Неравенства для некоторых новых квадратурных формул с весом

In 1938, Ostrowski proved the following interesting integral inequality which has received considerable attention from many researchers [1, 5, 6, 13-16].

Let f : [a, b] ^ R be eoiitiiinous on [a, b] and differentiable in (a, b) and its derivative f : (a, b) ^ R is bounded in (a, b), that is kf 0k^ := sup_xG( a,b ) |f (x)| <  to. Then for any x G [a, b]. we have the inequality

b j f (t) dt — f (x)(b — a) a

/ (b — a)²

V 4

+

x

-

a + b \ ²

2 )

kf0k∞

(1.1)

In [9]. the following results was obtained: If f : [a,b] ^ R is sudi that f ⁽ⁿ 1) is an absolutely continuous function and Yn 6 f ⁽ⁿ⁾(x) 6 Гп for all x G [a, b] for some constants Yn arid Г_п. then

b

^--a [ ^{f (a) + 4f} (^-a) ^{+ f (b)} ] ^— j ^{f (t)} dt

6 С п (Г п — Y n )(b — a)ⁿ⁺¹,

(1.2)

a where the constants Ci = -2. C2 = 162 and C3 = ^ are sharp in the sense that they cannot be replaced by smaller ones.

Very recently, V. N. Huy et al. [5, 6] have strengthened (1.1) and (1.2) by enlarging the number of knots. More precisely, they proved that b-a n

6 A a,b,m^{(S — s)}

(1.3)

f (x) dx--/ f (a + x i (b — a))

n a                    i=1

b j f (x) dx —

a

b-a n

n f (a + xi(b - a))

i =1

6 B a,b,m kf m ⁺¹k p ,

(1.4)

where s = inf _xG[_aib] f ^(m)(x) aiid S = sup x_{G [} a,b] f ^(m)(x). and

52 x k = 7++1 ^(V i = 1’ ^2,...^,m).                         (^L5)

Note that, (1.5) have the solutions only for n E [1, 9], n E N (see [17-19]).

On time scales,the Ostrowski type inequalities have been generalized in various ways. For example, [2, 3, 10]. It proved in [2] the following result on time scales: Let a,b,x,t E T, a < b and f : [a, b] ^ R be differentiable. M = sup_a<_x|f ^A(x)|. Then

b j f°(t)A(t) — f (x)(b — a) 6 M(h(x, a) + h(x, b)), a where hk(•, •) is defined in section 2.

In this paper, making use of the above theorem and some simple estimations, we obtain propose a new way of treating a class of quadrature formulas with weight involving n points and the L_p norin of m-th derivative on time scales where m,n E N arid 1 6 p 6 to.

2.    Preliminaries on time scales

A time scale is a nonempty closed subset of R and is denoted by T. We define the forward and backward jump operators a, p : T ^ T by

a(t) = inf{s E T : s > t},   p(t) = sup{s E T : s < t} (Vt E T), with inf 0 = supT arid sup0 = inf T. А г>oint t E T is called right-dense, right-scattered, left-dense and left-scattered if a(t) = t. a(t) > t. p(t) = t and p(t) < t. respectively. We now introduce the set Tk which is derived from the time scales T. as follows. If T has a left-scattered maximum m then Tk = T — {m}, otherwise Tk = T. The delta graininess function ^ : T ^ [0, to) is defined by

^(t) : = a(t) — t (V t E T ).

If f : T ^ R is a function then we define the function f° : T ^ R by f°(t) = f (a(t)) (Vt E T).

We say that a function f : T ^ R is delta dijji'rentiable at t E T^k if there exists a number f ^A (t) such tin it for all e >  0 there is a neighlmrliood U of t (i. e.. U = (t — 5,t + 5) П T for some 5 >  0) such that

If(a(t)) — f (s) — f^A(i)(a(i) — s)| 6 e|a(t) — s)| (Vs E U ).

We call f ^A ( t ) the delta decwatwe- of f at t.

For delta differentiable function f and g. the next formula holds:

(fg)^A(t) = f ^Ag ° (t) + f (t)g^A(t) = f ^Ag(t) + f ° (t)g^A (t).

A function f : T ^ R is said to be rd- continuous if it is continuous at right-dense points, and its left-side limits exist at left-dense points.

A function F : T ^ R is cal led a A-antideriinitive of f : T ^ R pros’ided F ^Л( t ) = f ( t ) holds for all t E T^k. The 11 the A-iiitegral of f is defiired by Rab f ( t )A t = F(b) — F ( a ) .

It is known that every rd-coiitinuous function f has an antiderivative.

The functions hk : T 2 ^ R are defined recursively as follows:

t ho (t, s) = 1,

hk+i(t, s) = j hk ( y,s )A y

( V s,t E T ).

s

Proposition 2.1. If a,b E T, then the assertions hold:

• 1.    If a 6 x 6 b then 0 6 hk (x,a) 6 hk (b, a);

• 2.    For a 6 b we 1 lave 0 6 hk+1(b, a) 6 (b — a)hk(b, a).

Now, we introduce a. useful result, which is well-known in the literature as Taylor’s formula, with the integral remainder.

Lemma 2.2 [1]. Assume f E Crd(T) aiid xo E T. Then for all x E (a,b) we have f (x) = Tr-1(f, xo, x) + Rr-1(f, xo, x)

where T_r-1 (f, xo, • ) is Taylor's polyilomial of degree r — 1. that is.

r -1

Tr-i(f,xo,x) = ^2 hk(x,xo)f^Л (xo)

k =0

and the remainder can be given by

x

Rr-i(f,xo,x) = j hr-i(x,o(t))f^ЛГ (t)At.

x 0

We have the Montgomery identity which is stated in the following lemma.

Lemma 2.3 [8]. Let a,b,s,t E T, a < b and f : T ^ R be differentiable. Then bb f (t) = Г— [ f"(s)As + -^ /p(t,s)f Л(s)As, b-a             b-a а                       а where s-a s-b

for a 6 s < t, for t 6 s 6 b.

3.    Main results

Let 1 6 m, n and 1 6 p 6 to, 0 6 ai 6 1 satis lies ^i-^i ai = 1. For each i = 1,..., n, let a 6 xi 6 b and we consider the following condition

Hi(xi,x2,... ,xn) = hi+i(b,a) ( V i = 1,2,...,m — 1),                (3.1)

where H i( x i , x 2 , ... ,X n ) = ( b — a ) Pn=_i a_k h i (x_k, a). and

b j hm(b,a(t)^t — a

x n ⁱ

(b — a)E i =1 a

a i h m—i (x i ,a(t)) At = 0.

(3.2)

We point out the fact that in the continuous case T = R and ai = ... = an = 1, conditions (3.1) and (3.2) become n ■         1

X Ук = i+1 (v i = 1,2,•••,m), k=i where xi = a + yi(b — a). Before stating our main result, let us introduce the following notations.

b

I ^(f ) = j ^{f (x)A} x, a

n

Q (f,n,m,x i ,...,X n ) = (b — a) X a i f (x i ). i =i

(3.3)

Note that, for the ease a i = ... = a n = n then

1 ⁿ

Q (f, n, m,xi,..., xn) = (b — a) X f (a + yi(b — a)) n       i=i are also known in [5, 6]. Now, we slightly improve [5, 6] with weights ak on time scales:

Theorem 3.1. Let a,b E T aiid f E Cm(T). Then, under conditions (3.1) and (3.2). we have

| I(f) — Q(f,n,m,x i ,... ,X n ) | 6 2(b — a) ² (T

-

s)h m-i (b, a),

where s = inf _xG[aib] f ^A m(x). T = (f ^A m ¹ (b) — f ^A m ¹ (a))/(b — a).

C Let ns first define

x

F ( x ) = j f ( x ) A x.

a

Then I(f) = F(b) — F(a). Applying Lemma, 2.2 to the function F(x) with x = b and xo = a. we get mb

F(b) = F(a) + X hk(b,a)FAk(a) + / hm(b,a(t))FAm+1 (t)At k=i which yields that

I (f )= X hk+i(b,a)f Ak (a)+ [ hm{b,r(t))f Am (t)At. k=0a

(3.4)

For each 1 6 i 6 n. applying Lernma 2.2 again to the function f (x) with x = xi aiid xo = a. we get m-i

f (x i ) = X h k (x i ,a)f ^A k(a) + / h m-i (x i ,a(t))f ^A m(t)At.

k=0

By applying to i = 1,..., n and then summing up, we deduce that

n              n m -1                         n ^{x i}

Xa i f(X i ) = £ £ a i h k (x i ,a)f^A(a) + £ / a i h m i=1             i=1 k=0                        i=1 _a

1 (x i ,^(t))f ^Am(t) At

m-1 n                         n ^xi

= XX a i h k (X i ,a)f ^Ak(a) + X / a i h m k =0 i =1                        i =1 _a

1 (x i ,^(t))f ^Am(t) At

m-1

= X b k=0

—Hk (X 1 ,X 2 , . .

-a

_{k           n} x i

. ,X n )f^A (a) + X / a i h m i=1 _a

1 (x i ,a(t))f ^Am(t) At.

Thus, m-1

=      Hk(X1 , X2, . . . , Xn k=0

Q(f,n,m,x i , ...,x n )

_{k                     n} x i

)f Ak(a) + (b - a) X / aihm i=1 a

i (x i ,a(t))f ^Am(t) At.

Then it follows from condition (3.1) that

Q(f,n,m,x i ,... ,X n )

m -1

= X kk+i(b,a)f Ak (a) + k=0

n ^{x i}

(b - a) X / a i h m- i i =1 _a

(x i ,a(t))f ^Am(t) At.

(3.5)

By (3.4), (3.5), we obtain that

I(f) - Q(f,n,m,x i ,... ,X n )

b x ni

(b - a)      a i h m -1

i =1 _a

j hm (b,a(t))f ^Am(t) At - a

(x i ,a(t))f ^Am(t) At

Then, by using condition (3.2), we have

I(f) - Q(f,n,m,x i ,... ,X n )

x n ⁱ

(b - a)E i=1 a

a i h m- i (x i , a(t))[f ^Am(t) - s] At

(3.6)

We estimate the first term of (3.6) as follows

b

| j h m (b,a(t))[f ^Am (t) - s] At | a

6 h _m

b

(b, a)   [fAm (t) - s] At

a

= h m (b,a) ( f ^Am 1 (b) - f ^Am 1 (a) - s(b - a) )

= (b - a)h m (b, a)(T - s) 6 (b - a)²h m-i (b, a)(T - s).

(3.7)

For the second one, we first have xi

/ ai h m-1

a

(x i , a(t)) f ^Am (t) - s] At 6 h

,m-1 (x i , a) j | f ^Am (t) - s ] At a

= h m- 1 (x i , a) ( f ^Am ¹ (b) - f ^Am ¹ (a) - s(b - a) ) 6 (b - a)h m- i (x i ,a)(T - s).

Hence, summing up the above inequalities with i = 1, 2,..., n, using the Proposition 2.6, it implies that

n

x i

n

(b - ")E /

i=1

a i h m— i (x i , a(t)) f ^Am(t) - s] At

a

n

(3.8)

• 6 (b - a)² (T - s) X a i h m— i (x i , a) = (b - a)(T - s) h m (b, a) i =i

6 (b - a)²(T - s) h m- i (b,a).

Combining relations (3.6), (3.7) and (3.8), we conclude that

I(f ) - Q(f,n,m,x i , ... ,x n ) | 6 2(b - a)²(T - s) h m- i (b, a)

and the proof of Theorem 3.1 is now completed. B

With the similar arguments as those used in the proof of Theorem 3.1, we also obtain the following theorem.

Theorem 3.2. Let a, b E T aiid f E Cm(T). Then, under conditions (3.1) and (3.2). we

• 1 1 f ) - Q(f, n m, x i ,..., x n )| 6 2(6 - a)² (S - T )h_m -i (6. a),

where S = sup x _G[ a,b] f ^Am(x). T = (f ^Am^-1 (b) - f ^Am^-1 (a))/(b - a).

Since s = inf x_G[a,b] f ^Am (x) 6 T = b-a J ab f ^Am (x)Ax 6 sup xG[a,b] f ^Am (x) = S, we have the following corollary.

Corollary 3.3. Let a,b E T aiid f E C m (T). Then, under conditions (3.1) and (3.2). we have

|I(f) - Q(f, n, m, x i ,..., x n )| 6 2(b - a)²(S - s)h_m -i (b, a),

Where S = SUP xG[a,b] f ^Am(x), s = inf xG[a,b] f ^Am(x)-

Now, we will give a new quadrature formulas with weight involving n points and Lp norm m-th derivative on time scales.

Theorem 3.4. Let 1 6 p 6 to , a,b E T and let f E C m (T). Then, under conditions (3.1).

• 11 (f) - Q(/,n,m,x i ,...,x „ )| 6 2h_m -i (6.^)(6 - _a) l ^q+1 > ^/q kf ^Am k p , where p + q = 1.

C AVe lurve known that

I(f) - Q(f,n,m,x i , ... ,x n )

b                                              n xi f hm(b,a(t))f Am (t)At - (b - a) X f aihm a                                      i=i a

i (x i ,a(t))f ^Am(t) At .

(3.9)

The first term of (3.9) can be estimated by using the Holder inequality as follows

b j hm(b,c(t))f Am (x) Ax a

b

b

6 (/

a

b

b

a

1/p

1/p

6 h m (b, a) (b - a)^1/q kf ^A" k p

(3.10)

a

a

6 h m- i (b,a)(b - a)^(q+1)/q kf ^Am k p .

Similarly, we deduce since xi G (0,1) and the Holder inequality that xi

αihm- a

i (x i ,a(t))f ^Am(t) At 6 a i

x i

( j [h m-1 (x i , ^(t))]^q At

a

x i

a

b

b

1/p

6 h m- 1

1 /p

= h m-1 (x i , a) (b — a)^1/q kf^Ak p .

a

a

Now, applying the above inequalities with i = 1, 2,..., n we get

n

x i

n

(b — ")E /

^a i ^h m- 1 ^(x i i ^ ^(t))f

(t) At

i=1

a

n

(3.11)

= (b - a)^(q+1)/q kf ^Am k p . X a i h m- 1 (x i , a) i =1

= h m (b, a) (b - a)^1/q kf ^Am k p 6 h m-1 (b, a) (b - a)^(q+1)/q kf ^Am k p .

Relations (3.9), (3.10) and (3.11) imply that

• 1 1 (f) - Q(f,n,m,x 1 , ...,x n ) | 6 2h m-1 (b,a) (b - a)^(q+1)/q kf ^Am k p

and thus Theorem 3.4 is completely proved. B

Next, we define the Chebyshev functional on a time scale by b      bb

T A (f,g) = b-a j f (x)g(x) Ax - (b - a) ₂ j f (x) Ax j g(x)Ax.

Then

b

TA(f,f) = w^ [ f2(x)Ax -b-a

a

(b - a)2

b j f (x)Ax

a

We also define ctaU) = (b - a) Тд(/,/)• Then, it should be noticed that in [15], N. Ujevic obtained the following result for the ease T = R: Let f : [a, b] ^ R Ire an absolutely continuous function, whose derivative f G L2(a,b). Then it holds that b-a

b

(b-У2 ^(f7).

f(a)+4f(bafa) + f(b) - /f(t)At 6

a

In this article, base on the result of N. Ujevic we will give a new quadrature formulas with weight involving n points and m-th derivative on time scales by using Chebyshev functional.

Theorem 3.5. Let a, b E T and let f E C m (T) be sueh that f ^Am E L² (a, b). Then, under conditions (3.1) and (3.2). we have

|I(f) — Q(f,n,m,x i ,... ,x n )| 6 2h_m-i(b,a^(b - a)³ o a U ^Am).

C lie have known that

I(f ) — Q(f,n,m,x i ,... ,xn )

b                                               _n x i

[ h m (b,^(t))f ^Am (t) At - (b - a) X f aihm

1 (x i ,^(t))f ^Am(t) At

_a                                      i =1 _a

Then, by using condition (3.2), we have

I (f) — Q(f,n,m,x i ,... ,x n )

b j hm(b,a(t))[f Am(t) — T] a

x n ⁱ

At - (b - a)     / aihm i=1 a

i(xi,a(t))[f ^Am(t) - T ] At ,

(3.12)

where T = (f Am 1 (b) - f Am 1 (a))/(b - a). The first term of (3.12) can be estimated by using the Holder inequality as follows

b j hm(b, a(t)) [f Am(t) - T] At a

b

b

1 / 2

6 h m (b, a)Vb - a ( j [ f ^Am (t) - T ] At

Combining this with the fact that

J [ f^Am(t) - T ] 2 At = J [ f^Am(t) ] 2 At - 2T j f^Am(t)At +(b - a)T²

= j [ f^Am (t) ] 2 At -      [/f^Am (t) At ] 2 = , д (f^Am)

aa we obtain that

b j hm(b,a(t))[f Am(t) - T]At a

6 h m (b,a)^(b - a)^ A (f ^Am)

(3.13)

Similarly, we deduce since xi G (a, b) and the Holder inequality that xi

α i h m-1

a

x i

6 ai^ I [hm-1 (xi, a(t))]2 At a b      1/2    b m

6 h m -1 ( x i , a )    A t         f ^A ( t ) - T

(x^(t)) [f Am(t) - T] At xi

1/2

a

1 / 2

)    = h m -i(x i ,a)Vb - a\J^A(f ^Am).

aa

Now, applying the above inequalities with i = 1, 2,.. ., n, we get

n

x i

n

(b - a)E /

i=1

a i h m -i(x i , a(t)) f ^Am(t) - T| At

a

= ^(.Ь - a)a^(f ^Am) X a i h m —i(x i ,a) = h m (b^^(b - а)а д (/^Am)      (   )

i =1

6 (b - a)h m —i(b,a)^(b - a)aA(f ^Am).

Relations (3.12), (3.13) and (3.14) imply that

I (f ) - Q(f,n,m,xi, -A ) 6 2(b - a)h m —i (b,a)^ (b - a)ffA(f ^Am)

and thus Theorem 3.5 is completely proved. B

Base on the inequality in (1.1), by using some simple estimations, we obtain some new quadrature formulas involving n knots on timо scales: For 0 6 xi 6 1. a + xi(b - a) G T. we put

1 ⁿ

Q(f,X1,X2,... ,Xn) = - E f(a + Xi(b - a))- n i=1

The next result of this paper can be described as follows.

Theorem 3.6. Let a,b G T a < b f : T ^ R be differentiable. and assume that fA is rd-continuous such that fA G L2(T). Then for 0 6 xi 6 1 with Vi' 1 xi = 2 we have the following estimate bb

^Q(f, xi^,x2, ...^,x n ) ^- b-Ta j ^f ° ^{(x) Ax + f} ^⁾_ afya j 2 h^{(s) As}

⁶ ^/ ^{(b - a)} ^ A^{(f A )} .

aa

C Put tk = a + xk(b - a), then it follows from Lemma 2.3 that f (a + xk(b - a)) -

b

E- / f ^CT(x)Ax = b - a

a

b

7-^ / p(t k ,s)f ^A(s) As b - a

a

b

= 77— /p(tk ,s) f ^A(s) - b - a

^{f (b) - f (a)} " b - a

A s +

b

1 f „   ff (b) - f (a)

7----- P(tk,s) 7-------- b-a            b-a

A s.

a

a

Since b2 - a2

-

a b

btb j p(t, s) As — J(s — a)As + J(s — b)As b       b     tb s + s + ^^(s)^ As— J^^^(s)As— ay As— by

a

I ~ ^(s)As — a(t — a) — b(b — t) — (t

2                                \

a

a a + b

-

t

As

b

1 z Z Л

) ⁽b ^- a) -   2 ^(s) ^A s,

a

we deduce that f (a + xk (b — a)) —

b

-^ / f ' (x) Ax — b - a

a

b

_1_ [ p(t k ,s) f A (s) — fM b - a                      b - a

a

As

f (b) — f (a) |        2 /       1 ) b 1

⁺ (b — a) 2 [^{( b — a)} Vk ^— 2) "J 2 ^ ^{(s) As}

a

Hence, bb f (a + xk(b — a)) — -^ / f CT(x)Ax + f^—f^ / 1 ^(s) As b — a                 (b — a)2 J 2

aa

b

• —         /p(t k , s) [f ^A (s) — f^{( b ) —} f(a) 1 As + [f (b) — f (a)] X^х— —     .

b — a                      b — a                             2

a

By applying to k — 1,... ,n and then summiiig up. since ^n=1 x— — 2. we obtain that bb

^Q(f,x i ^,x2 , ... ^,xn^{) —} b — a У ^f ^ ^{(x)Ax +} f ⁽(^Ь)_— af)⁽ 2 ^{a )} У 2 ^ ^(s)A s aa n^b

• —    ¹ X t A(     ^{f (b) — f (a)} A

• ^—    n(b — a)/L J ^{p(tk ,s) f (s)}       b — a ^A s.

k=1 _a

We first observe that

Г [,Az x   f (b) — f (a) l2 л Г '.a.    2,       1 Г Laz u12 z,Az f (s)--b—a—  As — f (s)J As — b—a   f (s)As  — CTA(f ), which yields bb

Q ^(f , ^xi , ^x2 , ...^,x n ) ^— b — a У ^{fa (x) Ax+ f} ((b)_—af)⁽ ₂ ^{a )} У 2 ^ ^{(s) As} aa b                          1b                                                     1

b1 n2 njb—^i V-afA) E (j [pftk • s)i2 As) k=1 a nb                  1

⁶ _n ( _b - a) V^-A^{(f A )} X (/^{(b—a) 2 As}) = v^b—a^ A f)• k =1 _a

The proof of Theorem 3.6 is now completed. B

Theorem 3.7. Let a, b E T aiid f E Cd (T). He a Iso let y = inf xGT f A(x) aiid T = f (b)-f (a)

b-a

. Then for 0 6 x i 6 1 with 522=1 x i = 2 we have the following estimate

b

b j 2 ^(s)As 6 (b — a)(T — y). a

Q(f,x i ,x 2 ,...,xn) —  --- f^а (x)Ax + -^

b-a            b-a

a

C Put tk = a + Xk(b — a), then it follows from Lemma 2.3 that f (a + Xk (b — a))

-

b-

b

- [ f' (x) Ax = a

a

1 b-a

b

ЬТ¹-- jp(t k , s) [ f ^A(s) — Y ] As + a

1 b-a

b

J p(t k ,s)f ^A(s)As a

b j p(tk ,s)Y As.

a

Combining this with the fact that

b

p(t, s) As = t

a

-

a+b 2

b

) ^(b - ^a) " j 2 ^ ^{(s) As}

a

we get f (a + xk(b — a)) —

b

^ / f ^CT(x)Ax b-a

a

b-

b a J p(tk,s)[f A(s) — y] As + b

a

γ

-a

b

^(b — a ^{) 2} ^^x k — 2^) ⁺ У 2 ^ ^{(s) As} '

a

Hence,

f (a + X k (b — a))

-

1 b-a

b j f"(x) Ax + b

a

γ

-

b aj 2 , >,

a

b

= v—a f^ ■ ^s) l f^{A (s)}

a and then by 52n=1 xk = 2 that

- y As + [f (b) - f (a)] x k

-

Q(f, X 1 ,X 2 , -A)

^^^.^^^^     ^^^^^^^^^^^^^^^^^^^^^™ b-

b a If"(x)Ax + b

a

γ

-

b

- j 2 ^(»> ^A s

a

n(b — a)

n ^b                                            n

X / p⁽t k , ^s) [ f^{A(s) —} Y ] ^{As + [f(b) —} f^(a)] X ^{(b — a)} k —1 a                                         k—1

x k

-

1 ⁿ f

= у—) E /pft k ^Т ^A<^{s) — y} I^v k=1 _a

Hence,

Q(f, X 1 ,X 2 , . . . ,X n ) -

bb

A- [ f '(x) Ax + ^ [^ ^(s) As b — a              b — a 2

aa nb

• 6 nbTa X /(b - a) [ f^A« — Y ] As = (b — a)(T — 7). k=1 _a

The proof of Theorem 3.7 is now completed. B

With the similar arguments as thosed used in the proof of Theorem 3.7 we also conclude the following result.

Theorem 3.8. Lei a,b E T aiid f E C_r 1 _d(T). He also lei Г = sup_xeT f ^A(x) and T = Mil. _{Theiг for} 0 6 x 6 1 _wilh Pn x = n _и._{с htwe} b-a                 " ^c           ^г—у      2

bb

Q(f,X 1 ,X 2 , . . . ,X n )

— Г— [ f " (x)Ax ' /I ^(s) As

6 (b — a)(' — T ).

b—a              b—a 2

aa

Acknowledgments. The authors thank the referee for useful remarks and comments that led to improvement of this paper.