DOUBLE INEQUALITIES OF NEWTON’S QUADRATURE RULE

. In this paper double inequalities of Newton’s quadrature rule are given. MSC 2000. 65D32.

The function ϕ is given by the relation (h denotes b−a 3 ):

MAIN RESULT
Under the assumptions of the quadrature formula (1.1) we have the next theorem: (2.1) f (4) (x) then inequalities are sharp.
To show that inequality (2.1) is sharp we consider the function f given by the relation f (x) = (x − a) 4 .It is easy to see that the equalities f (4) (x) = 24 and γ 4 = Γ 4 = 24, S 3 = 24 are obtained.
Calculating the three members of the inequality (2.1) under the given circumstances, we notice that these have the common value given by the expression 1 270 (b − a) 5 .Hence, we deduce that the inequality (2.1) is sharp.Another relation is given by the next theorem: Theorem 2.2.Under the assumptions of Theorem 2.1 we have: (2.13) f (4) (x) then the inequalities (2.13) are sharp.
To show that the inequalities are sharp we choose f (x) = (x − a) 4 and we follow the steps of the proof for Theorem 2.1.
The next theorem offers us inequalities which do not depend upon S 3 .
Proof.The inequalities (2.16) are easily deduced by (2.9) and (2.14) , respectively (2.12) and (2.15).To show that the inequalities are sharp we follow the steps of the proof for Theorem 2.1.
In use the next theorem is important: Theorem 2.4.Under the assumptions of Theorem 2.1 we have: (2.17) where x i = a + ih, h = b−a n , i = 0, 1, ..., n and x i , x i divide every interval [x i , x i+1 ] in three equal parts.
Proof.We divide each interval [x i , x i+1 ] in the equal parts by points x i , x i , then we use Theorem 2.1 on the interval [x i , x i+1 ] : By adding the formula we have got so far for i = 0, 1, ..., n − 1 and by noticing that h we get the relation we wanted.
From Theorem 2.2, following the steps done, we get: Theorem 2.5.Under the assumptions of Theorem 2.1 we have:  (2.20) Proof.Proving this theorem is similarly to proving Theorem 2.4.Then we use Theorem 2.3.
351840n 4 (b − a)5.Proof.Proving this theorem is similarly to proving Theorem 2.4.Then we use Theorem 2.2.Also Theorem 2.3 leads us to:Theorem 2.6.Under the assumptions of Theorem 2.1 we have: