A NEW PROOF OF SOME INEQUALITY CONNECTED WITH QUADRATURES

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Inequality Connected With Quadratures Szymon W ˛asowicz vol. 9, iss. 1, art. 7, 2008

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A NEW PROOF OF SOME INEQUALITY CONNECTED WITH QUADRATURES

SZYMON W ˛ASOWICZ

Department of Mathematics and Computer Science University of Bielsko-Biała

Willowa 2, 43-309 Bielsko-Biała, Poland EMail:[email protected]

Received: 04 November, 2007

Accepted: 02 January, 2008

Communicated by: I. Gavrea

2000 AMS Sub. Class.: Primary: 26A51, 26D15; Secondary: 41A55, 65D07, 65D32.

Key words: Convex functions of higher order, Inequalities, Quadrature rules, Splines, Spline approximation.

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1. Introduction

Recall that a real functionf defined on a real intervalI isn–convex (n ∈ N), if its divided differences involvingn+ 2points are nonnegative, i.e.

[x₁, . . . , x_n+2;f]≥0

for anyn+ 2distinct pointsx1, . . . , xn+2 ∈I (cf. [3]). Then 1–convexity reduces to an ordinary convexity.

In the recent paper [5] we established the order structure of a set of some six quadrature operators in the class of 3–convex functions mapping[−1,1]intoR. In this setting we have proved among others the inequality

(1.1) 1 3

f

−

√2 2

+f(0) +f √

2 2

≤ 1

12[f(−1) +f(1)] + 5 12

f

−

√5 5

+f

√ 5 5

between a three–point Chebyshev quadrature operator (on the left–hand side) and a four–point Lobatto quadrature operator (on the right–hand side). The proof was rather complicated. In its main part we needed to determine the inverse of a4×4 matrix with entries all of the form a+b√

2 +c√

5 +d√

10. This was done us- ing computer software. The only thing done precisely was the computation of the

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Theorem 1.1. Everyn–convex function mapping[a, b]intoRcan be uniformly ap- proximated on[a, b]by spline functions of the form

x7→p(x) +

m

X

i=1

ai(x−ci)ⁿ₊,

wherepis a polynomial of degree at mostn, a_i > 0, c_i ∈[a, b](i = 1, . . . , m) and y₊ := max{y,0},y∈R.

Observe that all spline functions of the above form are also n–convex. Indeed, ifx₁, . . . , x_n+2 ∈[a, b]are distinct, then by the properties of divided differences we have[x₁, . . . , x_n+2;p] = 0. Ifc∈R, then

B(c) := [x1, . . . , xn+2; (· −c)ⁿ₊]

is a value at the pointcof the B–spline on the knotsx₁, . . . , x_n+2. HenceB(c)≥ 0 (cf. [2]) and the function(· −c)ⁿ₊isn–convex. Obviously, the conical combination ofn–convex functions isn–convex by linearity of the divided differences.

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2. A New Proof of the Inequality (1.1)

Theorem 2.1. Iff : [−1,1]→Ris 3–convex then (1.1) holds.

Proof. By Theorem1.1it is enough to prove (1.1) only for polynomials of degree at most 3 and for any function of the formx7→(x−c)³₊,c∈R.

To show that (1.1) holds for polynomials of degree at most 3 (even with the equal- ity), we check it for the monomials1,x,x²,x³and use linearity.

Now letc∈R. Rearranging (1.1) we have to prove that ϕ(c) := (−1−c)³₊+ (1−c)³₊+ 5

−

√5 5 −c

3 +

+ √

5 5 −c

3 +

−4

−

√2 2 −c

3 +

+ (−c)³₊+ √

2 2 −c

3 +

≥0.

Obviouslyϕ(c) = 0forc6∈[−1,1]. We compute

ϕ(c) =







(c+ 1)³ for −1≤c <−

√2 2 ,

−3c³+ 3(1−2√

2)c²−3c+ 1−√

2 for −

√ 2

2 ≤c <−

√ 5 5 , 2c³+ 3(1 +√

5−2√

2)c²+ 1 +

√5 5 −√

2 for −

√5

5 ≤c <0,

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Then we can see thatϕ is an even function and it is enough to check thatϕ(c) ≥ 0 only for0≤c≤1. We have

ϕ⁰(c) =











−6c²+ 6(1 +√

5−2√

2)c for0≤c <

√5 5 , 9c²+ 6(1−2√

2)c+ 3 for

√ 5

5 ≤c <

√ 2 2 ,

−3(1−c)² for

√2

2 ≤c≤1.

Then

ϕ⁰(c) = 0 forc∈ {0,1 +√

5−2√ 2,1}, ϕ⁰(c)>0 forc∈(0,1 +√

5−2√ 2), ϕ⁰(c)<0 forc∈(1 +√

5−2√ 2,1).

Henceϕis increasing on[0,1 +√

5−2√

2)and decreasing on[1 +√

5−2√ 2,1].

Finally we compute

ϕ(0) = 1 +

√5 5 −√

2>0, ϕ(1 +√

5−2√

2) = (1 +√

5−2√

2)³+ 1 +

√5 5 −√

2>0, ϕ(1) = 0,

which proves thatϕ ≥0on[0,1]and finishes the proof.

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References

[1] R. BOJANICANDJ. ROULIER, Approximation of convex functions by convex splines and convexity preserving continuous linear operators, Rev. Anal. Numér.

Théor. Approx., 3(2) (1974), 143–150.

[2] C. DE BOOR, A Practical Guide to Splines, Springer–Verlag New York, Inc., 2001.

[3] T. POPOVICIU, Sur quelques propriétés des fonctions d’une ou de deux vari- ables réelles, Mathematica (Cluj), 8 (1934), 1–85.

[4] GH. TOADER, Some generalizations of Jessen’s inequality, Rev. Anal. Numér.

Théor. Approx., 16(2) (1987), 191–194.

[5] S. W ˛ASOWICZ, Inequalities between the quadrature operators and error bounds of quadrature rules, J. Ineq. Pure & Appl. Math., 8(2) (2007), Art. 42. [ONLINE:

http://jipam.vu.edu.au/article.php?sid=848].