nail-.,*.. > (f+!+ 4^Tf) jn^i-.^.i

Transcription

1 proceedings of the american mathematical society Volume 75, Number 2, July 1979 SOME INEQUALITIES OF ALGEBRAIC POLYNOMIALS HAVING REAL ZEROS A. K. VARMA Dedicated to Professor A. Zygmund Abstract. Let P (x) be an algebraic polynomial of degree n having all real zeros. We set '" = \\PAx)»(x)\\LÁaM In this work the lower and upper bounds of / are investigated under the assumptions that all the zeros of P (x) are inside [a, b] and outside [a, b], respectively. We restrict ourselves here with two cases, (1) u(x) (1 jt2)1/2> [", b] = [-1, 1]; (2) u(x) - e~x/2, [a, b] - [0, oo). Results are shown to be best possible. In this work we are concerned with the following theorems of P. Turan [3], P. Erdós [1], and A. K. Varma [5]. Theorem A. Let P (x) be an algebraic polynomial of degree n having all its zeros inside [ 1, +1]; then we have (P,(x) stands for the derivative of P (x).) Theorem B. Let P (x) be a polynomial of degree n satisfying the inequality \P (x)\ < 1 for 1 < x < + 1. Suppose Pn(x) has only real roots and no root inside [-1, +1]; then for -1 < x < + 1, P '(Jc) <\en. This is the best possible result. Theorem C. Let P (x) be an algebraic polynomial of degree n having all zeros real and inside [ 1, +1]; then we have where»al-.,+11 > (f +1 + )«2L2i-..+n> * > 13- (1-2) \\PnfLl{-l,+ lx= f J -I Ptix)dx. Further, if P (+ 1) = P ( 1) = 0 and «> 2 then under the above conditions nail-.,*.. > (f+!+ 4^Tf) jn^i-.^.i Received by the editors July 26, AMS (MOS) subject classifications (1970). Primary 26A60, 30A American Mathematical Society /79/ /$03.00

2 244 A. K. VARMA with equality holding for Pn(x) = (1 - x2)m, n = 2m. The above Theorem C is analogous to Theorem A in the Lx norm. In view of the above theorems it is natural to ask about lower bound (upper bound) of the expression K*«(*)liWW*)*.(*)lwi under the assumption that all the zeros of Pn(x) are all inside [a, b] (no roots inside [a, b], respectively). We limit ourselves here in this work to the above question when [a, b] = [0, oo), co(x) = e~x/2 and [a, b] = [ 1, +1], u(x) = (i - x2y/2. Throughout xk's denote the zeros of Pn(x). Here we prove the following theorems. Theorem 1. Let P (x) be an algebraic polynomial of degree n having all zeros inside [0, oo). Let Pn(0) = 0or k = \ xk i then with equality for P (x) = x". \\e-x/2pñ\lo, oo) > 2{2n"_ 1} IK^HLio, oo) (1.3) The following remark is needed concerning the above theorem. Let us choose P (x) = (x xxy, xx > 0. It can be shown easily that the ratio of r(p x)fe-*dx/ r(p (x))2e-*dx can be made arbitrary small for such polynomials by choosing xx arbitrary large. Thus some condition on the roots such as 2*_ i(**)- ' > 5 is natural. Next, we turn to the case when u(x) = (1 - x2)l/2, [a, b] = [ 1, +1]. Here we prove Theorem 2. Let P (x) be an algebraic polynomial of degree n having all zeros real and inside [ 1, +1]. Then for n > 2 we have lid - jc2),/2h 2 >(- + -_!_\\\(\ - x2y/2p f U x) ^ llzj-i.+ i]*^ 4 4(zz + 1) jllu } ^"Ikl-i.+ 'l with equality for P (x) = (1 - x2)m, n = 2m. The next two theorems are analogous to Theorem B of P. Erdôs [1]. (1.4)

3 SOME INEQUALITIES OF ALGEBRAIC POLYNOMIALS 245 Theorem 3. Let P (x) be an algebraic polynomial of degree < n having all real roots and no root inside the interval [ - 1, +1]; then we have (i-^),/2^wll-,+.i n(n + \)(2n + 3).., 1/2 2 * «p.+1) K1 - ^ f-(^l-». '5) vwrn equality for P (x) = (\ + x)n or Pn(x) = (1 - jc)". Theorem 4. Le/ *,, x2,..., x >e rea/ zeros o/ an algebraic polynomial P (x) of degree < n. Further, - oo < x, < 0, /' = 1, 2,..., «. 77ie«vwf«equality for P (x) = x". 2. Proof of Theorem 1. For the proof of this theorem we will need the following facts: P'Ax) -PA*) 2 L-. (2.1) fc=l * xk P>\x) - Pn(x)P;(x) = P2(x)! (2.2) fc-i (x-xk)2 (x - xx)2 = _ j + 2(x - x,) + (x, - xkf (x-xkf~ *-*k (x-xk)2' ^.^.^ ^i,, (24) ^C At A X\ fa ~ 1 X Xfc From (2.4) and the Cauchy-Schwarz inequality we have /,. Pn(x) \2 P2(x) " (xk - xx)2 P (x) - n^^- < n nk ' 2 --T (2-5) I x- xx tx _ f e, (;c _ f From (2.5) we immediately have PAx) " i;[pax) - n x X, dx r p2(x) $ (xk - xxy n- 2,-~2~e dx- (2-6) 'o (x - xxu i) *-> (*-**) Also, integrating by parts, we have 1 _ «j"e-xpn(x)p'n(x)dx = - - P2(0) + 2 Çe~xP2(x) dx. (2.7)

4 246 A. K. VARMA Now, we turn to the proof of Theorem 1. On integrating by parts, we have re-*p?(x)dx = -Pn(0)PM - rpn(x)(p:(x) - P'n(x))e~* dx, and an equivalent form 2( e-*p?(x)dx = P2(0) -i- + re-*(p?(x) - Pn(x)P:(x))dx J0 Zc = l xk J0 (2.8) + re-xpn(x)p^(x)dx. (2.9) 'o On using (2.2) and (2.7) we obtain 2re-*P'2(x)dx = Pt(<j)[-\+ 2 ^\+\re-xp2(x)dx J0 \ k-\ XkJ J0 + fxe-xp2(x) 2-~ dx. (2.10) 'o *=i(x-x,)2 Now, let us denote by xx the first nonnegative zero of P (x). Then we can write P (x) = (x- xx)qn_x(x), xx > 0, (2.11) where q -x(x) is a polynomial of degree < zz 1 having real zeros lying inside [0, oo). We set /. = Te-xP2(x) 2 l- 2dx =re-xq2_x(x) ± ^^dx, *=i (x - Xk) J0 Zc-1 (x - xk) and on using (2.3) we have (2.12) /* 0, /"OO - JL Ix=-n[ e-xq2_x(x)dx + 2/ (x - xx)q2_x(x) 2 (,e~xdx J0 J0 k~\ (X - Xk) 'o k-i (x-xj2 = -nf e-xq2_x(x)dx + 2 CP&x)qn_x(x)e-Xdx & (*i - xk? + [ e-xq2n_x(x)2 k- l (x - x ) dx, r, v_, 1 r, 7' =» rp'n\x)e-xdx - - C(P'n(x) - nqn_x(x))2e-*dx n J0 n J0 y (*1 - Xk) + /V'«g_,(*) (*' ^ dx; (2.13) 'O *-l i (x (r - xk) v,.r

5 SOME INEQUALITIES OF ALGEBRAIC POLYNOMIALS 247 on putting the values of /, from (2.13) into (2.10) we obtain 'o k-\ (x - xkf Now if i> (0) = 0 or SJ.i^)"1 obtain - \ /""(P-'to - nqn_x(x)fe-xdx. (2.14) n JQ > then on using (2.6) we immediately (2 " ll)l*>e~xp"2ix)dx > 2-{Xe~XP"2{x)dx' from which the proof of Theorem 1 is complete. We would like to remark that the ratio of Ce~xP'2(x)dxI 'O does indeed get smaller than n/2(2n re-xp2(x)dx '0 1) for polynomials P0(x) = (x xx)n for xx > 2/i. For such polynomials the last two terms in (2.14) are zero and "*-.(**r'<o. 3. Proof of Theorem 2. In order to prove Theorem 2, we need some results proved in [5]. Let Pn(x) be any polynomial of degree n; then if (1 - x2)p?(x)dx = nf P2(x)dx + S0, (3.1) (n + l)(2n + 3) f ' x2p2(x)dx = S2 - S0 + (n + 1) f ' P2(x)dx J-i J-i where + 2/ ' (( 1 - x2)p n (x) + nxpn (x))2dx (3.2) S2i = /' x2'p2(x) S ^~XÎ\dx, i = 0, 1. (3.3) J-i k = i(x-xk)2 For the proof of (3.1), (3.2) see (2.1) and (2.10) (with r = 1) in [5]. Equation (3.2) can be rewritten in the form (n + 1)(2«+ 3) f ' (1 - x2)p2(x)dx J-i = SQ- S2 + 2(n + \)2f P2(x)dx -if ((1 - x2)p x) + nxpn(x))2dx.

6 248 A. K. VARMA Therefore (n + l)(2n + 3)/'(l - x2)p2(x)dx < S0 - S2 + 2(n + l)2 j"_' P2(x)dx. From (3.1) and (3.4) we obtain (3.4) 2 f ' (1 - x2)pn'2(x) dx J-\ ^ J-\ n Ç P2(x) dx+s0 (n + l)(2/z + 3)f ' (1 - x2)p2(x) dx S0 - S2 + 2(/i + 1)2J* P2(x) dx (3.5) It is given that x* < 1, S0 > 0, S2 > 0 and S0 - S2 > 0 (here we use (3.3)). Hence it can be verified immediately that (s0 + nfp2(x)dxy \s0 - S2 + 2(zz + \)2 fp2(x)dx\ From (3.5) and (3.6) we obtain > n 2(n + l)2. (3.6) Ç(\ - x2)p?(x)dx > "^V^/'O - x2)p2(x)dx. This proves Theorem 2 as well. 4. Proof of Theorem 3. For the proof of Theorem 3 we need the following lemma. Lemma 4.1. Let P (x) be an algebraic polynomial of degree < zz having real roots and no root in the interval [ 1, +1]. Then we have /',<' - **«>* > ( fô(2vi 3)/>*> * <4-" Proof It follows from the well-known theorem [2] that for such P (x) we have Hence we have P (x)= 2 apq(\ + x)"(l - x)*, apq>0. (4.2) p + q~2n (' (1 - x^ix)^ = 2 apqf\\ + xr+l(l-xy+ldx. J-l p+q=2n J \

7 SOME INEQUALITIES OF ALGEBRAIC POLYNOMIALS 249 But f (i + xy+\\ - X)q+ldx _ 4(p +l)(q + 1) Ç _ (p + q + 3)(p + q + 2)J-iK Hence f(\ - x2)p2(x)dx ^ a»ap + 0(? + 0 /! =v^(,^+3)(,+,:2)/-/i+^-^ N2/i23W)+2)^.2/J>+^1-^. 2(2»+ 1) ry^ (n+l)(2/l + 3)J-1 "W This proves the lemma. Now we turn to the proof of the theorem. Since \xk\ > 1, k = 1, 2,..., n, it follows from (3.1) and (3.3) that if (1 - *2)P '2(x) x < nfp2(x)dx, (4.3) with equality holding for P (x) = (1 + x^l using (4.1) and (4.3) we obtain x)q, p + q = n. Now, on f(\ - x2)p?(x)dx f(l - x2)p?(x)dx fp2(x)dx f\\ - x2)p2(x)dx fp2(x)dx f(l - x2)p2(x)dx n (n + l)(2/i + 3) 2 ' 2(2/i + 1) This completes the proof of Theorem Proof of Theorem 4. On integrating by parts f00p;'(x)pn(x)e-xxdx = - rp x){p'n(x)x + P (x) - xpn(x)}e-xdx = -f xe-*(p;(x))2dx+±p (0) + \ f"xe-xp2(x)dx - "e-'p^dx. (5.1)

8 250 A. K. VARMA Since xk < 0, k = 1, 2,..., zz, we have / xe-'p^x) 2-1_ ^dx, < Ce-xP2(x) f<*> _...*,» ^ 2 1 dx J0 Zc-l (x - xky xt) jo 'o k-\ x - On using (5.1) and (5.2) and the identity we obtain 00 = / e-*pn(x)p (x)dx 'O 2 l r PM + yoxpn2(x)e-xdx. (5.2) >;2(x) = p (x)p;(x) + p2(x) 2 ' k-\ (x- Xk)2 2 [ P?(x)xe-Xdx < f xe-xp2(x)dx - ^ f "'e-xp2(x)dx. (5.3) 7q '0 ^ '0 It is easy to see that under the conditions of our theorem Hence similarly P2(x) = 2 akxk, ak > 0. k = 0 / e~xp2(x)dx = 2 a*f e-jtx*í c= 2 «**!> '0 *-0 '0 Zc-0 Ce'xxP2n(x)dx = 2 «*(* + 1)! y0 Zc = 0 < (2«+ 1) 2 «***- (2«+ 1) ( e~xp/;(x)dx. Zc = 0 '0 On using (5.3) and the above result it follows that From this Theorem 4 now follows. References 1. P. Erdós, On extremal properties of the derivatives of polynomials, Ann. of Math. (2) 41 (1941), G. G. Lorentz, 77ie degree of approximation by polynomials with positive coefficients, Math. Ann. 151 (1963) P. Turan, Über die Ableitung von Polynomen, Compositio Math. 7 (1939), A. K. Vanna, An analogue of some inequalities of P. Turón, Proc. Amer. Math. Soc. 65 (1976), _, An analogue of some inequalities of P. Turón. II, Proc. Amer. Math. Soc. 69 (1978), Department of Mathematics, University of Florida, Gainesville, Florida 32611