DOUBLE INEQUALITIES FOR QUADRATURE FORMULA OF GAUSS TYPE

. Double inequalities for the remainder term of the Gauss quadrature formula are given. These inequalities are sharp. It also will consider particular cases for n = 1, 2.


INTRODUCTION
In this work we will consider Gauss's quadrature rule(see [2]) where the remainder term for f : [a, b] → R, f ∈ C 2n [a, b], has the representation The nodes x i , i = 1, n from quadrature formula (1.1) are given by the relation (1.3) x i = a+b 2 + b−a 2 ξ i where ξ i is replaced by the roots ξ 1 , ξ 2 , ..., ξ n of Legendre polynomial (1.4) X n (ξ) = 1 2 n n! d n (1−ξ 2 ) n dξ n . and the coefficients C i , i = 1, n, from Gauss's formula shall be determined by putting the conditions that the quadrature formula must be accurate for any polynomial of degree 2n − 1.
The function ϕ(x), from remaining term expression, being symmetric to the line x = a+b 2 , is sufficient to provide the equation y = ϕ(x) in the intervals [a, .. to middle of interval (a,b). So the function ϕ(x) coincide on all the the intervals [a, .., with the relations: In the next paragraph we will establish double integral inequalities for the remainder term of the Gauss quadrature formula. We will also establish conditions under which the results are sharp. At the end of the paper we will analyze the particular cases of Gauss formula with one respectively two nodes.

MAIN RESULTS
In this section we prove the following theorems: the inequalities (5) are sharp.
Proof. Using (1.1) and (1.2), we obtain But it is well-known from [3] pp 283 that

From (2.3) and (2.4) it follows
On the other hand, we have By calculating the right member of the inequality(2.7), we get From (2.7), (2.8) and noting From (2.5), by using previous relation, result In the same way we have Using (2.6), (2.9), (2.12) and (2.13) we obtain the inequality The inequalities (2.1) follow from the inequalities (2.11) and (2.14).
To prove the second part of the theorem we consider the function f (x) = (x − a) 2n . I have f (2n) (x) = (2n)!, γ = Γ = (2n)! and S 2n−1 = (2n)!. It is easy to show that all the three members of the double inequality (2.1) are equal. This completes the proof.

From the inequalities (2.1) and (2.19) we have
2n+1 . Multiplying (2.20) with 1 2 results the double inequality (2.18). Considering the function f (x) = (x − a) 2n we will show that the inequality is sharp. This completes the proof.

PARTICULAR CASES
Using the results of the before paragraph, we will present the following double inequalities for the rest of Gauss quadrature formulas in the cases n=1, n=2.
The Gauss's formula with a single node (n = 1), also called the mid-point, has the form The function ϕ(x) is given by the relation . Applying Theorem 2.1 for n=1, we get a result established by Ujevic [3]. a, b), then we have the inequality Similar with Theorem 2.2 and 2.3, the Gauss's formula with a single node are given in the next theorems : Similar results, we obtain in the case of Gauss formula with two nodes. The results was established by the author in the paper [1].
Gauss's quadrature formula with two nodes has the following form The error R[f ] from the formula (3.7) is given by where the function ϕ has the form In this case we obtain the following theorems: