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, 1971. 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
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 0 + + cos nd = sin(n + 1)6 2 sin(0/2)
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,
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 + 0 + 1) 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
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 + 0 + 2) + 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), 22-25. MR 41 #2079. 2. G. Gasper, On the extension of Turán's inequality to Jacobi polynomials, Duke Math. J. 38 (1971), 415^128. 3. S. Karlin and G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/61), 1-157. MR 26 #539. 4. B. N. Mukherjee and T. S. Nanjundiah, On an inequality relating to Laguerre and Hermite polynomials, Math. Student 19 (1951), 47-48. MR 13, 649. 5. M. L. Patrick, Some inequalities concerning Jacobi polynomials, SIAM J. Math. Anal. 2 (1971), 213-220. 6. H. Skovgaard, On inequalities of the Turan type, Math. Scand. 2 (1954), 65-73. MR 16, 118. 7. G. Szegö, An inequality for Jacobi polynomials, Studies in Math. Anal, and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 392-398. MR 26 #2657. 8. -, Orthogonal polynomials, rev. ed., Amer. Math Soc. Colloq. Publ., vol. 23, Amer. Math. Soc, Providence, R.I., 1959. MR 21 #5029. Department of Mathematics, Northwestern University, Evanston, Illinois 60201