ON TURAN'S INEQUALITY FOR ULTRA- SPHERICAL POLYNOMIALS

Transcription

1 ON TURAN'S INEQUALITY FOR ULTRA- SPHERICAL POLYNOMIALS K. VENKATACHALIENGAR and s. K. LAKSHMANA RAO 0. Introduction. Some years ago P. Turan noticed the interesting inequality (0.1) An(X) = [Pn(x)]2 - Pn+l(x)Pn-l(x) 0,» 1, X \ g 1, where P (x) is the Legendre polynomial of order n and the above inequality has been extended in recent years to the case of some other orthogonal polynomials as well as the ordinary and modified Bessel functions [l; 2]. B. S. Madhava Rao and V. R. Thiruvenkatachar [3] deepened the inequality (0.1) by showing that d2 2 T d I2 (0.2) A (x) = - - Pnix) Z 0. 2 n(n + 1) L J The inequality (0.1) has also been generalized to the case of ultraspherical polynomials P M(x) in the form (0.3) A,.x(«) = [FnAx)]2 - Fn+i.x(x)Fn-i,x(x) ^ or < 0, according as \x} ^ or >1, where n,x(x) =PnMix)/Pnx\l). V. R. Thiruvenkatachar and T. S. Nanjundiah [4] and A. E. Danese [5] have obtained series of positive functions for A (x) and its analogue in the case of the ultraspherical, Laguerre and Hermite polynomials. In the present paper we deepen the above results on ultraspherical polynomials by determining the signs of the first and second derivatives of AnX)ix)=[PnX)ix)]2-Pn+1ix)Pnx!.lix) for various values of X and x. We notice first of all that danx)ix)/ can be represented as a numerical multiple of the Wronskian Pn+iix)dPnxllix)/ Pn-i(x)idPn+lix)/) and express it also via the Christoffel- Darboux formula as a series of functions of constant sign. The sign of da!nx)ix)/ for various values of X and x follows easily from this. We determine the sign of (X l)xida )ix)/) for a general value of n and X and show that id2/2)anx)ix) ^0 according as X>1 or 1/2 ^X<1, thus extending the result of B. S. Madhava Rao and V. R. Thiruvenkatachar to the case of ultraspherical polynomials. For integer values of n we obtain a series of positive functions for id2/2)anx\x). Next we show that (1-x2)AlM(x) decreases steadily in (0, 1) when KX^3/2. From the signs of da^ix)/, Received by the editors August 9, 1956 and, in revised form, April 1,

2 1076 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December d2anx)(x)/2 we prove the concavity of A,x(x) for 0 <X^ 1/2 and also arrive at inequalities for An,\(x) in the range \x\ : 1, which are not only simpler but also cover a wider range of X than the corresponding inequalities given by O. Szasz [2]. Finally we determine an integral representation for An(x). 1. Expressing da^\x)/ as a Wronskian. We append below the well known relations satisfied by the ultraspherical polynomials [6] for frequent use in the following. d2y dy m (1.1) (1 - x2) - (2X 4- l)x + 11(11 + 2X)y = 0, y = P (x) 2 (1.2) npn\x) - 2(m + X - l)xpnx-i(x) + (n + 2X - 2)Pn-i(x) = 0, w = 2 (1.3) -P»' (*) = /(r(x + r)/r(x))p( "!;) (*), r ^ n r ex) d (\) o (X) (1.4) npn (x) = x Pn (x)-pn-i(x), (X) d (x) d a) (1.5) (n + 2X)Pn (X) = Pn+l(x) - X Pn (x), (xi d <x) d (x> (1.6) 2(n + X)P (x) = Pn+i(x)-P -i(x), (1.7) (1 - x2) Pn\x) = - nxpt(x) + (n+2\- l)p»_i(*), (1.8) (1 - x2) Pn\x) = (» + 2X)*P,<iX>(.T) - («+ l)p (+i(*). Differentiation of (1.9) A? \x) = [PlX,(x)]2- P%i(x)P?2i(x) yields (1.10) An\x) = af\x) - Bn\x) where M(.) = P(:\x) ± P?(X) - PZ(X) y Pn+\(X)

3 i957l ULTRASPHERICAL POLYNOMIALS 1077 and We have then»)., rx)., d (X) (X) d <x> 8n (X) = Pn+l(x) Pn-l(x) - Pn (x) Pn (x). Pn+i(x) Pn-i(x) (1.11) an (x)+8n (x) = d (x) d (X). Pn+l(x) Pn-l(x) Also nan (x) (n + 2X)/3 (x) = 2(n + \)PnX\x) Pn\x) - npnli(x) P i(x) (X) d (x) - (n + 2\)Pn+i(x) P _i(*) = 2(«+ X)P^X)(^) - Pn\x) - (n +2\- l)p i(x) - Pn+\(X). (X), d (x>, s - (n + l)pn+i(x) Pn-i(x) - (2X - 1) \Pn+\(x) Pn-l(x) ~ Pnll(X) P +\(x)\. { ) The first three terms together are found to vanish on using (1.4), (1.5), (1.6) and we get nan\x) - (n + 2\)8n\x) = - (2X - 1)[«?'(*) + Bn\x)\, so that there follows the relation (1.12) (n + 2X - l)at(x) = (n+ l)^x>(x). Solving (1.11) and (1.12) for anx)(x), 8 )(x) and using (1.10), we have Pn+l(x) Pn~l(x) d (X) 1 X (1.13) An (x) = - d (x) d a,. n + X Pn+i(x) P _i(at) ax ax

4 1078 K. VENKA'I ACHALIENGAR AND S. K. LAKSHMANA RAO [December From (1.3) and (1.13) we can also express the first derivative (X) r dr (M ~12 dr (>,) dr (xi Ankx) m - Pn \X) - PUl(x) Pnll(x) l_r J r r of as a constant times the Wronskian of P _+r+i(x) and P^X-iix). 2. Series of functions of constant sign for danx\x)/. Using (1.2) in succession we can write (2.1) Pn+\(X) = anxpnll(x) + KP^x) + CnP^x), where (2.2) an = 4(«+ X)(w + X- 1) -' nvn + 1) («+ X)(»+2X-2)(» + 2\-3) nin + 1)(» + X - 2) Hence the determinant t/ ^ Ux, y) = P +\(x) ex) P^ix) (x) Pn+iiy) Pn-iiy) can be written in the form anx Pn-lix) + bnpn-lix) + CnPn-iix) Pn-\(x) anypnliiy) + bnpn-i(y) + cnpn-ziy) P i(y) and so we get.2 2 (X) (X) = an(x y )Pn-i(x)Pn-i(y) cnon-i(x, y) 2 2 (X) (X) (2.3) Bnix, y) + cjn-i(x, y) = o (x - y )Pn_1(x)Pn_1(y). On observing that we obtain Kix^ti PB+i(*) Pn-i(x) v-*x y - x Pn+iix) P»_i(x) from (2.3), the relation (n + X) d (X) n-2 + \ d a) m -A (x) + cn- A _2(x) = - 2aBx[PB_i(x)J2, 1 X 1 X

5 19571 ULTRASPHERICAL POLYNOMIALS 1079 which can be rewritten in the form m^ ''.Wn (»+2X-2)(«+ 2X-3) d (X) (2.4) A (x)- - An_2(x) n(n + 1) Solving this difference equation d (X), N 8(X- l)r(n+2x- 1) A (x) = -x (n + 1)! 8(n + X - 1) r Cx) l2 = - \ ^n (1 - \)x[pn2i(x)]\ n(n + 1) for da(n)(x)/ we obtain [(-+0/21 (n-2k+ 1)!(» - 2k + X 4-1) w, ' ^ -w-,lxii± n- l^»-»+i(*)j I i_i r(» 2k + 2X + 1) which is the desired series expansion. Differentiation of (2.5) gives a series for (d2/2)anx\x) which has however no particularly elegant form. In the case of Legendre polynomials (X = l/2), (2.5) becomes & 2x K»+l>/2] An(x) = Y (2n - U + 3)[Pn-2k+i(x)]2. n(n + 1) t»i Differentiating d2 this we see that 2 I(n+l)/2] =- E (2» - Ik + 3)P _24+i(x)[P -tt+i(*) + 2xP^2*+,(x)] n(n + 1) t-i which is the same as identity (0.2). From (2.5) it follows that (2.6) sgn a1x>(x)1 = sgn (X- 1)* (X > 0) which amounts to the following property of Anx'(x). The Turdn expression AjX)(x) is an increasing (a decreasing) function for positive (negative) values of x when X > 1. This is to be reversed when 0<X<1. From this we see that when X > 1, A (x) S AB (0) > 0 for all values of x; when 1/2 ^ X < 1, a1x>(x) S A? (l) SO for x g 1; when 0 < X g 1/2, C\x) ^ A^\l) ^ 0 for * gj 1.

6 1080 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December 3. Sign of (X l)xda^ (x) / for a general index n. For a general index n (which will be restricted in a way later on) we take P \x) as a solution of the differential equation (1.1) which remains regular at x= +1 (or at x= 1) either of which is a regular singular point of the differential equation. We stipulate that the value of the function as well as its slope at x= +1 are positive. Defining Aj,X)(x) as in (1.9) we can again derive the relation (1.13) as before. Multiplying this last relation by (1 x2)xh/2 and differentiating the result, we get ^-\a - xtm t a:\x)] L J 1 _ X Pn+l(x) Pn-l(x) = ^r~x~ 1[(1_,Y^P^m] ±\(l-xt1,2yp^(x)v L J \_ J Simplifying the second row of the determinant by means of the differential equation we are led to d Y 2 X+l/2 d (X) "I (1 - x) An (x) (3.1) l J = 4(l-X)(l-xY-1/2P X+>1(x)PBX_)1(x). Hence the extrema of (1 x2)x+ll2danx)(x)/ occur only at the zeros of Pn+i(x), Pnxii(x). The solution P X)(x) as defined above cannot have zeros on the real segment (1, oo). It increases from a positive value at x= 1. If it attains a maximum value at a point Xi (> 1), we have the conditions (dy/)xi = 0, (d2y/2)zl<q and from the differential equation we notice a contradiction if ra(w + 2X) is positive. Hence P X)(x) is increasing in x> 1 and consequently does not vanish for any x on (1, oc). Let us denote by (a): C*l, 6*2, «3, 03): 0,, 0,, ft, the zeros in ( 1, +1) of P +i(x) and Pfii(x) respectively. From (1.13) we have sgn An (x) \ = sgn PB_i(x) PB+i(x) LdX JI=a LW + X dx Jx=a From (1.7) we may write (I -a2) T^- Pn+\(x) 1 = (n + 2X) PnX\a) LdX J x=a

7 i957l ULTRASPHERICAL POLYNOMIALS 1081 and hence T d (X,, 1 T(X - 1)(» + 2\)PHX!i(a)Pn)(a)l sgn A (x) = sgn --. Lax Jx=a L (w + X)(l - a2) J Using the relation 2(n+\)aPnX)(a) = (n + 2\-l)Plnx±1(a) which follows from (1.2), we get d (X) r n sgn A (x) = sgn [(X - l)(n 4-2\)(n + 2X - l)/a]. In a similar way we observe that sgn [ Af'(x)l Lax JI=,(j = sgn [(X - l)nft/(» +!)] Combining the above two facts we see that whenever n S1 and X>0 [d >\) "I x A (x) = sgn (X - 1). J z=a,s Thus at all possible extrema of (1-x2)x+1/2(a*A^)(x)/a,x), Q^ l)x(danx)(x)/) is positive and hence (3.2) sgn (X - l)x An \x) = + 1, for n S 1, X > 0. For positive integer values of n this is the same as (2.6). 4. Sign of (d2/ox2)a;,x)(x). Differentiating (1.13) which also holds for general n we get D(X) I \ D(X) / \ Pn+l(x) Pn-l(x) d a), s 1 X -A (x) =- d2 (x) d2 (x) 2 «+ X - Pn+l(x) --P _l(x) 2 2 Replacing the second row elements by means of the equations d1 ex) d2 (xi d ai Pn+i(x) = X Pn (x) + (n + 2X + 1) Pn (*), ax2 2 d2 (x> d2 tx) d c\\ p _;(x) = x p '(*) - («-1) - pv(x) 2 2 (these follow on differentiating (1.5) and (1.4)) and replacing the first row elements by means of (1.7) and (1.8) we get

8 1082 K. VENKATACHAL1ENGAR AND S. K. LAKSHMANA RAO (December d2 (X), 1 - X - A (x) = - 2 n + X (n+2x)x.pnx)(x)-(l-x2) pt(x) «xpnx)(x) + (l-x2) P*\x) n + 1 n + 2X - 1 d2 rx) d (xj d2 (x) d (X) x P\ \x) + (ii+2\+l) Pn \x) x--pn\x)-(n-l) P:\x) 2 2 a form in which the right hand side involves only algebraic combinations of P x>(x) and its first and second derivatives. After simplifying this determinant we arrive at the equation (»+l)(«+2x-l) d2 (x,. Vd (x, "j2 -An (x) = 2 P (x) 2(X - 1) 2 \_ J (4.1) + (2X- l)x *[( PnX)(x)J - Pn(X,(x) ~ Pf (*)] Ii -p?\x)jlp?\x)\. ) (4.2) D?(x) = A^x) = \- PlX)(x)]2 - - pv+i(x) - P» (,), Lax J ax we can easily see that DnX)(x) =4X2Anx_+i')(x) and yd?\x) = 2x{x[( p1x)(x))2 - PIX)(X) ^PlX)(x)] - A)-ip.WW} We may therefore (4.3) write (4.1) in the form (»+l)(»4-2x-l) d2 w -An 2(X - 1) 2 (X) T d rx) "I2 d (x+d = 2 - p; \x) + 2x(2x - i)x - a;j; (*). Lax J Either of (4.1) and (4.3) is an extension of (0.2) to the case of ultraspherical polynomials. When n is a positive integer we may replace the last term on the right side of (4.3) by means of the series (2.5)

9 i957l ULTRASPHERICAL POLYNOMIALS 1083 and so arrive at the following series for d2abx>(x)/2 (»+l)(»+2x- 1) d2 (x,/ N - -A (x) 2(X - 1) 2 r d (x) "I r(«-i- 2X) = 2 PB \x) \2 + (2X - 1) - 16X2x2 \_ J nl "» 1 in - 2k) l{n - 2k + X + 1) u+1) 2^ -:-: LPB-2fc(x)J-. [ Tin - 2k + 2X + 2) From (3.2) it follows that \xdan\\l)ix)/>0 Using this in (4.3) we deduce that (a «d* a(xv ^ /+1 (X>1)' (4.5) sgn-ab (x) = < B 2 l-l (1/2 X < 1). for w^2, X>-l/2. Hence ABX)(x) is a convex function of x for X> 1 and a concave function of x for l/2gx<l. 5. Decreasing nature of (1 x2)a X)(x). From (3.1) we know that A X)(x) satisfies the differential equation r d'2 d i cm (5.1) ^(1 - x2) - (2X + l)x - + 4(1 -X)jAB (*) If (1 x2)anx)(x) is denoted by y, we get = 4(l-X)[P:X)(x)]2. (5.2) = (1 - x ) - An (x) - 2xA (x), -= (1 - x ) -A (x) \x AB ix) 2AB (x) We find from (5.1) and (5.2) that y satisfies the differential equation (5.3) [(1-^ i-(2x-3)^x-(4x-6)]^ = (4X - 6)x2ABM(x) + 4(1 - X)(l - x)[pn\x)]\ Now y(0)=a;x)(0) is positive when X>0 and y(l)=0. When X>1, dy/ is negative for all x > 1 and hence y decreases for x > 1 when X> 1. We shall now see that y steadily decreases even in (0, 1) when KXg3/2. On using (2.6) we have idy/)x=o = 0. From (5.3) we have (d2y/ix2),=o = (4X-6)AiX)(0)-r-4(l-X)[PiX)(0)]2 and so when

10 1084 K. VENKATACHALIENGAR AND S. K. LAKSHMANA RAO [December l<x 3/2, id2y/2)x=o is negative. Hence when KX 3/2, v reaches a maximum at x = 0 with the maximum value =ABX)(0) and decreases in the neighbourhood of x = 0. Also (dy/)x=i= 2A X)(1) is negative when X>l/2 and so y(x) is decreasing in the neighbourhood of x= 1 also when X> 1. If y(x) is not steadily decreasing in (0, 1) it must reach a minimum value at some point x = a<l, then increase to a maximum at some point x=j8 between a and 1 and then decrease again. We show that this possibility cannot occur when 1 <X 3/2. From the conditions for minimum, we have at x = a, (dy/) =0 and (d2y/2)>0 and from the differential equation (5.3) we get 2 / d2y\ i m (1 - a ) ( ~ ) = (4X - 6)yia) + (4X - 6)a AB \a) \ 2/x=a + 4(l-X)(l-a2)[PBX>(a)]\ The left hand side is positive while the right hand side is negative. We conclude that y cannot have a minimum anywhere in (0, 1). Therefore v(x) must steadily decrease in (0, 1) when 1 <X 3/2. It is to be expected that for large values of X, y(x) decreases first to a minimum and then attains a maximum in (0, 1) before decreasing again at and near x = 1. Eliminating (1 x2)d2anx)(x)/tix2 in d2y/2 by means of (5.1) we obtain = (2X - 3)x ABX)(x) + (4X - 6)ABX)(x) + 4(1 - X)[P X)(x)]\ 2 Using (3.2) we notice that sgn d2y/2= 1 when 1 <X 3/2. 6. Inequalities for AB,x(x). Let A.x(x) = [p1x)(x)/pbx)(1)]2- [P(n+\ix)/Pn+\il)].[P iix)/p (l)]. From the following well-known relations (see (4), (5)) 2 (X1 '',A n (6.1) A / \ (1 ~ x)dn AB,x(x) =-= (X) 4X \ <M-1>,. - (1 - x )An_i ix) where ^B,x = n(w + 2X) [PBX)(1)]2, and the conclusions section, we have of the previous (a) A,x(x) is steadily decreasing in (0, 1) for 0 < X 1/2; d2 (b) sgn-a,x(x) = - 1 when 0 < X 1/2, 2

11 1957l ULTRASPHERICAL POLYNOMIALS 1085 showing the concavity property of A,x(x). From (a) we can immediately deduce the well known inequality viz., the Turan expression A,x(x)>0 in x <1 when 0<Xgl/2. From (6.1) and the concluding results of (2) we can prove Turin's inequality in full for X>0. We have seen in (3.2) that (3.2) sgn (X- l)x A X)(x) = + 1 for n S 1, X > 0 L J and in (4.5) that d2 (x) (+1 XI>I1 (4.5) sgn-an (x) = < n S 2. 2 l-l 1/2 S X < 1 From the identity D X) (x) = 4X2A xjt"11) (x) it therefore follows that d rx) (6.2) sgn Xx Dn (x) = +1 (n S 2, X > - 1/2), These show the monotonic increasing-decreasing behaviour and the convexity-concavity property of D X)(x) ior various values of X, and enable us to write the following chain of inequalities: When X > 0, Z? X)(0) ^ >lx>(x) ^ A!X)(0)(1 - x/a) + DnX\a) x/a when ^ Dnk\a), -1/2 < X < 0, D X,(0) ^ DnX\x) S DnX)(0)(l - x/a) + D (o) x/a S DnX\a) 0 ^ x ^ a. Taking a = 1 and multiplying the above inequalities by (1 x2) we get the following inequalities for A,x(x) in the range x ;S1, for w^2. A,x(0)(l - x2) =S A,x(x) ^ ran,x(0)(l - x) + (1 - x2) L 2X + 1_ (6.4) = (1 " x2)/(2\ +1) (X > 0), ^fri = l^ttl + A».x(0)(l - *)}(1 - x2) g An.x(x) 2X + 1 (2X + 1 ) g ABlX(0)(l - x2) (-1/2<X<0).

12 1086 K. VENKATACHALIENGAR AND S. K LAKSHMANA RAO [December These are simpler than the inequalities for A,x(x) in O. Szasz's paper [2] and are at the same time an improvement over the corresponding ones in [4]. 7. Integral representation for A (x). From (3.1) we have (7.1) ±\il - x2)^ L AlX)(x)l + 4(1 - X)(l - *tx'\vix) L ox J = 4(1 - X)(l - x2)x-1/2[p lx)(x)]2. If we denote [(l-x2)*-1'2ank)(x)]/x by/ X)(x), then (1 - x2)x+m~ A«\x) = [x{l _ J)f?\x)] + (2X + l)x2f*\x). Hence from (7.1) we get "»(1 - *2)Ax)l + (2X + l)x2 ~fn\x) + 6x/ X>(x) 2 L J which may also be written in the form,, 2vX-I/2r (X),. -,2 = 4(1-X)(1-X) [Pn \x)], d T 2 2 -x+3/2 d (X) "1 rr,(x)/ \i2 x (1 - x ) / (x) = 4(1 - X)x[Pn (x)j. L For Legendre functions (X = l/2) this becomes (7.2) -f-^(l - *2) [*»(*)/*]) = 2x[Pn(x)]2. \ / Hence x2(l x2)(d[ah(x)/x]/) is an increasing (a decreasing) function for positive (negative) values of x and vanishes at x= +1. From (7.2) we have in succession and ~ (An(x)/X) = 2 Ct[Pnit)]2dt, x2(l x2) J 1 (7.3) A (x) = 2x r U Cl[Pnit)]*dt, J 1 «-(l u-) J 1 the latter giving a positive integral representation for A (x). Using the relation

13 i957) ULTRASPHER1CAL POLYNOMIALS 1087 T (1-X2)[^-Pn(x)]2 d, ^ Lax J., - (l-x2)[p (x)] = - 2x[Pn(x)]2, L n(n + 1) we may also write A (x) in the form f r d f} (l-l2)\- Pn(t)\ C r n \-dt J (7.4) A (x) = x [Pn(t)]2 +-;-\r2dt. J x { n(n + 1) References 1. G. Szego, On an inequality of P. Turin concerning Legendre polynomials. Bull. Amer. Math. Soc. vol. 54 (1948) pp O. Szasz, Inequalities concerning orthogonal polynomials and Bessel functions, Proc. Amer. Math. Soc. vol. 1 (1950) pp B. S. Madhava Rao and V. R. Thiruvenkatachar, On an inequality concerning orthogonal polynomials, Proceedings of the Indian Academy of Sciences, Sect. A vol. 29 (1949) pp V. R. Thiruvenkatachar and T. S. Nanjundiah, Inequalities concerning Bessel functions and orthogonal polynomials, ibid. Sect. A vol. 33 (1951) pp A. E. Danese, Explicit evaluations of Turin expressions, Annali di Matematica Pura ed Applicata, Serie IV vol. 38 (1955) pp G. Szegb, Orthogonal polynomials, (1939) pp Mysore University Indian Institute and of Science