An extremal problem for real algebraic polynomials
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AN EXTREMAL PROBLEM FOR REAL ALGEBRAIC POLYNOMIALS ˇ Milovanovic ˇevic ´1 and Igor Z. ´2 Milan A. Kovac 1
University of Niˇs, Faculty of Electronic Engineering, Department of Mathematics Aleksandra Medvedeva 14, P.O. Box 73, 18000 Niˇs, Serbia E-mail: [email protected]
2
University of Niˇs, Faculty of Electronic Engineering, Department of Mathematics Aleksandra Medvedeva 14, P.O. Box 73, 18000 Niˇs, Serbia E-mail: [email protected] (Received October 10, 2011; Accepted June 5, 2012) [Communicated by Gy¨ orgy Petruska]
Abstract Let Gn be the set of all real algebraic polynomials of degree at most n, positive on the interval (−1, 1) and without zeros inside the unit circle (|z| < 1). In this paper an inequality for the polynomials from the set Gn is obtained. In one special case this inequality is reduced to the inequality given by B. Sendov [5] and in another special case it is reduced to an inequality between uniform norm and norm in the L2 space for the Jacobi weight.
1. Introduction Let Gn be the set of all real algebraic polynomials Pn (x) of degree at most n which are positive on the interval (−1, 1) and without zeros inside the unit circle (i−1) (i−1) (|z| < 1). The subset of the set Gn for which Pn (−1) = Pn (1) = 0, i = (m) 1, . . . , m, will be denoted by Gn (m ≤ [n/2]). Lemma 1.1. If the polynomial Pn (x) belongs to the set Gn , it can be written in the form Pn (x) =
n X
ak (1 − x)k (1 + x)n−k ,
ak ≥ 0
(k = 0, 1, . . . , n).
(1)
k=0
Mathematics subject classification numbers: 26D05; 41A44. Key words and phrases: real algebraic polynomials, inequalities, norm. 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
168
ˇ ´ and I. Z. ˇ MILOVANOVIC ´ M. A. KOVACEVI C
Proof. It is sufficient to consider the following two special cases: (1) P (x) linear: P (x) =
P (−1) P (1) (1 − x) + (1 + x). 2 2
(2) P (x) quadratic with conjugate-complex pair of zeros α ± iβ on or outside the unit circle, i.e., α2 + β 2 ≥ 1. So, we have P (x) = (x − α − iβ)(x − α + iβ) = x2 − 2αx + α2 + β 2 , and x2 − 2αx + α2 + β 2 = a(1 − x)2 + b(1 − x)(1 + x) + c(1 + x)2 , where a=
P (−1) ≥ 0, 4
b=
α2 + β 2 − 1 ≥ 0, 2
c=
P (1) ≥ 0. 4
Remark 1.1. Note that algebraic polynomials that are positive on the interval (−1, 1) with only real zeros belong to the set Gn . Let w(x; α, β) = (1−x)α (1+x)β , α, β ≥ −1, q = max(α, β) and s = min(α, β).
2. An extremal problem for polynomials which belong to Gn Let the polynomial Pn (x) belong to the set Gn . Then, for some fixed r = 1, 2, . . . , the polynomial Pn (x)r also belongs to the set Gn and according (1) we have Pn (x)r =
rn X
ck (1 − x)k (1 + x)rn−k ,
ck ≥ 0 (k = 0, 1, . . . , rn).
(2)
k=0
If mk = max {(1 − x)k (1 + x)rn−k }, |x|≤1
we have m0 = mrn = 2rn ,
mk =
and hk (α, β) =
Z
1
2rn k k (rn − k)rn−k (rn)rn
(k = 1, 2, . . . , rn − 1)
(3)
w(x; α, β)(1 − x)k (1 + x)rn−k dx
−1
2rn+α+β+1 Γ(k + α + 1)Γ(rn − k + β + 1) , = Γ(rn + α + β + 2)
(4)
AN EXTREMAL PROBLEM FOR REAL ALGEBRAIC POLYNOMIALS
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for k =
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