AN INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS

Transcription

1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 2, April 1972 AN INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS GEORGE GASPER Abstract. For Jacobi polynomials P^^Hx), a, ß> 1, let RAX) = ^" /^ > A»M = KM - Rn^(x)R, AX). We prove that 2(n + a + l)(n + p) with equality only for x= ±1. This shows that the Turan inequality A (a)^0, -1^*^1, holds if and only if j9j a>-l. 1. Let /? (*; a, ß)=P(n'-ß)(x)lPtJl-,)(l) where /^'"'(x) is the Jacobi polynomial [8] of order (a, ß), a, /?> 1, and let An(x; a, ß) = R2n(x; a, 0) - R^x; a, ß)Rn+1(x; a, /S). In [3, p. 153] Karlin and Szegö raised the question of whether or not the Turan inequality (1) An(x; a, ß) > 0, -I <x <\,n^l, holds for/3^a> 1. From u-u -, «^ (-+"ïï"+ -r (n + a + l)(n + 0) \ n / \» 7 it is obvious that (1) fails for /?<oc. Later [7] Szegö showed that (1) holds for /5^ <x, <x> 1, but unfortunately his method fails for the triangle /={(<*, ß):o:<ß< -a,-1 < cc<0}. Recently the author [2] proved (1) for the set V={(a,ß):ß^x> -l,(ß-x)(o: + ß)(4ß2 + 4*+ l)ä0}, Received by the editors July 14, AMS 1970 i«a/y>c, classifications. Primary 33A50, 33A65, 42A04; Secondary 33A10, 42A52. Key words and phrases. Inequality, Turan inequality, Jacobi polynomials, ultraspherical polynomials, orthogonal polynomials, Dirichlet kernel. 435 American Mathematical Society 1972

2 436 GEORGE GASPER [April which contains most of U, but again the method employed failed for the set U V. In this note we give a proof of (l)for the "best possible" range ß ^ a > 1 and also establish the following stronger result. Theorem. Let a > 1, ß > 1 and n ^ 1. Then (2) \n(x; a, ß)> (ß~ g)(1 ~ X) R2n(x; a, ß), -1 < x < 1, P; - 2(n + a + l)(n + ß) - ~ with equality only for x= ± 1. The method used to prove (2) is a modification of that used by Mukherjee and Nanjundiah [4] for Laguerre and Hermite polynomials, and by Skovgaard [6] for ultraspherical polynomials and Bessel functions. It depends on the observation that if qn(x) is a polynomial of degree n with simple zeros xx, x2,, x, then n' ~ = I(x-xk)-\ Qn *=1 (an\ _ In Qnan _ V/ \-2 I, Z \x ~ xk), aj a *=i where a prime denotes differentiation with respect to x. These identities were used almost a hundred years ago by Laguerre (see [8, p. 120]) to derive bounds for the zeros of the classical polynomials. Before proving (2), let us first point out some interesting special cases. Using P(n*J)(-x)=(-l)nPinß-*)(x), we find that (2) may be rewritten in the form {tn a){ltßtll«a*a' «- R- x>a' «**«(*; a> 0 (3) (" + a + *)(" + 0) ^_ia-ml+x),<,<!, - 2(n + a + l)(n + 8) r' ~ ~ where equality holds only for x=±\. Since sin(n + )0 R (cos0;,- ) i\ (2«+ l)sin(0/2) [8, p. 60], it follows from (3) that the Dirichlet kernel Dn(6) = I + cos cos nd = sin(n + 1)6 2 sin(0/2)

3 1972] INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS 437 satisfies >2,flN n,mr> /m ^ 2^ + C0S ö) n (2n + l)2 with equality only when cos 9= ± ]. If we let pn(x; X)=P{n')(x)/P(nx)0) where P[k)(x) is the ultraspherical polynomial [8, p. 81] of order X, X~> \, then from [8, p. 59] we have pth(x; X) = Rn(2x2 - M-i-l), Pin+Ax; X) = xrn(2x2-1 ; X - \, \). Hence (2) and (3) yield, for X> \ and l íx<l, p\n(x; X) - p2n_2(x; X)p2n+2(x; X) ^ - - -^ - p2b(*; A), (2n l)(2n + 2/ + 1) pl+i(x; X) - Pin-i(x'> %Ws(*; X)^ -,.', *.. pl+i(*; ). (2/7 + l)(2n + 2X + 1) (2n + 1)(2h + 2X- 1), i>a«(x; Â) - p2 _2(x; A)p2n+2(x; X) (In - 1X2*1 + 2A + 1) > -pl(x;x), _ (2«- l)(2n + 2A + 1) (2«+ 3)(2» + 2A - 1) 2,n f. «, n 0, 1V.,,,,,«, JWjX*; A) -p2b_i(x; A)p2n+3(x; X) (2n + l)(2n + 2/1+1) ;. 4(A - l)x2, (2«+ l)(2n + 2À + 1) with equality only for x= l, 0, 1. This sharpens the inequalities in [2, Corollaries 1 and 2]. For other types of inequalities for orthogonal polynomials see Askey [1] and Patrick [5]. 2. In proving the theorem we may assume that l<x<l, for from Rn(\ ; a, ß) = 1, K (-l; a, ß) = (-l)»(n + ^ (" + "Y* it is easy to see that equality holds in (2) when x=±l Since consecutive orthogonal polynomials qn(x) and qn+1(x) cannot have common zeros [8, 3.3], we may also assume that x is not a zero of Rn(x; a., ß). Fix ws>l, a>-l, ß>-\ and let Rn=Rn(x; a, ß), A =A (x; «., ß). From [8, p. 72] we have Rn-i = ÁnR'n + ß R, RB+1 = CnR'n + DBRB,

4 438 GEORGE GASPER [April where An, Bn, Cn, Dn are functions of x defined by 2n + ai + ß,1..2 B = 2n + «+ ß / ß-x \ i4«- (1 2n(ii + /?) Using 2n + a. + ß + 2 2(n + a + l)(n + a ) i 2n +<x + ß + a-j8 + 2(n + a + 1) ( \ ' 2n + a + /S + 2) (R-nl^n)' = (^n^k R'n)IRn 2(n + 8) \ 2n + a + ß! ' and the differential equation [8, p. 60] satisfied by R we obtain Hence (1 - x2)r;2 =[«- + («+,? + 2)x]R R' - n(n + a + ß + 1)R2 - (1 - x2)(r'jrjr2. A = R2 - (AnR'n + BnRn)(CnR'n + DnRn) = [1 - BnDn + n(n + x + ß + 1)(1 - x2)"m CJR2 + [- D. - R C + (1 - x2t\ß - a - (a + ß + 2)x)AnCn]RnR'n + (RnlRn)'A CnRn (ß - a)(i - x) 2(n + a + [)(n + j8) R (* + (2n + oc + S)(2n + a + ß + 2)) -(1-^@,4 C Í?2 1 -X2 Now recall [8, 3.3] that all zeros of Rn(x; a, ß) are real and simple and are located in the open interval ( 1, 1). Thus ni n / ni \i n ~ = I(x- xkr\ p )= - i(x - xkr2, R *-i \RJ fc=i where xx, x2,, xn are the zeros of Rn, and so (4) A = (ß-m-x) _2 2(n + a + l)(n + ß) (l-x2rmrf ^E(k,n;a,ß) (2n + a + /3)(2n + a + /5 + 2) i (x - x*)2

5 1972] INEQUALITY OF TURAN TYPE FOR JACOBI POLYNOMIALS 439 with E(k, n; a, ß) = (2n + a + ß)(2n + a + ß + 2)(1 - xxk) + (a2 - ß2)(x - xk). To complete the proof it suffices to observe that the expression AnCnE(k, n; a, ß) in (4) is (strictly) positive since,4bc >0 and E(k, n; a, ß) = 2[2(n - l)(n + a + ß + 2) + 2(a + 1) + (0 + l)(a + ß + 2)](1 - xx,) + (a2 - ß2)(l + x)(l - x,) = 2[2(n - 1)(«+ a ) + 2(ß + 1) + ( 32 - oc2)(l - x)(l + x,) > 0 + (a + l)(a + ß + 2)](1 - xx,) for l<x, xk<l. References 1. R. Askey, An inequality for the classical polynomials, Nederl. Akad. Wetensch. Proc. Ser. A 73=Indag. Math. 32 (1970), MR 41 # G. Gasper, On the extension of Turán's inequality to Jacobi polynomials, Duke Math. J. 38 (1971), 415^ S. Karlin and G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/61), MR 26 # B. N. Mukherjee and T. S. Nanjundiah, On an inequality relating to Laguerre and Hermite polynomials, Math. Student 19 (1951), MR 13, M. L. Patrick, Some inequalities concerning Jacobi polynomials, SIAM J. Math. Anal. 2 (1971), H. Skovgaard, On inequalities of the Turan type, Math. Scand. 2 (1954), MR 16, G. Szegö, An inequality for Jacobi polynomials, Studies in Math. Anal, and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962, pp MR 26 # , Orthogonal polynomials, rev. ed., Amer. Math Soc. Colloq. Publ., vol. 23, Amer. Math. Soc, Providence, R.I., MR 21 #5029. Department of Mathematics, Northwestern University, Evanston, Illinois 60201