t) =
B−A 2
+ t −
A+B . 2
Now, both these functions attain their maximum values at the ends of the interval and their minimums at the midpoints. That is, they are symmetric and convex.
9
Thus,
(3.15)
θ
sup h1 (t) = h1 (A) = h1 (B) = (B − A) ,
t∈[A,B]
sup h2 (t) = h2 (A) = h2 (B) = B − A,
t∈[A,B]
θ = 2 B−A , inf h1 (t) = h1 A+B 2 2 t∈[A,B] B−A and inf h2 (t) = h2 A+B = 2 . 2 t∈[A,B]
Using (3.14) and (3.15) then from (3.2) and (3.3), on taking α (·) and β (·) at either of their extremities gives i (b − a) 1 h nq+1 nq+1 (x − a) + (b − x) ≤ nq + 1 nq + 1
nq+1
Qn (q, x) ≤ QU n (q, x) = and
M (x) ≤ M U (x) =
1 [b − a + |x − a|] ≤ b − a, 2
where the coarsest bounds are obtained from taking x at its extremities. The following corollary holds. Corollary 3. Let the conditions of Theorem 2 hold. Then the following result is valid for any x ∈ [a, b]. Namely, Z n b i X 2−k h k k−1 k (3.16) (b − x) + (−1) (x − a) f (k−1) (x) f (t) dt − a k! k=1 h i k k−1 k + (x − a) f (k−1) (a) + (−1) (b − x) f (k−1) (b) ≤
h i kf (n) k∞ −n n+1 n+1 2 (x − a) + (b − x) , n! f (n) i1 k kp 2−n h nq+1 nq+1 q (x − a) + (b − x) 1 n! (nq+1) q n kf (n) k1 −n b−a 2 + x − a+b , n!
a+x 2
2
2
f (n) ∈ L∞ [a, b] , f (n) ∈ Lp [a, b] with p > 1,
1 p
+
1 q
= 1,
f (n) ∈ L1 [a, b] .
Proof. Taking α (·) and β (·) at their respective midpoints, namely α (x) = and β (x) = x+b 2 in (3.1)-(3.3) and using (2.3) readily gives (3.16)
Remark 6. Corollary 3 could have equivalently been proven using (2.9) and (2.10) following essentially the same proof of Theorem 2 from using identity (2.9). The more general setting however, allows greater flexibility and, it is argued, is no more difficult to prove.
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P. CERONE AND S.S. DRAGOMIR
Corollary 4. Let the conditions on f of Theorem 2 hold. Then the following result for any x ∈ [a, b], is valid. Namely, for any γ ∈ [0, 1] Z n k b i X (−1) h k (k−1) k f (t) dt − (1 − γ) rk (x) f (x) + γ sk (x) a k!
(3.17)
k=1
≤
kf (n) k∞ f (n) ∈ L∞ [a, b] , (n+1)! H1 (γ) G1 (x) , kf (n) k 1 1 q q p f (n) ∈ Lp [a, b] 1 Hq (γ) Gq (x) , n!(nq+1) q p > 1, p1 + 1q = 1, (n) kf k1 n ν (x) , f (n) ∈ L1 [a, b] , n!
where
(3.18)
nq+1 Hq (γ) = γ nq+1 + (1 − γ) , nq+1 nq+1 Gq (x) = (x − a) + (b − x) , a+b , ν (x) = 21 + γ − 12 b−a 2 + x− 2 k k−1 k rk (x) = (b − x) + (−1) (x − a) , k k−1 k and sk (x) = (x − a) f (k−1) (a) + (−1) (b − x) f (k−1) (b) .
Proof. Take α (·) and β (·) to be a convex combination of their respective boundary points as given by (2.13) then from (3.1)-(3.3) and using (2.14) and (2.3) readily produces the stated result. We omit any further details.
Remark 7. It is instructive to note that the relative location of α (·) and β (·) is the same in Corollary 4 and is determined through the parameter γ as defined in (2.13). Theorem 2 is much more general. From (2.13) it may be seen that α (x) = β (x) = x is equivalent to γ = 1, giving trapezoidal type rules while α (x) = a, β (x) = b corresponds to γ = 0 which produces interior point rules. Taking γ = 0 and γ = 1 reproduces the results of Cerone et al. [9] and [10] respectively. Taking γ = 12 in (3.17) produces the optimal rule while keeping x general and thus reproducing the result of Corollary 3. Following the discussion in Remark 5 and as may be ascertained from (3.18) the optimal rules, in the sense of providing the tightest bounds, are obtained by taking γ and x at their respective midpoints. The following two corollaries may thus be stated.
11
Corollary 5. Let the conditions on f of Theorem 2 be valid. Then for any γ ∈ [0, 1], the following inequalities hold (3.19)
Z b f (t) dt a
−
≤
where
(3.20)
n k X (−1)
k!
k=1
k
(1 − γ) rk
a+b 2
f
(k−1)
a+b 2
k
+ γ sk
kf (n) k∞ a+b , f (n) ∈ L∞ [a, b] , (n+1)! H1 (γ) G1 2 kf (n) k 1 1 q q p a+b , f (n) ∈ Lp [a, b] 1 Hq (γ) Gq 2 q n!(nq+1) p > 1, p1 + 1q = 1, (n) kf k1 n a+b ν , f (n) ∈ L1 [a, b] , n! 2
a+b 2
Hq (γ) is as given by (3.18), nq+1 = 2 b−a , Gq a+b 2 2 1 1 ν a+b = b−a , 2 2 2 + γ− 2 i k h k−1 rk a+b = b−a 1 + (−1) 2 2 h i and s a+b = b−a k f (k−1) (a) + (−1)k−1 f (k−1) (b) . k
2
2
Proof. The proof is trivial. Taking x = the result.
a+b 2
in (3.17)-(3.18) readily produces
Remark 8. It is of interest to note from (3.20) that
(3.21)
rk
a+b 2
=
2
0,
b−a k 2
,
k odd k even
so that only the evaluation of even order derivatives are involved in (3.19). Further, for f (k−1) (a) = f (k−1) (b) then (3.22)
sk
a+b 2
=f
(k−1)
(a) rk
a+b 2
=f
(k−1)
(b) rk
a+b 2
so that only evaluation of even order derivatives at the end points are present.
12
P. CERONE AND S.S. DRAGOMIR
Corollary 6. Let the conditions on f of Theorem 2 hold. Then the following inequalities are valid Z n k b X (−1) −k a+b a + b a + b f (t) dt − 2 rk f (k−1) + sk (3.23) a k! 2 2 2 k=1 kf (n) k∞ −(n−1) b−a n+1 , f (n) ∈ L∞ [a, b] , (n+1)! · 2 2 kf (n) k q1 p b−a b−a n −n ≤ · 2 , f (n) ∈ Lp [a, b] 1 nq+1 2 q n!(nq+1) with p > 1, p1 + 1q = 1, (n) kf k1 b−a n , f (n) ∈ L1 [a, b] , n! 4 where rk a+b and sk a+b are as given by (3.20). 2 2
Proof. Taking γ = 12 in Corollary 5 will produce inequalities with the tightest bounds as given in (3.23). Alternatively, taking γ = 21 and x = a+b 2 in Corollary 4 will produce the results (3.23). The results (3.21) and (3.22) together with the discussion in Remark 8 are also valid for Corollary 6. The following are Taylor-like inequalities which are of interest (see [12] and [14] for related results). Corollary 7. Let g : [a, y] → R be a mapping such that g (n) is absolutely continuous on [a, y]. Then for all x ∈ [a, y] n i X 1 h k k−1 k (3.24) (β (x) − x) + (−1) (x − α (x)) g (k) (x) g (y) − g (a) − k! k=1 h i k k−1 k + (α (x) − a) g (k) (a) + (−1) (y − β (x)) g (k) (y) ≤
where
kg(n+1) k∞ ˜ Qn (1, x) , g (n+1) ∈ L∞ [a, b] , n! g(n+1) h i q1 k kp ˜ Q (q, x) , g (n+1) ∈ Lp [a, b] n n! with p > 1, p1 + 1q = 1, kg(n+1) k 1 ˜ n M (x) , g (n+1) ∈ L1 [a, b] , n!
˜ n (q, x) = Q
˜ (x) = M
1 h nq+1 nq+1 (α (x) − a) + (x − α (x)) nq + 1 i nq+1 nq+1 + (β (x) − a) + (y − β (x)) ,
y − a a + x x + y + α (x) − + β (x) − 2 2 2 a + y a + x x + y + x − + α (x) − + β (x) − . 2 2 2 1 2
13
Proof. The proof follows from Theorem 2 on taking f (·) ≡ g 0 (·) and b = y so that β (x) ∈ (x, y]. Alternatively, starting from (2.15) and (2.16) and, following the proof of Theorem 2 with b replaced by y and f (·) replaced by g 0 (·) readily produces the results shown and the corollary is thus proven. Remark 9. Similar corollaries to 3, 4, 7 and 8 could be determined from the Taylor-like inequalities given in Corollary 6. This would simply be done by taking specific forms of α (·), β (·) or values of x as appropriate. Remark 10. If in particular we take α (x) = a and β (x) = y in (3.24) then for any x ∈ [a, y]
n i X 1 h k k−1 k (3.25) g (y) − g (a) − (y − x) + (−1) (x − a) g (k) (x) k! k=1
≤
:
en (x, y)
h i kg(n+1) k∞ n+1 n+1 (x − a) + (y − x) , g (n+1) ∈ L∞ [a, y] , (n+1)! g(n+1) h i1 k kp nq+1 nq+1 q = , g (n+1) ∈ Lp [a, y] (x − a) + (y − x) n!(nq+1) q1 with p > 1, p1 + 1q = 1, (n+1) kg n k1 x+y , g (n+1) ∈ L1 [a, y] . n! 2 −a
Merkle [30] effectively obtains the first bound in (3.25). It is well known (see for example, Dragomir [14]) that the classical Taylor expansion around a point satisfies the inequality (3.26) ≤ where
(3.27)
En (y) ≤
n k X (y − a) (k) g (a) g (y) − k! k=1 Z y 1 n (n+1) := En (y) , (y − u) g (u) du n! a
(y−a)n+1 (n+1)! (y−a)
n+ 1 q
1 n!(nq+1) q n
(y−a) n!
(n+1)
g
∞
,
(n+1)
g
, p
(n+1)
g
, 1
g (n+1) ∈ L∞ [a, y] , g (n+1) ∈ Lp [a, y]
with p > 1, p1 + 1q = 1, g (n+1) ∈ L1 [a, y] ,
for y ≥ a and y ∈ I ⊂ R. Now, it may readily be noticed that if x = a in (3.25), then the classical result as given by (3.26) is regained. As discussed in Remark 5 the bounds are convex so that a coarse bound is obtained at the end points and the best at the midpoint.
14
P. CERONE AND S.S. DRAGOMIR
Thus, taking x = a+y 2 gives h i k−1 n 1 + (−1) X a + y k g (y) − g (a) − (3.28) 2−k (y − a) g (k) k! 2 k=1 a+y ,y ≤ en 2 kg(n+1) k∞ −n n+1 (y − a) , g (n+1) ∈ L∞ [a, y] , (n+1)! 2 kg(n+1) k n+ 1 p −n = (y − a) q , g (n+1) ∈ Lp [a, y] 1 2 q n!(nq+1) with p > 1, p1 + 1q = 1, (n+1) k1 3 n kg n (y − a) , g (n+1) ∈ L1 [a, y] . n! 4 The above inequalities (3.28) show that for g ∈ C ∞ [a, b] the series h i k−1 ∞ 1 + (−1) X a+y k (k) g (a) + (y − a) g k!2k 2 k=1
converges more rapidly to g (y) than the usual one ∞ k X (y − a)
k=0
k!
g (k) (a) ,
which comes from Taylor’s expansion (3.26). It should further be noted that (3.27) only involves the odd derivatives of g (·) evaluated at the midpoint of the interval under consideration. Remark 11. If α (x) = β (x) = x in (3.24), then for any x ∈ [a, y] n i X 1 h k (k) k−1 k (k) (x − a) g (a) + (−1) (y − x) g (y) (3.29) g (y) − g (a) − k! k=1
≤
en (x, y) ,
where en (x, y) is as defined by (3.25). See Cerone et al. [10] for related results. 4. Perturbed Rules Through Gr¨ uss Type Inequalities
In 1935, G. Gr¨ uss (see for example [32]), proved the following integral inequality which gives an approximation for the integral of a product in terms of the product of integrals. Theorem 3. Let f, g : [a, b] → R be two integrable mappings so that φ ≤ h (x) ≤ Φ (x) and γ ≤ g (x) ≤ Γ for all x ∈ [a, b], where φ, Φ, γ, Γ are real numbers. Then we have 1 (4.1) |T (h, g)| ≤ (Φ − φ) (Γ − γ) , 4 where Z b Z b Z b 1 1 1 (4.2) T (h, g) = h (x) g (x) dx − h (x) dx · g (x) dx b−a a b−a a b−a a
15
and the inequality is sharp, in the sense that the constant a smaller one.
1 4
cannot be replaced by
For a simple proof of this fact as well as for extensions, generalisations, discrete variants and other associated material, see [32], and the papers [6], [11], [13], [15] and [22] where further references are given. A premature Gr¨ uss inequality is embodied in the following theorem which was proved in the paper [29]. It provides a sharper bound than the above Gr¨ uss inequality. The term premature is used to denote the fact that the result is obtained from not completing the proof of the Gr¨ uss inequality if one of the functions is known explicitly. See also [2] for further details. Theorem 4. Let h, g be integrable functions defined on [a, b] and let d ≤ g (t) ≤ D. Then 1 D−d |T (h, g)| ≤ [T (h, h)] 2 , (4.3) 2 where T (h, g) is as defined in (4.2). The above Theorem 4 will now be used to provide a perturbed generalised three point rule. 4.1. Perturbed Rules From Premature Inequalities. We start with the following result. Theorem 5. Let f : [a, b] → R be such that the derivative f (n−1) , n ≥ 1 is absolutely continuous on [a, b]. Assume that there exist constants γ, Γ ∈ R such that γ ≤ f (n) (t) ≤ Γ a.e. on [a, b]. Then the following inequality holds Z n b i X 1 h (4.4) |ρn (x)| : = Rk (x) f (k−1) (x) + Sk (x) f (t) dt − a k! k=1 (n−1) (b) − f (n−1) (a) n θ n (x) f − (−1) · n+1 b−a Γ−γ 1 · I (x, n) ≤ 2 n! n+1 n Γ−γ (b − a) √ · ≤ · √ , (n + 1)! 2n + 1 2 where (4.5)
I (x, n) =
1 ˆ n (2, x) √ n2 (b − a) Q (n + 1) 2n + 1 4 X n 2 zi zj [zin − (−zj ) ] + (2n + 1) i=1 j>i
Z = {α (x) − a, x − α (x) , β (x) − x, b − β (x)} , zi ∈ Z, i = 1, ..., 4, ˆ Qn (·, x) = (2n + 1) Qn (·, x) with Qn (·, x) being as defined in (3.2), n
n
θn (x) = (−1) z1n+1 + z2n+1 + (−1) z3n+1 + z4n+1 , and Rk (x), Sk (x) are as given by (2.3).
16
P. CERONE AND S.S. DRAGOMIR
Proof. Applying the premature Gr¨ uss result (4.3) by associating f (n) (t) with g (t) and h (t) with Kn (x, t), from (2.2) gives Z b n (4.6) Kn (x, t) f (n) (t) dt (−1) a ! Z b f (n−1) (b) − f (n−1) (a) n − (−1) Kn (x, t) dt b−a a ≤
1 Γ−γ [T (Kn , Kn )] 2 , 2
(b − a)
where from (4.2) 1 T (Kn , Kn ) = b−a
Z
b
Kn2 a
(x, t) dt −
Z
1 b−a
b
Kn (x, t) dt
a
!2
.
Now, from (2.2), 1 b−a
(4.7)
Z
b
Kn (x, t) dt
"aZ
=
# Z b n n (t − α (x)) (t − β (x)) dt + dt n! n! a x 1 n+1 n n+1 (x − α (x)) + (−1) (α (x) − a) (b − a) (n + 1)! n+1 n n+1 (b − β (x)) + (−1) (β (x) − x)
:
=
=
and
1 b−a
1 θn (x) (b − a) (n + 1)!
1 b−a
(4.8) = =
x
Z 1
b a
Kn2 (x, t) dt "Z x
2
(b − a) (n!)
(t − α (x))
1
2
(b − a) (n!) (2n + 1) (b − β (x))
=
a
2n+1
(b − a) (n!) (2n + 1)
dt +
Z
b
x
(x − α (x)) 2n+1
+ (β (x) − x)
1 2
2n
2n+1
(t − β (x))
2n
+ (α (x) − a)
dt
#
2n+1
ˆ n (2, x) Q
on using (3.2). Hence, substitution of (4.7) and (4.8) into (4.6) gives Z b (n−1) (n−1) f (b) − f (a) θ (x) n n · (4.9) Kn (x, t) f (n) (t) dt − (−1) a (n + 1)! b−a ≤
Γ−γ 1 · J (x, n) , 2 n!
17
where 2
(4.10) =
(2n + 1) (n + 1) J 2 (x, n) 2 ˆ n (2, x) − (2n + 1) θ2 (x) . (n + 1) (b − a) Q n
Now, let (4.11)
A = α (x) − a, X = x − α (x) , Y = β (x) − x and B = b − β (x) ,
then (4.7) and (4.8) imply that ˆ n (2, x) = A2n+1 + X 2n+1 + Y 2n+1 + B 2n+1 Q and n
n
θn (x) = (−1) An+1 + X n+1 + (−1) Y n+1 + B n+1 . Hence, from (4.10) and using the fact that b − a = A + X + Y + B, (4.12) = =
2 ˆ n (2, x) − (2n + 1) θ2n (x) (n + 1) (b − a) Q ˆ n (2, x) + (2n + 1) (A + X + Y + B) Qn (2, x) − θ2n (x) n2 Q
ˆ n (2, x) + (2n + 1) n2 Q
4 X i=1 j>i
n 2
zi zj [zin − (−zj ) ]
after some straight forward algebra, where Z = {A, X, Y, B} , zi ∈ Z, i = 1, ..., 4. J(x,n) √ Substitution of (4.12) into (4.10) gives I (x, n) = (n+1) as presented by 2n+1 (4.5). Utilising identity (2.1) in (4.6) gives (4.4) and the first part of the theorem is proved. The upper bound is obtained by taking α (·), β (·), x at their end points since I (x, n) is convex and symmetric. The second term for I (x, n) is then zero and ˆ n (2, x) < 2 (b − a)2n+1 and hence after some simplification, the theorem is comQ pletely proven. Corollary 8. Let the conditions of Theorem 3 hold. Then the following result is valid, Z n b i X 2−k h (4.13) f (t) dt − rk (x) f (k−1) (x) + sk (x) a k! k=1 f (n−1) (b) − f (n−1) (a) n −n n n −2 [1 + (−1) ] (A + B ) · b−a ih i Γ − γ 2−2(n+1) h 2 n−1 · 4n + 1 + (−1) (2n + 1) A2(n+1) + B 2(n+1) ≤ 2 n! 2 n−1 n+1 + 4n + 2 (2n + 1) AB A2n + B 2n + 4 (2n + 1) (−1) (A − B) , where rm (x) and sm (x) are as given by (3.18) and A = x − a, B = b − x.
x+b Proof. Let α (x) = a+x 2 and β (x) = 2 in (4.4), readily giving the left hand side of (4.13). Now, for the right hand side. Taking A = x − a, B = b − x, we have ˆ n (2, x) = 2−2n A2n+1 + B 2n+1 Q
18
P. CERONE AND S.S. DRAGOMIR
and 4 X i=1 j>i
n 2
zi zj [zin − (−zj ) ]
=
h i n−1 2−2(n+1) A2(n+1) + B 2(n+1) 1 + (−1) 2 n−1 n +2AB An + (−1) B
so that from (4.5) and using the fact that b − a = A + B, √ (4.14) (n + 1) 2n + 1I (x, n) −2(n+1) = 2 4n2 (A + B) A2n+1 + B 2n+1 n−1 + (2n + 1) A2(n+1) + B 2(n+1) 1 + (−1) 2 n−1 n n +2AB A + (−1) B nh ih i n−1 = 2−2(n+1) 4n2 + 1 + (−1) (2n + 1) A2(n+1) + B 2(n+1) o n−1 n+1 + 4n2 + 2 (2n + 1) AB A2n + B 2n + 4 (2n + 1) (−1) (AB) . A simple substitution in (4.4) of (4.14) completes the proof.
Corollary 9. Let the conditions of Theorem 3 and Corollary 8 hold. Then the following inequality results, Z n b X a+b a+b a+b 2−k (k−1) (4.15) rk f + sk f (t) dt − a k! 2 2 2 k=1 h i n −2 · 4−n (1 + (−1) ) f (n−1) (b) − f (n−1) (a) ≤
where rm
2(n+1) h i Γ−γ b−a n−1 8n2 + 3 (2n + 1) 1 + (−1) , n! 4 a+b and sm a+b are as given in (3.20). 2 2
Proof. The proof follows directly from (4.13) with x = a+b 2 so that A = B = giving for the braces on the right hand side 2(n+1) h i b−a n−1 2 8n2 + 3 (2n + 1) 1 + (−1) . 4
b−a 2 ,
Some straight forward simplification produces the result (4.15).
Remark 12. It may be noticed (See also Remark 8) that only even order degrees are involved, in (4.15), at the midpoint while this is only the case at the endpoints if the further restriction f (k−1) (a) = f (k−1) (b) is imposed. Further, if n is odd, then there is no perturbation arising from the Gr¨ uss type result (4.15). Theorem 6. Let the conditions of Theorem 3 be satisfied. Further, suppose that f (n) is absolutely continuous and is such that
(n+1)
f
:= ess sup f (n+1) (t) < ∞. ∞
t∈[a,b]
19
Then (4.16)
b−a 1
|ρn (x)| ≤ √ f (n+1) · I (x, n) , ∞ n! 12
where ρn (x) is the perturbed interior point rule given by the left hand side of (4.4) and I (x, n) is as given by (4.5). Proof. Let h, g : [a, b] → R be absolutely continuous and h0 , g 0 be bounded. Then Chebychev’s inequality holds (see [34, p. 207]) 2
|T (h, g)| ≤
(b − a) √ 12
sup |h0 (t)| · sup |g 0 (t)| .
t∈[a,b]
t∈[a,b]
Mati´c, Peˇcari´c and Ujevi´c [29] using a premature Gr¨ uss type argument proved that p (b − a) (4.17) |T (h, g)| ≤ √ sup |g 0 (t)| T (h, h). 12 t∈[a,b]
Thus, associating f (n) (·) with g (·) and K (x, ·) , from (2.2), with h (·) in (4.17) produces (4.16) where I (x, n) is as given by (4.5). Theorem 7. Let the conditions of Theorem 3 be satisfied. Further, suppose that f (n) is locally absolutely continuous on (a, b) and let f (n+1) ∈ L2 (a, b). Then
b−a
(n+1) 1 |ρn (x)| ≤ (4.18)
f
· I (x, n) , π 2 n! where ρn (x) is the perturbed generalised interior point rule given by the left hand side of (4.4) and I (x, n) is as given in (4.5). Proof. The following result was obtained by Lupa¸s (see [34, p. 210]). For h, g : (a, b) → R locally absolutely continuous on (a, b) and h0 , g 0 ∈ L2 (a, b), then 2
|T (h, g)| ≤
(b − a) kh0 k2 kg 0 k2 , π2
where kkk2 :=
1 b−a
Z
a
b
2
|k (t)|
! 12
for k ∈ L2 (a, b) .
Mati´c, Peˇcari´c and Ujevi´c [29] further show that b−a 0 p |T (h, g)| ≤ kg k2 T (h, h). (4.19) π
Now, associating f (n) (·) with g (·) and K (x, ·), from (2.2) with h (·) in (4.19) gives (4.18) where I (x, n) is as found in (4.5). Remark 13. Results (4.16) and (4.18) are not readily comparable to that obtained in Theorem 3 since the bound now involves the behaviour of f (n+1) (·) rather than f (n) (·). Remark 14. Premature results presented in this section may also be obtained, producing bounds for generalized Taylor-like series expansion by taking f ≡ g 0 and b = y. See also Mati´c et al. [29] for related results.
20
P. CERONE AND S.S. DRAGOMIR
5. Applications in Numerical Integration Any of the inequalities in Sections 3 and 4 may be utilised for numerical implementation. Here we illustrate the procedure by giving details for the implementation of Corollary 4. Consider the partition Im : a = x0 < x1 < ... < xm−1 < xm = b of the interval [a, b] and let the intermediate points ξ = ξ 0 , ..., ξ m−1 where ξ j ∈ [xj , xj+1 ] for j = 0, 1, ..., m − 1. Define the formula for γ ∈ [0, 1], (5.1) Am,n (f, Im , ξ)
=
m−1 n XX
j=0 k=1
+γ
k
h
k
(−1) k!
Akj f (k−1)
k (1 − γ) rk ξ j f (k−1) ξ j k−1
(xj ) + (−1)
Bjk f (k−1)
(xj+1 )
i
,
where
(5.2)
k−1 k rk ξ j = Bjk + (−1) Aj Aj = ξ j − xj , Bj = xj+1 − ξ j , and hj = Aj + Bj = xj+1 − xj for j = 0, 1, ..., m − 1.
The following theorem holds involving (5.1).
Theorem 8. Let f : [a, b] → R be a mapping such that f (n−1) is absolutely continuous on [a, b] and Im be a partition of [a, b] as described above. Then the following quadrature rule holds. Namely, (5.3)
Z
a
b
f (x) dx = Am,n (f, Im , ξ) + Rm,n (f, Im , ξ) ,
where Am,n is as defined by (5.1)-(5.2) and the remainder Rm,n (f, Im , ξ) satisfies the estimation (5.4)
≤
|Rm,n (f, Im , ξ)| m−1 P kf (n) k∞ H1 (γ) An+1 + Bjn+1 , j (n+1)! j=0 " #1 q (n) P nq+1 nq+1 kf kp Hq (γ) m−1 Aj + Bj , 1 n! (nq+1) q j=0 kf (n) k 1 1 n 1 × n! 2 + γ− 2 n xj +xj+1 ν(h) + max − ξ , j 2 2 j=0,...,m−1
for
f (n) ∈ L∞ [a, b] ,
for
f (n) ∈ Lp [a, b] , p > 1,
for
1 p
+
1 q
= 1,
f (n) ∈ L1 [a, b] ,
where Hq (γ) is given by (3.18), ν (h) = max {hj |j = 0, ..., m − 1} , and the rest of the terms are as given in (5.2).
21
Proof. Apply Corollary 4 on the interval [xj , xj+1 ] to give Z n k xj+1 X (−1) k (5.5) f (t) dt − (1 − γ) rk ξ j f (k−1) ξ j xj k! k=1 h i k−1 k (k−1) k k (k−1) +γ Aj f (xj ) + (−1) Bj f (xj+1 )
≤
H1 (γ) (n+1)!
(n) n+1 f (t) A + Bjn+1 , j
sup
t∈[xj ,xj+1 ]
1 hR p i p1 Anq+1 +Bjnq+1 q xj+1 (n) j f (u) du , nq+1 xj
Hq (γ) n!
i n h 1 1 ( 2 +|γ− 2 |) R xj+1 f (n) (u) du hj + ξ − j n! 2 xj
n
xj +xj+1 2
,
where the parameters are as defined in (5.2) and Hq (γ) is as given in (3.18). Summing over j from 0 to m − 1 and using the generalised triangle inequality gives (5.6)
|Rm,n (f, Im , ξ)|
≤
m−1 P
H1 (γ) (n+1)!
Hq (γ) n!
sup
j=0 t∈[xj ,xj+1 ]
m−1 P R
xj+1 xj
(n) n+1 f (t) A + Bjn+1 , j
1 (n) f (u) p du p
Anq+1 +Bjnq+1 j nq+1
j=0 n P R xj+1 (n) ( 1 +|γ− 1 |) m−1 f (u) du hj + ξ j − 2 n! 2 2 xj j=0
Now, since
q1
sup
t∈[xj ,xj+1 ]
, n
xj +xj+1 2
(n)
f (t) ≤ f (n) , the first inequality in (5.4) readily fol∞
lows. Further, using the discrete H¨older inequality, we have !1 ! p1 m−1 p X Z xj+1 Anq+1 + Bjnq+1 q j (n) f (u) du nq + 1 xj j=0 ≤
1 nq + 1
q1
m−1 X
× =
.
j=0
m−1 X
j=0
Z
xj+1
xj
Anq+1 + Bjnq+1 j
p (n) f (u) du
q1 q
q1
1
(n) m−1
f q X p Anq+1 + Bjnq+1 1 j q (nq + 1) j=0
and thus the second inequality in (5.4) is proven.
! p1 p p1
22
P. CERONE AND S.S. DRAGOMIR
Finally, let us observe from (5.5) that ! n m−1 X Z xj+1 hj xj + xj+1 (n) + ξ j − f (u) du 2 2 xj j=0 n m−1 Z xj+1 X hj xj + xj+1 (n) + ξ j − (u) ≤ max f du j=0,...,m−1 2 2 j=0 xj n
xj + xj+1 ν (h)
(n) + max ξ j − ≤
f
. j=0,...,m−1 2 2 1
Hence, the theorem is completely proved.
Remark 15. Following the discussion in Remark 5, coarser upper bounds to those in (5.4) are obtained by taking ξ j at either extremity of its interval, giving q1
(n)
(n) m−1 m−1
f H (γ)
f X X p q ∞ , H1 (γ) hn+1 , hnq+1 1 j j (n + 1)! n! q (nq + 1) j=0 j=0
(n)
f 1 n ν (h) n! for f (n) belonging to the obvious Lp [a, b], 1 ≤ p ≤ ∞. These are uniform bounds relative to the intermediate points ξ. Corollary 10. Let the conditions of Theorem 8 hold. Then we have Z b f (x) dx = Am,n (f, Im ) + Rm,n (f, Im ) , a
where
Am,n (f, Im )
=
m−1 n XX
j=0 k=1
+γ k hkj with
h
k
(−1) k! f
(k−1)
k
(1 − γ) rk (δ j ) f (k−1) (δ j )
(xj ) + (−1)
k−1
f
(k−1)
i (xj+1 ) ,
hkj xj + xj+1 k−1 and rk (δ j ) = 1 + (−1) 2 2 and the remainder Rm,n (f, Im ) satisfies the inequality δj =
|Rm,n (f, Im )|
≤
P hj k kf (n) k∞ m−1 2 , (n+1)! 2 j=0 " # q1 P hj nq+1 kf (n) kp Hq (γ) m−1 2 2 , 1 n! (nq+1) q j=0 n kf (n) k1 1 1 n ν(h) , n! 2 + γ− 2 2
for f (n) ∈ L∞ [a, b] ,
for f (n) ∈ Lp [a, b] p > 1, for f
1 p
(n)
+
1 q
= 1,
∈ L1 [a, b] .
23
Proof. The proof is trivial from Theorem 8. Taking ξ j = h Bj = 2j and the results stated follow.
xj +xj+1 2
gives Aj =
6. Concluding Remarks Taking γ = 0 in Corollary 1 gives a generalised Ostrowski type identity which has bounds given by Corollary 4 with γ = 0, reproducing the results of Cerone and Dragomir [4]. This gives a coarse upper bound as discussed in Remark 5 since the bound is convex and symmetric in both γ and x. Let the identity be denoted by Mn (x) which is produced from taking a Peano kernel of (t−a)n n! , t ∈ [a, x] (6.1) kM (x, t) = (t−b)n n! , t ∈ (x, b].
Further, taking γ = 1 in Corollary 1 gives a generalised Trapezoidal type identity with bounds given by Corollary 4 with γ = 1 reproducing the results of Cerone and Dragomir [3]. This choice of γ again gives the coarsest bound as discussed in Remark 5. Let the resulting identity be denoted by Tn (x), which results from taking a Peano kernel of n
(x − t) . n! It was shown in Cerone and Dragomir [4] that kkM (x, t)kq = kkT (x, t)kq . Let (6.2)
kT (x, t) =
IL (x) = λMn (x) + (1 − λ) Tn (x) which is obtained from the kernel k (x, t) = λkM (x, t) + (1 − λ) kT (x, t) where kM (x, t) and kT (x, t) are given by (6.1) and (6.2) respectively. The best one can do for q > 1, q 6= 2 with such a kernel when determining bounds is to use the triangle inequality and so (6.3)
kk (x, t)kq
≤ =
λ kkM (x, t)kq + (1 − λ) kkT (x, t)kq kkM (x, t)kq = kkT (x, t)kq .
The results thus obtained would be given by, for f (n) ∈ Lp [a, b], p ≥ 1, Z n n b X X k k (k−1) f (t) dt − λ (−1) rk (x) f (x) − (1 − λ) (−1) sk (x) a k=1
≤
kf (n) k∞ (n+1)! G1 (x) , kf (n) k p 1 Gq (x) , n!(nq+1) q kf (n) k1 b−a + x− n!
2
k=1
for f (n) ∈ L∞ [a, b] , for f (n) ∈ Lp [a, b] , p > 1,
a+b , 2
1 p
+
1 q
= 1,
for f (n) ∈ L1 [a, b] ,
where rk (x), sk (x), Gq (x) are given by (3.18) and it should be noted that the bound is independent of λ. This result would be no more difficult to implement than (3.17) in Corollary 4, with the best bounds resulting from γ = 12 . For q = 1, 2 or infinity, kk (x, t)kq may be evaluated explicitly without using the triangle inequality
24
P. CERONE AND S.S. DRAGOMIR
at which stage comparison with the results of Corollary 4 would be less conclusive. This will not be discussed further here. In the application of the current work to quadrature, if we wished to approxiRb mate the integral a f (x) dx using a rule Q (f, Im ) with bound E (m), where Im is a uniform partition for example, with an accuracy of ε > 0, then we require mε ∈ N where mε ≥ E −1 (ε) + 1,
with [w] denoting the integer part of w ∈ R. The approach thus described enables the user to predetermine the partition required to assure the result to be within a certain error tolerance. This approach is somewhat different from that commonly used of systematic mesh refinement followed by a comparison of successive approximations which forms the basis of a stopping rule. See [1], [26] and [28] for a comprehensive treatment of traditional methods. References
[1] K.E. ATKINSON, An Introduction to Numerical Analysis, Wiley and Sons, Second Edition, 1989. [2] P. CERONE and S. S. DRAGOMIR, Three point quadrature rules involving, at most, a first derivative, submitted, RGMIA Res. Rep. Coll., 4, 2(1999), Article 8. [ONLINE] http://rgmia.vu.edu.au/v2n4.html [3] P. CERONE and S. S. DRAGOMIR, Trapezoidal type rules from an inequalities point of view, Accepted for publication in Analytic-Computational Methods in Applied Mathematics, G.A. Anastassiou (Ed), CRC Press, New York. [4] P. CERONE and S. S. DRAGOMIR, Midpoint type rules from an inequalities point of view, Accepted for publication in Analytic-Computational Methods in Applied Mathematics, G.A. Anastassiou (Ed), CRC Press, New York. [5] P. CERONE, S. S. DRAGOMIR and J. ROUMELIOTIS, An inequality of Ostrowski type for mappings whose second derivatives are bounded and applications , Preprint. RGMIA Res. Rep Coll., 1(1) (1998), Article 4, 1998. [ONLINE] http://rgmia.vu.edu.au/v1n1.html [6] P. CERONE, S. S. DRAGOMIR and J. ROUMELIOTIS, An inequality of Ostrowski-Gr¨ uss type for twice differentiable mappings and applications, Preprint. RGMIA Res. Rep Coll., 1(2) (1998), Article 8, 1998. [ONLINE] http://rgmia.vu.edu.au/v1n2.html [7] P. CERONE, S. S. DRAGOMIR and J. ROUMELIOTIS, An Ostrowski type inequality for mappings whose second derivatives belong to Lp (a, b) and applications , Preprint. RGMIA Res. Rep Coll., 1(1) (1998), Article 5, 1998. [ONLINE] http://rgmia.vu.edu.au/v1n1.html [8] P. CERONE, S. S. DRAGOMIR and J. ROUMELIOTIS, On Ostrowski type for mappings whose second derivatives belong to L1 (a, b) and applications , Honam Math. J., 21 (1999), No. 1, 122-132. Preprint. RGMIA Res. Rep Coll., 1(2), Article 7, 1998. [ONLINE] http://rgmia.vu.edu.au/v1n2.html [9] P. CERONE, S. S. DRAGOMIR and J. ROUMELIOTIS, Some Ostrowski type inequalities for n-time differentiable mappings and applications , Preprint. RGMIA Res. Rep Coll., 1 (2) (1998), 51-66. [ONLINE] http://rgmia.vu.edu.au/v1n2.html [10] P. CERONE, S.S. DRAGOMIR, J. ROUMELIOTIS and J. SUNDE, A new generalization of the trapezoid formula for n-time differentiable mappings and applications, RGMIA Res. Rep. Coll., 2 (5) Article 7, 1999. [ONLINE] http://rgmia.vu.edu.au/v2n5.html [11] S.S. DRAGOMIR, A Gr¨ uss type integral inequality for mappings of r-H¨ older’s type and applications for trapezoid formula, Tamkang Journal of Mathematics, accepted, 1999. [12] S.S. DRAGOMIR, A Taylor like formula and application in numerical integration, submitted. [13] S.S. DRAGOMIR, Gr¨ uss inequality in inner product spaces, The Australian Math. Gazette, 26 (2), 66-70, 1999. [ONLINE] http://rgmia.vu.edu.au/v2n5.html [14] S.S. DRAGOMIR, New estimation of the remainder in Taylor’s formula using Gr¨ uss’ type inequalities and applications, Mathematical Inequalities and Applications, 2, 2 (1999), 183194.
25
[15] S.S. DRAGOMIR, Some integral inequalities of Gr¨ uss type, Indian J. of Pure and Appl. Math., accepted, 1999. [ONLINE] http://rgmia.vu.edu.au/v1n2.html [16] S.S. DRAGOMIR and N. S. BARNETT, An Ostrowski type inequality for mappings whose second derivatives are bounded and applications , Preprint. RGMIA Res. Rep Coll., 1(2) (1998), Article 9, 1998. [ONLINE] http://rgmia.vu.edu.au/v1n2.html [17] S.S. DRAGOMIR, P. CERONE and A. SOFO, Some remarks on the midpoint rule in numerical integration, submitted, 1999. [ONLINE] http://rgmia.vu.edu.au/v2n2.html [18] S.S. DRAGOMIR, P. CERONE and A. SOFO, Some remarks on the trapezoid rule in numerical integration, Indian J. of Pure and Appl. Math., (in press), 1999. Preprint: RGMIA Res. Rep. Coll., 2(5), Article 1, 1999. [ONLINE] http://rgmia.vu.edu.au/v2n5.html [19] S.S. DRAGOMIR, Y.J. CHO and S.S. KIM, Some remarks on the Milovanovi´ c-Peˇ cari´ c Inequality and in Applications for special means and numerical integration., Tamkang Journal of Mathematics, 30 (1999), No. 3, 203-211. ˇ ´ and S. WANG, The Unified treatment of trape[20] S.S. DRAGOMIR, J.E. PECARI C zoid, Simpson and Ostrowski type inequality for monotonic mappings and applications, Preprint: RGMIA Res. Rep. Coll., 2(4), Article 3, 1999. [ONLINE] http://rgmia.vu.edu.au/v2n4.html [21] S.S. DRAGOMIR and A. SOFO, An integral inequality for twice differentiable mappings and applications, Preprint: RGMIA Res. Rep. Coll., 2(2), Article 9, 1999. [ONLINE] http://rgmia.vu.edu.au/v2n2.html [22] S.S. DRAGOMIR and S. WANG, An inequality of Ostrowski-Gr¨ uss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules, Computers Math. Applic., 33, 15-22, 1997. [23] S.S. DRAGOMIR and S. WANG, A new inequality of Ostrowski’s type in L1 norm and applications to some special means and some numerical quadrature rules, Tamkang Journal of Mathematics, 28, 239-244, 1997. [24] S.S. DRAGOMIR and S. WANG, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and to some numerical quadrature rules, Appl. Math. Lett., 11, 105-109, 1998. [25] S.S. DRAGOMIR and S. WANG, A new inequality of Ostrowski’s type in Lp norm, Indian Journal of Mathemaitcs, 40 (1998), No. 3, 299-304. [26] H. ENGELS, Numerical Quadrature and Cubature, Academic Press, New York, 1980. [27] A. M. FINK, Bounds on the derivation of a function from its averages, Czech. Math. Journal, 42 (1992), No. 117, 289-310. [28] A.R. KROMMER and C.W. UEBERHUBER, Numerical Integration on Advanced Computer Systems, Lecture Notes in Computer Science, 848, Springer-Verlag, Berlin, 1994. ´ J.E. PECARI ˇ ´ and N. UJEVIC, ´ On new estimation of the remainder in gener[29] M. MATIC, C alised Taylor’s formula, M.I.A., Vol. 2 No. 3 (1999), 343-361. [30] M. MERKLE, Remarks on Ostrowski’s and Hadamard’s inequality, Univ. Beograd. Publ. Elecktrotechn Ser. Mat., 10 (1999), 113-117. ´ and J.E. PECARI ˇ ´ On generalisations of the inequality of A. Os[31] G.V. MILOVANOVIC C, trowski and some related applications, Univ. Beograd. Publ. Elecktrotechn Ser. Fiz., 544-576. ´ J.E. PECARI ˇ ´ and A.M. FINK, Classical and New Inequalities in [32] D.S. MITRINOVIC, C Analysis, Kluwer Academic Publishers, 1993. ´ J.E. PECARI ˇ ´ and A.M. FINK, Inequalities for Functions and Their [33] D.S. MITRINOVIC, C Integrals and Derivatives, Kluwer Academic Publishers, 1994. ˇ ´ F. PROSCHAN and Y.L. TONG, Convex Functions, Partial Orderings, and [34] J.E. PECARI C, Statistical Applications, Academic Press, 1992. ´ [35] J. SANDOR, On the Jensen-Hadamard inequality, Studia Univ. Babes-Bolyai, Math., 36 (1), 9-15, 1991. [36] F. QI, Further generalisations of inequalities for an integral, Univ. Beograd. Publ. Elecktrotechn Ser. Mat., 8 (1997), 79-83.
26
P. CERONE AND S.S. DRAGOMIR
School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia E-mail address:
[email protected] E-mail address:
[email protected] URL: http://melba.vu.edu.au/~rgmia/SSDragomirWeb.html
