PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 58 (72), 1995, Slobodan Aljan»cić memorial volume AN ESTIMATE FOR COEFFICIENTS OF

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1 PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série, tome 58 (72), 1995, Slobodan Aljan»cić memorial volume AN ESTIMATE FOR COEFFICIENTS OF POLYNOMIALS IN L 2 NORM. II G. V. Milovanović and L. Z. Ran»cić Dedicated to the memory of Professor S. Aljan»cić Abstract. Let P n be the class of algebraic polynomials P (x) = P n k=0 a kx k of degree at most n and kp k dff =( R R jp (x)j2 dff(x)) 1=2, dff(x) is a nonnegative measure on R. We determine the best constant in the inequality ja k j»c n;k (dff)kp k dff,fork =0; 1;... ;n,when P 2P n and such thatp (ο k )=0,k =1;... ;m. The cases C n;n(dff) andc n;n 1(dff) were studed by Milovanović and Guessab [6]. In particular, we consider the case when the measure dff(x) corresponds to generalized Laguerre orthogonal polynomials on the real line. 1. Introduction Let P n be the class of algebraic polynomials P (x) = n P k=0 a k x k of degree at most n. The first inequality oftheformja k j»c n;k kp k was given by Markov [3]. Namely, ifkp k = kp k 1 = max x2[ 1;1] jp (x)j and T n (x) = n-th Chebyshev polynomial of the first kind, then Markov proved that n P t n;ν x ν denotes the ρ jtn;k j kpk 1 if n k is even; ja k j» jt n 1;k j kpk 1 if n k is odd: (1.1) For k = n (1.1) reduces to the well-known Chebyshev inequality ja n j»2 n 1 kp k 1 : (1.2) AMS Subject Classification (1991): Primary 26 C 05, 26 D 05, 33 C 45, 41 A 44. *Research partly supported by Science Fund of Serbia under grant 0401F.

2 138 G. V. Milovanović and L. Z. Ran»cić Using a restriction on the polynomial class like P (1) = 0 or P ( 1) = 0, Schur [8] found the following improvement of (1.2) ja n j»2 n 1 cos ß 2n kp k 1 : 4n This result was extended by Rahman and Schmeisser [7] for polynomials with real coefficients, which have at most n 1 distinct zeros in ( 1; 1). Similarly in L 2 norm, kp k = kp k 2 = Z 1 1 jp (x)j 2 dx 1=2 Tariq [10] improved the following result of Labelle [2] (2k 1) ja k j» k + 1 1=2 [(n k)=2] + k +1=2 kp k 2 : (1.3) k! 2 [(n k)=2] for P 2 P n and 0» k» n, the symbol [x] denotes as usual the integral part of x. Equality in this case is attained only for the constant multiplies of the polynomial [(n k)=2] k + ν 1=2 ( 1) ν (4ν +2k +1) ν P m (x) denotes the Legendre polynomial of degree m. Under restriction P (1) = 0, Tariq [10] proved that ja n j» ; P k+2ν (x); n 1=2 (2n)! 2n +1 n +1 2 n (n!) 2 kp k 2 ; (1.4) 2 with equality case P (x) =P n (x) 1 n 1 n 2 (2ν +1)P ν (x): Also, he obtained that with equality case ja n 1 j» (n2 +2) 1=2 n +1 P (x) = 1=2 (2n 2)! 2n 1 2 n 1 ((n 1)!) 2 kp k 2 ; (1.5) 2 2n +1 n 2 +2 P n(x) P n 1 (x)+ 1 n 2 n 2 (2ν +1)P ν (x): +2

3 An estimate for coefficients of polynomials in L 2 norm. II 139 In the absence of the hypothesis P (1) = 0 the factor (n 2 +2) 1=2 =(n + 1) appearing on the right-hand side of (1.5) is to be dropped. This result was extended by Milovanović and Guessab [4] for polynomials with real coefficients, which have m zeros on real line. In this paper we consider more general problem including L 2 norm of polynomials with respect to a nonnegative measure on the real line R. The generalized Laguerre measure is is also included. 2. Main results Let dff(x) be a given nonnegative measure on the real line R, with compact or infinite support, for which all moments μ k = R R xk dff(x), k = 0; 1;..., exist and are finite, and μ 0 > 0. In that case, there exist a unique set of orthonormal polynomials ß n ( ) =ß n ( ; dff), n =0; 1;..., defined by ß n (x) =b (n) n (dff)x n + b (n) n 1 (dff)xn b (n) 0 (dff); b(n) n (dff) > 0; (ß n ;ß m )=ffi nm ; n; m 0; (f;g) = For P 2P n,we define Z R f(x)g(x) dff(x) (f;g 2 L2 (R)): (2.1) kp k dff = p (P; P) = Z 1=2 jp R (x)j2 dff(x) : (2.2) Also, for ο k 2 C, k =1;... ;m,we define a restricted polynomial class P n (ο 1 ;... ;ο m )=fp 2P n j P (ο k )=0; k =1;... ;mg (0» m» n): In the case m = 0 this class of polynomials reduces to P n. The case m = n is trivial. If ο 1 = = ο k = ο (1» k» m) then the restriction on polynomials at the point x = ο becomes P (ο) =P 0 (ο) = = P (k 1) (ο) =0. Let i=1 (x ο i )=x m s 1 x m 1 + +( 1) m 1 s m 1 x +( 1) m s m s k denotes elementary symmetric functions of ο 1 ;... ;ο m, i.e., s k = ο 1 ο k for k =1;... ;m: (2.3) For k =0wehave s 0 = 1, and s k = 0 for k>mor k<0.

4 140 G. V. Milovanović and L. Z. Ran»cić Theorem 2.1. Let P 2P n (ο 1 ;... ;ο m ) and s 1,..., s m be given by (2.3). If the measure d^ff(x) is given by d^ff(x) = and kp k dff is defined by(2.2), then ja n k j» k k jx ο k j 2 dff(x) (2.4) ( 1) k i s k i^b(n m j) n m i 2 1=2 kp k dff; (2.5) for k =0; 1;... ;n, ^b μ ν = b μ ν (d^ff), ν =0; 1;... ;μ, are the coefficients in the orthonormal polynomial ^ß μ ( ) =ß μ ( ; d^ff). Inequality (2.5) is sharp and becomes an equality if and only if P (x) is a constant multiple of the polynomial k ^ß n m j (x) k ( 1) k i s k i^b(n m j) n m i 1 Y A m (x ο k ): Proof. At first we consider the inner product (2.1). Then the polynomial P (x) = np P a ν x ν 2 P n can be represented in the form P (x) = n ff ν ß ν (x; dff); ff ν =(P; ß ν ), ν =0; 1;... ;n. Then we have a n k = k ß ν ( ) =ß ν ( ; dff). ff n i b (n i) n k (dff) = ψp; k b (n i) n k (dff)ß n i! Suppose now thatp 2P n (ο 1 ;... ;ο m ). Then we can write P (x) =Q(x) ; k =0; 1;... ;n; (2.6) (x ο k ); (2.7) Q(x) =a 0 n mx n m + a 0 n m 1 xn m a 0 0 2P n m. Also, we have i=1 (x ο i )=x m s 1 x m 1 + +( 1) m 1 s m 1 x +( 1) m s m s k, k = 0; 1;... ;m, denotes elementary simmetric functions (2.3). putting this in (2.7), we obtain P (x) = n m m a 0 i( 1) ν s ν x m+i ν = n k=0 a n k x n k ; Now,

5 An estimate for coefficients of polynomials in L 2 norm. II 141 a n k = k a 0 n m i( 1) k i s k i ; k =0; 1;... ;n; (2.8) and a 0 k = 0 for k<0andk>n m. Now, the corresponding equalities (2.6) for polynomial Q in the measure d^ff(x), given by (2.4), become i a 0 n m i = Q; ^b(n m j) n m i ^ß n m j ; i =0; 1;... ;n m; (2.9) ^ß ν ( ) =ß ν ( ; d^ff). and According to (2.7), we have 0 k a n k = ( 1) k i s k i ^b (μ) ν W n m (x) = Q; i 1 ^b(n m j) n m i ^ß n m j A =(Q; Wn m ) (2.10) k i ( 1) k i s k i k ^ß n m j (x) k ^b(n m j) n m i ^ß n m j (x) ( 1) k i s k i^b(n m j) n m i = 0 for ν<0. Now, using Cauchy inequality we get C n;n k = kw n m k d^ff = Z ja n k j»c n;n k kqk d^ff k k ( 1) k i s k i^b(n m j) n m i kqk 2 d^ff = R jq(x)j2 d^ff(x) = jp R (x)j2 dff(x) =kp k 2 dff we obtain inequality (2.5). Q The extremal polynomial is x 7! W n m (x) m (x ο k ). Λ Z 2 1=2 : Since Remark 2.1. For k =0andk = 1 Theorem 2.1 gives the results obtained by Milovanović and Guessab [4] (see also [6, pp ]). Consider now the generalized Laguerre measure dff(x) =x ff e x dx, ff> 1, on (0; +1). With ~ L (ff) n (x) we denote the generalized orthonormal Laguerre poly- of x k in L ~ (ff) (x) isgiven by nomial. The coefficient b (n) k b (n) k n =( 1) n k n k (ff + k +1)n k p n! (n + ff +1) : As a direct corollary of Theorem 2.1, we have:

6 142 G. V. Milovanović and L. Z. Ran»cić Corollary 2.2. Under restriction P (i) (0) = 0, i =0; 1;... ;m 1, wehavethat A n;k = ja n k j» p A n;k kp k dff ; 1 (n m k)! (n + m k + ff +1) k n + m j + ff n m j k j k j for n k m, anda n;k =0for n k<m. The equality is attained if and only if P (x) is a constant multiple of the polynomial x m k ^b(n m j) ~ n m k L (ff+2m) n m j (x): REFERENCES [1] A. Giroux and Q.I. Rahman, Inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 193 (1974), [2] G. Labelle, Concerning polynomials on the unit interval, Proc. Amer. Math. Soc. 20 (1969), [3] V.A. Markov, On functions deviating least from zero in a given interval, Izdat. Imp. Akad. Nauk, St. Petersburg 1892 (Russian) [German transl. Math. Ann. 77 (1916), ]. [4] G.V. Milovanović and A. Guessab, An estimate for coefficients of polynomials in L 2 norm, Proc. Amer. Math. Soc. 120 (1994), [5] G.V. Milovanović and L.Z. Marinković, Extremal problems for coefficients of algebraic polynomials, Facta Univ. Ser. Math. Inform. 5 (1990), [6] G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore-New Jersey-London-Hong Kong, [7] Q.I. Rahman and G. Schmeisser, Inequalities for polynomials on the unit interval, Trans. Amer. Math. Soc. 231 (1977), [8] I. Schur, Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall, Math. Z. 4 (1919), [9] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., [10] Q.M. Tariq, Concerning polynomials on the unit interval, Proc. Amer. Math. Soc. 99 (1987), Elektronski fakultet (Received ) Katedra za matematiku p.p. 73 Yugoslavia grade@efnis.elfak.ni.ac.yu

AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS

AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS W. M. SHAH A. LIMAN P.G. Department of Mathematics Department of Mathematics Baramulla College, Kashmir National Institute of Technology India-193101