ELSEVIER

Available online at www.sciencedirect.com MATHEMATICAL AND

SCIENCE @DIRECT" COMPUTER MODELLING

Mathematical and Computer Modelling 41 (2005) 463-483 www.elsevier.com/locate/mcm

Generalization of Bullen-Simpson's 3/8 Inequality

M. MATIC FESB, Mathematics Department, University of Split

R. Bo~kovida BB, 21000 Split, Croatia mmat ic©f esb. hr

J. PECARIC Faculty of Textile Technology, University of Zagreb

Pierottijeva 6, I0000 Zagreb, Croatia pecaric©hazu, hr

A. VUKELIC Faculty of Food Technology and Biotechnology, Mathematics Department

University of Zagreb, Pierottijeva 6, I0000 Zagreb, Croatia avukelic©pb£.hr

(Received March 200~; revised and accepted June 200~)

Abs t r ac t - -Gene ra l i z a t i on of Bullen-Simpson's 3/8 inequality for (2r)-convex functions is given, by using some Euler-type identities. A number of inequalities, for functions whose derivatives are either functions of bounded variation or Lipschitzian functions or functions in Lp-spaces, are proved. (~) 2005 Elsevier Ltd. All rights reserved.

Keywords - -Bul len-S impson ' s 3/8 inequality, Quadrature formulae, Functions of bounded varia- tion, Lipschitzian functions.

1. I N T R O D U C T I O N

For any convex function f : [0, 1] ~ R, the following pair of inequalities, usually referred to in the literature as Hadamard's inequMities, hold:

(1) f ~ f(t) dt ~_ 2

If f is concave, the inequalities are reversed. In [1], Hammer showed, by a simple geometric argument, that for convex functions the absolute value of error in the midpoint quadrature rule is always smaller than absolute value of the error in the trapezoidal rule, i.e., the following inequalities are valid for a convex function f:

0<_ f ( t ) d t - f < 5 [f (0)+ f (1 ) ] - f ( t )dt . (1.2)

0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. Typeset by .AA/}S-TEX doi:lO.lO16/j.mcm.2004.06.019

464 M. MATIC et al.

An elementary analytic proof of (1.1) and (1.2), but stated on the interval [-1, 1], was given in [2].

The trapezoid rule is the simplest example of a closed quadrature rule, while the midpoint rule is the simplest open quadrature rule [3]. The next simplest such pair consisting of Simpson's rule and the dual Simpson formula was considered in [4]. Another pair consisting of an open and a closed quadrature rule is based on Simpson's 3/8 formula [5]

o f(t dt g

and the Maclaurin formula

1 1

which hold in this form for some rl and ~ from [0, 1], for any function f with continuous fourth derivative f(4) on [0, 1]. If additionally, f(4) is nonnegative, tha t is, if f(4)(t) > 0, Vt e [0, 1], then from the above formulas, we immediately have

1 1

(1.5) -< l [f(O)+ 3f ( 1 ) + 3f ( 2 ) +

In the case when f(4) exists, the condition f(4)(t) > 0, Yt E [0, 1] is equivalent to the requirement that f is a 4-convex function on [0, 1]. However, a function f may be 4-convex although f(4) does not exist.

In [2], Bullen proved that, if f is 4-convex, then (1.5) is valid. Moreover, he proved that the Maclaurin quadrature rule is more accurate than the Simpson's 3/8 quadrature rule, that is, we have

i 1

-8<1 I f ( 0 ) + 3 f ( 1 ) + 3 f ( ~ ) q - f ( l ) ] - f 0 1 f ( t ) d t ,

provided f is 4-convex. We shall call this inequality Bullen-Simpson's 3/8 inequality. We recall tha t a function f : [0, 1] ~ R is said to be n-convex on [0, 1], for some n >__ 0, if for

any choice of n + 1 distinct points xo, Xl,.. . , xn from [0, 1], we have

f [x0, z l , . . . , zn] >__ 0,

where fix0, x l , . . . , xn] is nth-order divided difference of f . In the case when the above inequality is reversed, f is said to be n-concave on [0, 1]. If f is n-convex (n-concave), then f(~-2) exists and is a convex (concave) function in the ordinary sense. Especially, 2-convex (2-concave) function f is convex (concave) in the ordinary sense. Also, if f(n) exists, then f is is n-convex (n-concave) if and only if f(~) > 0 (f(~) < 0). For some further details on n-convexity, see, for example, [6].

In the recent papers [7,8], Dedid et al. considered a generalizations of (1.3) and (1.4), based on the well-known Euler formula for expanding n times differentiable function f in terms of Bernoulli polynomials (see, for example, [9, p. 17]. Before stating the basic results from [7,8], we recall that a sequence of Bernoulli polynomials (B~(t))k>o is uniquely determined by the following identities:

B'k(t ) = kBk_l(t), k >_ 1, Bo(t) = 1, (1.7)

and Bk(t + 1) -- Bk(t) = kt k-~, k >_ O. (1.8)

Bullen-Simpson's 3/8 Inequality 465

The values Bk = Bk (0) are usually called the Bernoulli numbers. For some further details on the Bernoulli polynomials and the Bernoulli numbers, see, for example, [10] or [11]. Also, we need a sequence (B~(t))k>O of periodic functions of period 1, related to the Bernoulli polynomials as

B;(t) = Bk(t), O < t < l and B;(t + l) = B;(t), t E R.

It is easy to see that B~(t) = i and B~(t) is a discontinuous function with a jump of - 1 at each integer. Also, since Bk(1) = Bk(O) = Bk for k k 2, the functions B~(t) are continuous for k > 2. Moreover, from (1.7), we get

B•'(t) = kB~_l(t), k k 1,

for every t 6 R, when k _> 3, and for every t E R \ Z when k = 1, 2. Now we consider a function f : [0, 1] ~ R, such that f (~- l ) is a continuous function of bounded

variation, for some n _> 1. Denote

where [a] is the greatest integer less than or equal to a. For such function f , the following identities were obtained in [7]:

and

/0 1[ ] f ( t) dt = ~ f(0) + 3 f + 3f + f(1) + TS( f ) + a~(f)

1 81 If(0)+3f (1)+3f (2)+ f(1)] + TS(f)+ c~(f). o f ( t dt =

The perturbations TS(f) and TS(f) are de~ned as ToS(f) = Ts( f ) = 0 and, for r~ _> 2,

TSm(f) = 81~2 ~ 1 (1-3 2-2k) B2k [f(2k-1)(1)- f(2k-1) (0)] k=2

while the remainders a l ( f ) and a2(f) are given by

i i ~01 crn(f) - 8 (n!) G s (t) d/( '~-l)(t)

and

where, for all n _> 1,

g~(f) = 1 j£o I F s (t) df(n-1)(t),

GS(t):2B (1-t)+3B (1-t)+3B (3-t ) , F[(t) a~(t) -~

Further, under the same assumptions on f , the following two identities were obtained in [8]:

i 1 fo f ( t ) d t = - g [3f ( 1 ) + 2 f ( 1 ) + 3 f ( 5 ) ] + T M ( f ) + p ~ ( f )

(1.9)

(1.10)

(1.11)

(1.12)

466 M. MATId et al.

and

/01 (1) 3f + 2f + 3f + T~ (f) + p2n(f). (1.13) f(t) dt : §

Here, the perturbations T S ( f ) and T y ( f ) are de~ned as To"(/ ) = T y ( f ) = 0 and, for m > 2,

Tr ~M(f) = -81f i ~ 1 (1-21.2k) (1-3 2-2k) B2k [f(2k-1) (1) -- f(2k-1) (0)] k-=2

(1.14)

while the remainders p~(f) and p2n(f) are given by

and

where, for all n _> 1,

I___L_ f l GM(t) df(n-1)( t) pl(f) s (n!) .Io

1 f lFM( t ) d f ( n - 0 ( t ) , 8 (n!) J0

The aim of this paper is to establish a generalization of the inequalities (1.5) and (1.6) for a class of (2r)-convex functions and also to obtain some estimates for the absolute value of difference between the absolute value of error in the Maclaurin quadrature rule and the absolute value of error in Simpson's 3/8 quadrature rule. Let us define

We shall make use of the following seven-point quadrature formula

fo f(t) ~ D(O, 1), dt

obtained by adding the Simpson 3/8 and the Maclaurin quadrature formulae. It is suitable for our purposes to rewrite the second inequality in (1.6) in the form

£o i f(t) dt < D(O, 1). (1.15)

As we mentioned earlier, this inequality is valid for any 4-convex function f and we call it the Bullen-Simpson's 3/8 inequality.

It should be noted that each continuous n-convex function on [0, 1] is the uniform limit of the sequence of corresponding Bernstein's polynomials (see, for example, [6, p. 293]). Also, Bernstein's polynomials of continuous n-convex function are also n-convex functions. Therefore, when stating our results for a continuous (2r)-convex function f , without any loss in generality, we assume that f(2~) exists and is continuous. Actually, those results are valid for any continuous (2r)-convex function f .

Bullen-Simpson's 3/8 Inequality 467

2. EULER BULLEN-SIMPSON 3/8 FORMULAE

We consider the sequences of functions (Gk(t))k>l and (Fk(t))k>l defined for t E R by

Ca(t) := CSk(t) + aM(t), Fk(t) := FS(t) + FM(t),

where cS(t), aM(t), FS(t), and FM(t) are defined as in the Introduction. So we have

Gk(t) = 2B;(l - t) + 3B; ( 1 - t) + 3B; (~ - t)

+ 2 B ; ( 1 - t ) + 3 B ; ( 2 - t ) + 3 B ~ ( ~ - t ) , tER ,

and

where

Fk(t) = Ok(t) - [ ~ , t e R ,

= B k ( 0 ) + 3 B k ( 1 ) + 3 B k ( 1 ) + 2 B k (2 )+3Bk (2 )+3Bk ( 5 ) + Bk(1).

For any function f : [0, 1] --~ R, such that f(n-D exists on [0, 1] for some n > 1, let D(0, 1) be defined as in the Introduction. Further, we define To(f) = Tl ( f ) := 0 and, for 2 < m _< [n/2],

1 Tin(f) := ~ [T~ (f) + T2 ( f ) ] ,

where TS(f) and TmM(f) are given by (1.11) and (1.14), respectively. It is easy to see that

In the next lemma, we establish two formulae which play the key role in this paper. We call them the Euler Bullen-Simpson 3/8 formulae.

LEMMA 1. Let f : [0, 1] ~ R be such that f(n-1) is a continuous function of bounded variation on [0, 1], for some n > 1. Then we have

where r = [n/2] and

Also,

1 f(t) dt = D(O, 1) + Tr(f) + r~(f), (2.2)

1 f01 ~"(f) - 16 (n!) a . (t) df("- l )( t ) .

1 f( t) dt = D(0, 1) + Ts(f) + T~(f), (2.3)

w h e r e s = [ (n - 1 ) / 2 ] a n d

f0 ,-~(I) = 10~-,~!) F~ (t) df("-l)(t).

PaOOF. First we multiply formulas (1.9) and (1.12) by the factor 1/2 and then add them up to obtain the identity (2.2). The identity (2.3) follows analogously from (1.10) and (1.13). |

468 M. MATI6 eta[.

REMARK 1. In the case when f : [0, 1] -+ R is such that f(n) exists and is integrable on [0, 1], then the Riemann-Stieltjes integral f~ H(t)df(n-i)(t) is equal to the Riemann integral

f~ H(t)f(n)(t)dr. Therefore, if f(~) exists for some n > 1 and is integrable on [0, 1], then (2.2)

and (2.3) reduce to

fo 16 (n!) G,~ (t) f (n) (t) at (2.4) fo f(t) dt = D(0, 1) + T~(f) + - -

and ~o i 1 ~i f(t) at = D(0, 1) + T8 (f) + 16 (n!------~ Fn (t) f(n)(t) at. (2.5)

REMARK 2. The interval [0, 1] is used for simplicity and involves no loss in generality. The results which follow will apply, without comment, to any interval that is convenient.

Namely, it is easy to transform the identities (2.2) and (2.3) to the identities which hold for any function f : [a, b] -* R, such that f (n - i ) is a continuous function of bounded variation on

[a, b], for some n >_ 1. We get

fab ( b - a ) ~ b ( t - a ) f(t)dt=D(a,b)+T~(f)+ -~(~.) G~ ~-a df(~- l ) ( t ) (2.6)

and

f(t)dt=D(a,b)+¢8(f)+ T6-((~.) Fn ~-a df(n-1)(t)'

where

b-a D(a,b) := 16

[f(a)+3(5a + 3 (2a

while T0(f) = T l ( f ) = 0 and

(2.7)

for 2 <_ m <_ [n/2].

3. B U L L E N - S I M P S O N ' S 3 / 8 I N E Q U A L I T Y F O R ( 2 r ) - C O N V E X F U N C T I O N S

In this section, we use the Euler Bullen-Simpson 3/8 formulae established in Lemma 1 to extend the Bullen-Simpson's 3/8 inequality for (2r)-convex functions. First, we need some properties of

the functions Gk(t) and Fk(t) defined in the previous section. First note that it is enough to know the values of the functions Gk(t) and Fk(t), k >_ 1 only

on the interval [0, 1/2]. Namely, the functions B~($) are periodic with period 1 so that for

0 < t < 1/2, we have

+ 2 B ; (1 - t)

= Gk (t)

Bullen-Simpson's 3/8 Inequality 469

and

Fk (t + l ) = Gk (t + l ) -- [~k = Gk (t) -- Bk = Fk (t) .

Since B1(t) --- t - (1/2), we get/~1 = 0 and

- 1 , t = 0,

-16 t + 1, t E ( 0 , 6 ] ,

Gl( t )=Fl( t )= -16 t + 4, t E ( 6 , ~ ] , (3.1)

Further, for k > 2, the functions B~(t) are continuous and the same is true for Gk(t) and Fk(t), k > 2. Also, we have for k > 2,

Gk(O)=Gk(2 ) = Gk(1) ----/~k

and

For example, for k -- 2 and k = 3, we have B2(t) --- t 2 - t + (1/6) and B3(t) = t 3 - (3/2)t2 + (1/2)t, so that by direct calculation, we g e t / ~ =/~3 -- 0 and

1 1 C 2 ( t ) = F 2 ( t ) = 16t 2 - 8 t + 1 , t e ~ , ~ ,

16 t2 - 14t + 3 , t E ( 1 , ~ ] .

16t3 +3t2 ' ( 1 )

G3(t) = F3(t) = -16 t 3 + 12t ~ -- 3t + ,

-16t3 + 21t2 - 9t + ( 5 ) ,

(3.2)

(3.3)

The Bernoulli polynomials of even order are symmetric and those of odd-order skew-symmetric about 1/2, that is, [10, 23.1.8],

B k ( 1 - t ) =(--1)kBk(t), O < t < l , k>__l. (3.4)

Also, we have

and

1 B k ( 1 ) = B k ( 0 ) = B k , k_>2, B I ( 1 ) = - B I ( 0 ) =

Therefore, for r > 1, we have

B2r-1 = 0, r _~ 2.

/~2~-1 = 0 (3.5)

470 M. MATI6 et al.

and ~ ~ + ~ ( 1 ) ~ (1)÷~ (~)~o~ (~)~,~ (~) Also, we have [10, 23.1.21, 23.1.23, 23.1.24],

and

B 2 r ( 1 ) : - ( 1 - 2 1 - 2 r ) B2r,

which gives the formula

( 3 ) 1 (1 - 31-2~) B2~'

( 1 ) 1 (1 _ 21_2~) (1 _ 31_2r) B2r ' B2~ =

/)2r ---- -21-2~ (1 - 32-2r) B2~,

Now, using (3.5) and (3.6), we get

F2~-l(t) = a2~_l(t) ,

and

r > l .

r > _ l ,

(3.6)

(3.7)

We use the formula [10, 23.1.10]

m - - 1

i----0

Setting m = 3 , x = 1/12, we get

~(1) = ~ ( ~ ) ~ [~(~)÷~(1~)~(~)]

r_>0, m > l .

PROOF.

and

(3.9)

F2r(t) = G2~(t) + 21-2r (1 - 32-2r) B2r, r > 1. (3.8)

Further, as we pointed out earlier, the points 0 and 1/2 are the zeros of Fk(t), k >_ 2. As we shall see below, 0 and 1/2 are the only zeros of Fk(t) in [0, 1/2] for k = 2r, r >_ 2, while for k = 2r - 1, r _> 2, using (3.4), we easily get

~ ~(~)_-~ ~(~) 0 We shall see that 0, 1/4, and 1/2 are the only zeros of F2~_l(t) = G2~_l(t), in [0, 1/2] for r _> 2. Also, note tha t for r _> 1, we have

The values G2~(1/4) and F2~(1/4) can also be evaluated exactly.

LEMMA 2. F o r r > 1, w e h a v e

~ (1)--~ ~/1-~ ~/(~-~ ~

Bullen-Simpson's 3/8 Inequality 471

which can be rewritten as

B2r(1)+B2r(5)=(31-2r-1)B2~(~) ,

since B2r(3/4) = S2r(1/4), by (3.4). Now we have (see [10, 23.1.22])

(1)= and using again (3.4), we get

= 2 (32-2r - 1) B2~ ( ~ )

- - 21-2~ (1 - 2 ~-2~) (1 - 3 ~-2~) B2~

Also using (3.6), we get (3.9),

Now we prove that the functions Gk (t) and Fk (t) are symmetric for even k and skew-symmetric for odd k, about 1/4.

LEMMA 3. For k >_ 2, we have

Gk(~--t)=(--1)kGk(t), 0 < t < 1,

and FkI~--tl=(--1)kFk(t ), O~_t~_ 1.

PROOF. As we already noted, the functions B~ (t) axe periodic with period 1 and continuous for k _> 2. Also, from (3.4), we get

B•(1-t)=(-1)kB;(t), t e R , k_>2.

Therefore, for k _> 2 and 0 < t < 1/2, we have

= (--1)kGk(t),

which proves the first identity. Further, we have/~k = (--1)k/~k, k _> 2, since (3.5) holds, so that

Fk(1 t ) = G k ( 2 - t ) - / ~ k = ( - x ) k [ G k ( t ) - / ~ k ] = ( - 1 ) k F k ( t ) '

which proves the second identity. |

Note that the identities established in Lemma 3 are valid for k -- 1, also, except at the points 0, 1/6, 1/3, and 1/2.

472 M. MATI6 et al.

LEMMA 4. For r >_ 2, the function a2 r - i ( t ) has no zeros in the interval (0, 1/4). The sign of this

function is determined by 1

(--1) G2r-l( t) > 0, 0 < t < 4"

PROOF. For r = 2, G3(t) is given by (3.3) and it is easy to see that

1 - . (3.10) as( t ) > 0, 0 < t < 4

Thus, our assertion is true for r = 2. Now, assume that r _> 3. Then 2r - 1 >_ 5 and G2r-l( t) is a continuous and twice differentiable function. Using (1.7), we get

= - 1)a2 _2(t)

and

agr_ l ( t ) = (2r - 1)(2r - 2)a2~_3(t). (3.11)

We know that 0 and 1/4 are the zeros of a2~-l( t ) . Let us suppose tha t some a, 0 < a < 1/4, is also a zero of a2~-l( t ) . Then inside each of the intervals (0, a) and (a, 1/4), the derivative G~r_l(t) must have at least one zero, say/31, 0 </31 < a, and/32, a </32 < 1/4. Therefore, the second derivative a'2'r_l(t) must have at least one zero inside the interval (/31,/32). Thus, from the assumption that a2~-l( t ) has a zero inside the interval (0, 1/4), it follows that (2r - 1)(2r - 2)G2,_a(t) also has a zero inside this interval. From this, it follows that G3(t) would have a zero inside the interval (0, 1/4), which is not true. Thus, C2~-1 (t) cannot have a zero inside the interval (0, 1/4). Further, if a2~-3(t) > 0, 0 < t < 1/4, then from (3.11), it follows that C2r-1(t) is convex on (0, 1/4), and hence, G2~-l(t) < 0, 0 < t < 1/4, while in the case when C2r-3(t) < 0, 0 < t < 1/4, we have that G2~-l(t) is concave, and hence, a 2 r - l ( t ) > 0, 0 < t < 1/4. Since

(3.10) is valid, we conclude that

1 ( - 1 y G 2 r _ l ( t ) > o, 0 < t < 5" !

COROLLARY 1. For r >_ 2, the functions ( - 1 ) r - l F2r(t) and (--1)r- lc2r( t ) are strictly increasing

on the interval (0, 1/4), and strictly decreasing on the interval (1/4, 1/2). Consequently, 0 and 1/2 are the only zeros of F2+(t) in the interval [0, 1/2] and

max ]F2,(t)] = 22-2. (1 - 2 -2~) (1 - 32-2.) [B2r[, r __ 2. te[0,1]

Also, we have m a x [ a 2 r ( t ) l : 21 -2 r (1 -- 32-2r) IB2~l, r _ 2. te[o,1]

PROOF. Using (1.7), we get

[(-1)~-lF2~(t)] ' = [(-1)~-1C2~(t)] ' = 2 r ( -1)*a2r_ l (t)

and ( -1 ) rC2r - l ( t ) > 0 for 0 < t < 1/4, by Lemma 4. Thus, (-1)~-lF2~(t) and (-1)~-1C2~(t) are strictly increasing on the interval (0, 1/4). Also, by Lemma 3, we have F2~(1/2 - t) = F2r(t), 0 < t < 1/2 and G2~(1 /2 - t) = a2~(t), 0 < t < 1/2, which implies that (-1)~-lF2~(t) and (-1)~-lG2r( t ) are strictly decreasing on the interval (1/4, 1/2). Further, F2r(0) = F2,(1/2) = 0, which implies that IF2r(t)l achieves its maximum at t = 1/4, that is,

m a x I F 2 r ( t ) , = F2r (1) l=22-2r(l-2-2r)(l-32-2r)IB2rl. te[o,l]

Also, F

max JG2r(t)l = max 4 JG2r (0)[ te[0,1] l

which completes the proof.

___ 21-2~(i _ 32-2~)]B2~I,

Bullen-Simpson's 3/8 Inequality 473

COROLLARY 2.

ALso, we have

and

Assume r > 2. Then we have

fo x JG2~-~(t)l d t = 2 3-2r (1 -- 2 -2r) (I - 3 2-2~)

j~0 1 B2v IF2~(t)] dt = = 21-2r (1 - 3 2 -2 r ) IB2~]

~o I 2 B2~[ = (1 - 3 2-2r) ]Bur] la2~(t)l at _< 2 ~-2r

PROOF. Using (1.7), it is easy to see that

G'(t) = -mGm_l(t), m > 3. (3.12)

By (3.7), we have fo ~ ]F2~_l(t)J dt = fo ~ JG2~_l(t)] dt. Now, using Lemmas 3 and 4 and (3.12), we get

jr01 ~01/4 at --~r G2r(t)]l/4 [G2~-l(t)] dt = 4 G2~-l(t) = 4

- - - G 2 ~ ( 1 ) (0) 2 3 - 2 ~ ( 1 - 2 - 2 r ) ( 1 - 3 2 - 2 ~ ) 2 - G2r = [B2r]

- - 7 ' 7" '

which proves the first assertion. By Corollary 1, F2~(t) does not change the sign on the interval (0, 1/2). Therefore, using (3.8) and (3.12), we get

~oilF2r(t)[dt=2 foi/2F2~(t) dt =2 ~l/2[G2r(t)-B2~] dt

] 27" 1 ~- 1 G2r+l(t)]lo/2 ~B2r = 2 - = [/~2~ = 21-2~ (1 - 3 2-2~) JB2~J

This proves the second assertion. Finally, we use (3.8) again and the triangle inequality to obtain

~ 1 IG2~(t)l dt = .~1 F2~(t) + dt

_< ]F2~(t)[ dt + = 2 = 2 2-2r (1 - 3 2-2r) [B2~],

which proves the third assertion. |

In the following discussion, we assume that f : [0, 1] --* R has a continuous derivative of order n, for some n > 1. In this case, we can use formulae (2.4) and (2.5) from Remark 1, so that remainders ~-nl(f) and ~'n2(f) are given as

Tl(f) = 1 jO 1 16(n[-----~ Gn(s)f ('~) (s) ds (3.13)

and

1 f0x T ~ ( f ) = 16(n!---~ F,~(s)f ('~) (s) ds. (3.14)

474 M. MATId et al.

LEMMA 5. I f f : [0,1] --* R is such that f(2r) is continuous on [0, 1], for some r > 2, then there exists a point ~ E [0, 1], such that

1 2_2r (1 - 3 2-2r) B2r/(2~)(~). (3.15) ¢ ~ ( / ) = 8(2r)!

PROOF. Using (3.14) with n = 2r, we can rewrite T2r(f) as

1 ~-2r(f) = (-1)~-1 16(2r)! Jr, (3.16)

where

f0 1

Jr = ( -1 )~- lF2r(s ) f (Rr ) ( s )ds .

If m = min f(2~)(t),

to[o,1]

then m < f(2~)(s) < M, 0 < s < 1.

On the other side, from Corollary 1, it follows that

= max ~ "'l(2r)tt),, M tel0,1]

(3.17)

(-1)~-lF2~(s) > 0, 0 < s < 1,

which implies

/01 /0 m ( -1 ) r - lF2r ( s ) ds < Jr <- M ( -1)~- lF2r(s ) ds.

Similarly, as we have already calculated in the proof of Corollary 2, we get

fo F2r(s) ds = --/~2r = 21-2r (1 -- 3 2-2r) B2r,

so that m( -1 ) r -121 .2 r (1 - 3 2-2r) B2r < Jr < M(-1 ) r -121-2 r (1 - 3 2-2r) B2r.

By the continuity of f(2r) (s) on [0, 1], it follows that there must exist a point ~/ C [0, 1], such that

Jr = ( - 1 ) r - 121 - 2r (1 - 3 2-2r) B~J(~r) (v) .

Combining this with (3.16), we get (3.15). |

Now, we prove the main result.

THEOREM 1. Assume f : [0, 1] --* R is such that f(2r) is continuous on [0, 1], for some r >_ 2. I f

f is a (2r)-convex function, then for even r, we have

1 1

(3.18) ),o, +

while for odd r, we have reversed inequalities in (3.18).

PROOF. Let us denote by LHS and RHS, respectively, the left-hand side and the right-hand side in the second inequality in (3.18). Then we have

LHS = p2r(f)

Bullen-Simpson's 3/8 Inequality 475

and

RHS - LHS = --2T~r(f),

where P~r(f) and T22r(f) are determined, respectively, by the identities (1.13) and (2.3). Dedid et al. [8] proved that, under given assumption on f , there exists a point ~ E [0, 1], such that

p ~ . ( f ) _ 1 8(2r)! (1 - 21-2r) (1 - 32-2r) B2,f(2r)(~). (3.19)

Also, by Lemma 5, we know that

1 1-2. (1 3 2-2") --2T~r(f) = --8(2r)!2 - B2rf(2")(7/), (3.20)

for some point 77 E [0, 1]. Finally, we know that [10, 23.1.15]

( - 1 ) r - l B 2 , > 0, r = 1, 2 , . . . . (3.21)

Now, if f is a (2r)-convex function, then f(2r)(~) > 0 and f(2r)(7) -> 0, so that using (3.19)-(3.21), we get the inequalities

LHS _> 0, RHS - LHS >_ 0, when r is even,

LHS < 0, RHS - LHS _< 0, when r is odd.

This proves our assertions. 1

REMARK 3. In the case when r = 2, we have B4 = - 1 / 3 0 and formula (3.15) reduces to

T~( f ) : - 1 1036S0/(4)(~) •

Note that, in this case, the result stated in Theorem 1 reduces to Bullen~s result that we mentioned in the Introduction.

THEOREM 2. Assume f : [0, 1] ~ R is such that f(2r) is continuous on [0, 1], for some r >_ 2. I f f is either a (2r)-convex or (2r)-concave function, then there exists a point 0 E [0, 1], such that

(: - - 3,-,.),.. [:(..-:>(:)- :(,.-,(o)].

PROOF. First, consider the case when f is (2r)-convex, that is, f(2r)(t) >_ 0, 0 < t < 1. By Corollary 1, we get

O ~ _ ( - 1 ) r - l F 2 r ( S ) ~ _ ( - 1 ) ~ - l F 2 r ( 1 ) , 0 < s < l .

Therefore, if Jr is given by (3.17), then

(1)/0 ' o <_ Jr <__ (-1)r-IF2. /¢2.)(.) ds

---- (--1)r-lF2r (1) [f(2r-1)(1)--f(2r-D(O)].

So, there must exist a point t~ E [0, ], such that

Jr~-O(-l)r-iF2r (1) [f(2r--l)(1)--f(2r--1)(O)].

476 M. MATIO et al.

Combining this with (3.16) and using (3.9), we get (3.22). The argument is the same when f is (2r)-concave since in that case, f(2r)(t) _< 0, 0 < t < 1 and we get

( - -1 ) r - lF2r(1) [f(2r-1)(1)--f(2r-D(O)] <_Jr<_O.

REMARK 4. If we approximate f~ f( t) dt by

I2~(f) := D(0, 1) + T r - l ( f ) ,

then the next approximation will be I2~+2(f). The difference

A2r(f) = I2~+2(f) - I2r(f)

is equal to the last term in I2~+2(f), that is,

1 (1 - 3 ~-2~) B2~ [S(2~-1)(1) - f(2"-1)(0)]. A 2 " ( f ) - 8(2r)[ 2-2~

We see that, under the assumptions of Theorem 2,

= ( 1 - 2

THEOREM 3. Assume f : [0, 1] --* R is such that f(2r+2) is continuous on [0, 1], for some r >_ 2. If f is either (2r)-convex and (2r + 2)-convex or (2r)-concave and (2r + 2)-concave function, then the remainder T~r(f ) has the same sign as the first neglected term A2r(f) and

PROOF. We have

that is,

A2r(f) q- T22r+2(f) = r2r(f),

A2r(f ) = T2r(f) -- TZ2r+2(f). (3.23)

By (3.14), we have

r~r(f) - 16(2r)! F2r(s)f (2r)(s) ds

and

fo - r~r+2(f ) - 16(2r + 2)! [-F2r+2(s)] f(2~+2)(s) ds.

Under the assumptions made on f , we have for all s E [0, 1], either

f(2~)(s) _> 0 and f(2~+2)(s) >_ 0

or f(2r)(s) ~ 0 and f(2r+2)(s) < O.

Also, from Corollary 1, it follows that for all s C [0, 1],

( -1)r - lF2~(s) > 0 and ( -1)~- l [ -F2r+2(s)] > 0.

We conclude that T2~(f) has the same sign as -%2~+z(f ). Therefore, because of (3.23), A2r(f) must have the same sign as T2~(f) and --T2~+2(f). Moreover, it follows that

[T2r(f)l <: [A2r(f)] and [-T2r+2(f)l _< [A2r(f)l •

|

Bullen-Simpson's 3/8 Inequality 477

4. S O M E I N E Q U A L I T I E S R E L A T E D TO E U L E R B U L L E N - S I M P S O N 3//8 F O R M U L A E

In this section, we use the Euler Bullen-Simpson 3//8 formulae established in Lemma 1 to determine the absolute value of difference between the absolute value of error in the Maclaurin quadrature rule and the absolute value of error in the Simpson's 3/8 quadrature rule. We do this by proving a number of inequalities for various classes of functions.

First, let us denote

f 1 81[f(O)+3f(1)+3f(2)+f(1)] R s := /, f ( t ) d t -

and

By the triangle inequality, we have

I[RM[ - IRsll _< [RM + Rs[.

Now, if we define R := RM + R s , then

f0 g = f ( t ) dt - D(O, 1), (4.1)

where D(0, 1) denotes the same expression as in the Introduction.

THEOREM 4. Let f : [0, 1] --* R be such that f(n-1) is an L-Lipschitzian function on [0, 1] for

some n >_ 2. I f n = 2r - l , r >_ 2, then

Ifo i (f) 1 ~01 f ( t ) dt - D(O, 1) - T~_I -< 1 6 ( 2 r - - i)! [G2._l ( t ) l dt . L (4.2)

= 2-~r (1 - 2-~r) (1 - 3 ~ - ~ ) I B ~ I . L. (2r)!

I f n = 2r, r > 2, then

I/o I f ( t ) dt - D(O, 1) - T ~ _ i ( f ) < ~

and also

1 fol 16(2r)! [F2~(t)[ d t . L

2 - ~ (1 - 3 ~-2~) Is2~l. L, 8(2r)[

(4.3)

fo i T r ( f ) 1 fo 1 f ( t ) dt - D(0, 1) - < 16(2r)! [G2r(t)[ d t . L (4.4)

2-~r ( 1 - 32 -~ ) < -f(2r~. IB2~I'L"

PROOF. For any integrable function ~ : [0, 1] ~ R, we have r / o 1

o eb( t )d f (~- ! ) ( t ) <_ Icb(t)I d t . L, (4.5)

since f(,~-l) is L-Lipschitzian function. Applying (4.5) with if(t) = G2r-l(t), we get

I t fo i df(2r-2) (t) I ~ i i 6 ( 2 r - 1)! G2~-i(t) < 16(2r - 1)! IG2~-I (t)l d t -L .

Applying the above inequality and the identity (2.3), we get the inequality in (4.2). Similarly, we can apply the inequality (4.5) with @(t) = F2r (t) and again the identity (2.3) to get the inequality in (4.4). Finally, applying (4.5) with @(t) -- G2T(t) and the identity (2.2), we get the first inequality in (4.4). The equalities in (4.2) and (4.3) and the second inequality in (4.4) follow from Corollary 2. |

478 M. MATId et al.

COROLLARY 3. Let f : [0, 1] --* R be a given function. I f f" is L-Lipschitzian on [0, 1], then

f ( t ) dt - D(O, 1) < 13824 L,

I f f ' " is L-Lipschitzian on [0, 1], then

f01 1) 1 f ( t ) dt - D(O, <_ 103680 L,

[RI _< 69-~L.

1 IRI <_ 51840 L.

PROOF. The first pair of inequalities follow from (4.2) with r = 2, while the second pair follow from (4.3) with r = 2. |

REMARK 5. If f is L-Lipschitzian on [0, 1], then, as above

f o l f ( t ) d t - D ( O , 1) <_ ~-~ fol IGl( t ) l d t ' L .

Since ~01 25 [Gl(t)] dt = ~--~,

we get

fo 1 f ( t ) dt - D(O, 1) _< 5~6" L and

If f ' is L-Lipschitzian on [0, 1], then

IRI ~ 2-~8" L.

/o I /01 f ( t ) dt - D(O, 1) _< ~-~ [F2(t)] d t . L.

Since

we get

fo t 1

W2(t)[ dt = ~--~,

~o 1 1 1 f ( t ) dt - D(O, 1) < ~-~- L and IRI - 3--~" L.

THEOREM 5. Let f : [0, 1] --* R be such that f(n-1) is a continuous function of bounded variation on [0, 1] for some n > 2.

I f n = 2r -- l, r >_ 2, then

[fo 1 f ( t ) dt - D(O, 1) - T , - l ( f ) <_ 1

max IG2r_l(t)]. V 1 (f(2r-2>) . 16(2r - 1)! te[o,1]

(4.6)

I[ n ----- 2r, r >_ 2, then

i f ( t ) dt - D(O, 1) - -< 16(2r)! te[o,1]max IF2r(t)[. Vo 1 f(2r-t)

= IB rl. Vo

(4.7)

Bullen-Simpson's 3/8 Inequality 479

Also, we have

fo i f ( t ) - D(O, 1 ) - T~(f) < dt 1

2-" (: - 3'- 018 .1 v0: (I (4.s)

Here V~(f ('~-i)) denotes the total variation o f f (~-:) on [0, 1].

PROOF. If @ : [0, 1] ~ R is bounded on [0, 1] and the Riemann-Stieltjes integral fo i ~(t) df('~-l)(t) exists, then

fo i df(~-l)( t ) < max l~(t)l" v2 (f(n-l)) . (4.9) ~ ( t ) - te[o,i]

We apply this estimate to if(t) = G2r-: (t) to obtain

[ 1 ~o i df(2r_2)(t ) 1 max IG2r-: (t)l-Vo: (f(2r-2)) 16(2r - 1)[ G2r-1 (t) < 16(2r - 1)! t~[0,i]

which is just the inequality (4.6), because of the identity (2.3). Similarly, we can apply esti- mate (4.9) with @(t) = F2~(t) and use the identity (2.3) and Corollary 1 to obtain (4.7). Finally, (4.8) follows from (4.9) with O(t) = G2r(t), the identity (2.2) and Corollary 1. I

COROLLARY 4. Let f : [0, 1] --* R be a given function. If f " is a continuous function of a bounded variation on [0, 1], then

fo 1 dt 1) < 14 f(t) - D ( O , 6---~voi(f"), I R I < 1 V2 (f") - - 3072 "

If f '" is a continuous function of bounded variation on [0, I], then

fo 1) 51 v l (ira) 1 (f,,,) f(t) dt - D(0 , < , IRI < Vo: • - 5 296 - 27648

PROOF. From explicit expressions (3.3), we get 1

max ]G3(t)[ = - t~[0,1] 64'

so that the first pair of inequalities follow from (4.6) with r -- 2. The second pair of inequalities

follow from (4.7) with r = 2. II

REMARK 6. If f is a continuous function of bounded variation on [0, 1], then, as above,

/0 I: f(t) at - D(0, 1) _< -~ t~moa~,i] IGi(t)l • Vo'(I ).

Since 5 m a x IGl( t ) ] = ~,

t~[o,i]

f o i f ( t ) d t _ D(0, 1) I < 5 . Vol(f) a n d ]R[ <__ 5 . V~(f).

we get

If f ' is a continuous function of bounded variation on [0, 1], then

i f(t) dt - D(0, < - - max [F2(t)[ • Vo ~ ( f ' ) . - 3 2 t¢[o,1]

Since

we get

1 max IF2(t)l-- ~,

t~[o,il

fo I I 1 f(t) dt - D(O, 1) <_ 2-ff~. Vo i (f ' ) and 1

IRI <_ ~ . v 1 ( f ' ) .

480 M. MATI6 et al.

THEOREM 6. Assume (p, q) is a pair of conjugate exponents, that is, 1 < p, q < 0% 1/p+ 1/q = 1 or p = 0% q = 1. Let f : [0, 1] ~ R be an R-integrable function such that f(n) e Lp[0, 1] for some n > 1.

If n = 2 r - l, r >_ l, then

/ o l f ( t ) dt D ( 0 , 1 ) - T r - l ( f ) < K ( 2 r - 1,p) p. (4.10) f(2r-1)

I[ n = 2r, r >_ 1, then

fo 1 f( t) dt - D(0, 1) - T~-l( f ) <_ K*(2r, p) f(2r) v" (4.11)

ALso, we have

Here

and

o f( t) d t - D(O, 1) - T~(f) <_ K(2r, p) f(2r) p"

K(n,p) = ~ Iv.(t)I ~ dt

(4.12)

,[/01 ]l/q IF.(t)l q dt

PROOF. Applying the H51der inequality, we have

]16(21r _ l )," /ol G2r-l (t) f(2r-1)(t) at

1 1/q

--< 16(2r1 fro 'G2r-l (t)lq dt] " f(2r--1) p= K ( 2 r - l'P) f(2r-1) p - l ) !

The above estimate is just (4.10), by the identity (2.6). The inequalities (4.11) and (4.12) are obtained in the same manner from (2.5) and (2.4), respectively. |

REMARK 7. For p ~ oc, we have

1/o /o 16n! Ic,~(t)l at and K*(n, ee) = 1 16n---~ IN,~(t)] at.

The results established in Theorem 6 for p = ee coincide with the results of Theorem 4 with L = ILf('~)ll~. Moreover, by Remark 5 and Corollary 3, we have

f ( t ) d t - D ( O , 1 ) - T ~ _ l ( f ) _< K ( 2 r - 1 , ~ ) , r = 1,2,

where 25 1

K(1, c~) = 576' K(3, c~) = 13824"

Also, we have

f ( t ) d t - D ( O , 1 ) - T ~ _ l ( f ) <_K*(2r,~) , r = 1,2,

Bullen-Simpson's 3/8 Inequality 481

where "-; 1

K*(2, co) = 768'

REMARK 8. Let us define for p = 1,

1 - - m a x I G f f t ) l K(n, 1) = 16n] tE[o,1]

1 K*(4, c~) = 103680"

1 and K*(n, 1 ) = i6n! max [Fn(t)[. re[o,1]

Then, using Corollary 4, Remark 6, and Theorem 5, we can extend the results established in Theorem 6 to the pair p = 1, q = o0. This means that if we set 1/q = 0, then (4.10) and (4.11) hold for p = 1. Also, by Corollary 4, we have

where

Also, we have

where

1 T~-l(f) f ( t )d t - D(0 ,1) - _< K(2r- 1, 1) f ( 2 ~ - O Jl'

5 1 K(1, 1) = 4-'8' K(3, 1) = 614---'4"

] / l f ( t ) d t - D(O, 1 ) - T r _ l ( f ) l <_ K*(2r, 1)' f (2r ) I1 '

r = 1, 2,

r -- 1, 2,

1 1 K*(2, 1) = 288' K*(4, 1) = 55296"

REMARK 9. Note that K*(1,p) = K(1,p) , for 1 < p < c~, since Gl(t) = Fl(t) . Also, for 1 < p < co, we can easily calculate K(1,p) . We get

1-~ [3q+1 +4q+l + 5q+l]l/q K ( 1 , p ) = t j ' 1 < ; < co.

So, from (4.10) with r = 1, we get the following inequality:

[ 3q+1 -}- 4q+1 + 5q+l 1 1/q

In the limit case when p --+ 1, that is, when q --* co, we have

~__~ [3q+l-{-4q+1+ 5q+l11/q 5 lim L 4(q ~ i ) 3 ~ 'T J = - - = K(1,1) . q-*o~ 48

At the end of this section, we prove an interesting Grfiss type inequality related to the Euler Bullen-Simpson's 3/8 identity (2.4). To do this, we use the following variant of the key technical result from the recent paper [12].

LEMMA 6. Let F, G[0, 1] ~ R betwo integrable functions. If, for some constants m, M C R,

m<_F(t)<_M, 0 < t < l ,

and

then

/ 1 c ( t ) dt = o,

1F(t)G(t) <_ - - - 7 - IG(t)J dt. (4.13)

482 M. MATIC et al.

THEOREM 7. some n _> 1. Assume that

m~ <_ f( 'q (t) _< Mn, 0 < t < 1,

for some constants rnn and M,~. Then

I/01 f ( t) dt - D(O, 1) - _< ~ C ~ - ,

where k = [n/2] and

c ~ = l ab ( t ) [ dr, n >_ 1.

Suppose that f : [0, 1] --* R is such that f (~) exists and is integrable on [0, 1], for

Moreover, i fn = 2r - 1, r _> 2, then

fo dt D(O, 1) - Tr- l ( f ) f(t)

2-'" (1 - 2 -2") (1 - 3 2-2") IB2. L (M2r-1 - m2.-1). 2 [(2r)[]

<

(4.14)

(4.15)

PROOF. We can rewrite the identity (2.4) in the form

/01 1 f( t) at - D(0, 1) - Tk(f) = 16(n!) F(t)a(t) at, (4.16)

where F(t) = f(n)(t), G(t) = Gn (t), O < t < l.

In [13, Lemma 20) ], it was proved that for all n _> 1 and for every 7 E R,

£ 1 B~*(~ - t) dt 0,

so that we have

f o l G ( t ) d t = fo 1 I 2 B ~ ( I - s ) - t - 3 B ~ ( 1 - s ) - t - 3 B ~ ( 1 - s )

Thus, we can apply (4.13) to the integral in the right-hand side of (4.16) and (4.14) follows immediately. The inequality (4.15) follows from (4.14) and Corollary 2. |

REMARK i0. For n = 1 and n = 2, we have already evaluated

fo i 25 /o i 1 C 1 = IGl(t)l at = ~--g, C2 = [G2(t)l dt = ~-~,

so that we have

and

f01 1) 25 f ( t ) dt - D(0, _ 1-i-~ (M1 - ml)

f o l f ( t ) d t - D ( 0 , 1 ) <_ 1 5 - ~ ( M 2 - m 2 ) .

For n = 3, we apply (4.15) with r -- 2 to get the

f0 i dt D(0, 1) / ( t )

inequality

1 _< 27648(M3 - m 3 ) .

Bullen-Simpson's 3/8 Inequality 483

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Euler bitrapezoid formulae, Nonlinear Studies 8 (3), 343-372, (2001). 4. M. Matid~ J. Pe~arid and A. Vukelid~ On generalization of Bullen-Simpson's inequality, (submitted). 5. P.J. Davis and P. Rabinowitz, Methods of Numerical Integration~ New York, (1975). 6. J. Pe~arid, F. Proschan and Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications,

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(2003). 9. V.I. Krylov, Approximate Calculation of Integrals, Macmillan, New York, (1962).

10. M. Abramowitz and I.A. Stegun, Editors, Handbook o/Mathematical Functions with Formulae, Graphs and Mathematical Tables, Applied Math. Series, Volume 55, Fourth printing, National Bureau of Standards, Washington, (1965).

11. I.S. Berezin and N.P. Zhidkov, Computing Methods, Volume I, Pergamon Press, Oxford, (1965). 12. M. Matid, Improvement of some inequalities of Euler-Griiss type, Computers Math. Applie. 46 (8/9), 1325-

1336, (2003). 13. Lj. Dedid, M. Matid and J. Pe~arid, Some inequalities of Euler-Griiss type, Computers Math. Applic. 41

(7/8), 843-856, (2001).