(1.1) maxj/'wi <-7-^ n2.

Transcription

1 proceedings of the american mathematical society Volume 83, Number 1, September 1981 DERIVATIVES OF POLYNOMIALS WITH POSriTVE COEFFICIENTS A. K. VARMA Abstract. Let P (x) be an algebraic polynomial of degree n with positive coefficients. We set _ IK(*M*)kjO,oc) In this work upper bounds of / are investigated. We restrict ourselves here with the case a(x) = xa/2e~*/2. Results are shown to be best possible. 1. The problems dealt with in the present paper are connected with the classical inequalities of A. Markov [5], P. Erdos [1], G. G. Lorentz [3], [4], G. Szego [7] P. Turan [8]. First, we describe them briefly. Theorem A (A. A. Markov). If fix) is a polynomial of degree n, such that \f(x)\ < 1/or a < x < b, then (1.1) maxj/'wi <-7-^ n2. a<x<b O a Further, this bound is sharp, as can be seen by the example where Tn is Chebyshev's polynomial. fix) = F (2(* - a)/ (b-a)- 1) For special polynomials, one can sometimes improve the estimate (1.1). Next, two theorems are of this kind. Theorem B (P. Erdos). Let fix) be a polynomial of degree n satisfying the inequality \f(x)\ < 1 for -1 < x < 1. Suppose fix) has only real roots no root inside [-1, +1]; then for -1 < x < 1, (1.2) \f'(x)\<{en. This is the best possible result. Inequalities of the same type (as in Theorem B) hold for the wider class of polynomials with positive coefficients in 1 + x, 1 x; that is, for the polynomials (1.3) fix) = 2 akl(\ + x)k(\ - x)', akl > 0, -1 < x < 1. k + l<n These polynomials were introduced by G. G. Lorentz [3] studied extensively by J. T. Scheick [6]. The Lorentz theorem can be stated as follows. Received by the editors April 21, Mathematics Subject Classification. Primary 26A60; Secondary 30A40. American Mathematical Society /81/ /S02.50

2 108 A. K. VARMA Theorem C (G. G. Lorentz). There exists a constant C > 0 such that for each polynomial fix) of the form (1.3), (1.4) Jjjflf < Cn, n = 1, 2,..., for the uniform norm on [-1, +1]. In 1964 G. Szego [7] studied the order of magnitude of /' / / polynomials/of degree < n for the norm (1.5) 11/11 = sup \f(x)e-*\ x>0 on (0, oo ). More precisely, he proved the following. for unrestricted Theorem D (G. Szego). Let fix) be a polynomial of fixed degree n not vanishing identically. Then (1.6) Jr<Cn> «= 2, 3,..., holds (for the weighted uniform norm as defined by (1.5) on (0, oo)). In 1968 G. G. Lorentz [4] considered the problem of G. Szego for the special polynomials with positive coefficients in x, (1.7) fix) = 2 akxk, ak > 0, *=o the norm of a function/on (0, oo) is given by / = sup \fix)e^*\ x>0 where to increases on (0, oo). Then the Lorentz theorem can be stated as follows. Theorem E (G. G. Lorentz). Let w(x) satisfy the inequalities u(x) u(0) < Axo)'(x), x > 0, u'(y) < Au'(x),y < x,for some constant A > 0. Then for some constant C > 0, the inequality is valid. ii/ii.jif; <C7r 7, pn(x) = x", Other related interesting results are due to A. Zygmund [10], Hille, Szego Tamerkin [2] P. Turan [8], where similar problems are considered either in L2 norm or even in Lp norm. We may note that Lorentz raised the problem of determining the best constant in (1.6) for (1.9). Now we state the main theorem of this paper. Let S [a, b] be the set of all polynomials whose degree is n whose roots are all real no root inside (a, b). It is easy to see that if pn E S [0, oo) then it can be expressed in the form n (1.8) P (x) = 2 akxk, ak>0fork = 0,\.n. k = 0

3 DERIVATIVES OF POLYNOMIALS 109 We now state Theorem 1. Let P (x) be an algebraic polynomial of degree n with nonnegative coefficients. Then for a > (VS l)/2 (1.9) r(p^x))2x"e-' dx < f- rp2(x)x"e-x dx, (2n + a)(2n + a \) Jq equality holding for P (x) = x". For 0 < a < \ we have (1.10) C(P'n(x))2x«e-* dx < i- rp2(x)x"e-x dx. (2 + «)(1 + a) ^o Moreover (1.10) «also best possible in the sense that for Pn(x) = x" + Xx the expression on the left can be made arbitrarily close to the expression on the right-h side by choosing X positive sufficiently large. Corollary. Let Pn(x) be an algebraic polynomial of degree n with nonnegative coefficients satisfying (1.8) with an > 0. Then for a > (VS l)/2 we obtain (1.11) f P2(x)xae-X dx > (an)2t(2n + a + 1), equality holding for P (x) = x" (an = 1). Proof of this Corollary is as follows. Repeated application of the inequality (1.9) gives at once (1.12) / (Pk(x)) xae-x dx T,- / p2ax)> )xae-x dx n2(n -\Y- (n - k +1)' < (2n + a)(2n + a - 1) (In + a - 2(k - l))(2n + a - 1-2(k - 1)) ' Next, we note that (1.13) r(p<r\x)fxae-* dx = (a )2(«!)2r(a + 1). Now, we substitute k = n in (1.12) use (1.13) which leads to (1.11). 2. Proof of Theorem 1. Let Pn(x) be any algebraic polynomial of degree n with positive coefficients. Then we may write n-\ (2.1) Pn(x) = anx» + Pn_x(x), P _x(x) = 2 akxk, ak > 0. k-0 Using (2.1) (2.2) [ xke-x dx = T(k + 1)

4 110 A. K. VARMA we obtain (2.3) r(p;(x))2e-*x" dx = C{PÍ_ x)fe-'x' (2.4) fcc(p (x))2e'xxa dx = ö2r(2n + a + 1) 'o Next we set (2.5) K = dx + a2n2t(2n + a - 1) + 2na ("i»1i_i(je)**+«"v" dx, + i"(^-iw)vxx" î/x + 2a f F _1(x)e-j:x',+a x. '0 'o n2 " (2n + a)(2n + a - 1) (2.6) A = 2«r 0Fn'_1(x)x"+a-1e-JC x - 2Z» f P,I_,(x)x',+ae-JC dx. From (2.3)-(2.6) we may conclude that r (P '(x))v*x«dx - bn r(pn(x))2e-xx" (2.7) = \,fl + f V '-,(*))V*x«dx-bn C{Pn-X(x))2e-xx«dx. J0 '0 First, we claim that for a > 0 (2.8) \, < 0, n = 1, 2,.... From (2.1), (2.2) (2.6) we obtain dx (2-9) \ = *"-rr 2 %ftb,r(* + n + a - 1), at > 0, (2n + a)(2n + a - 1) ^q where ikn = (2n + a)(2n + a - l)k - n(k + n + a)(k + n + a 1) = (k - n){n(n - k) + (2a - l)n + a(a - I)], 0 < k < n - 1. Clearly for a > 0 we have (2.10) ju^ < -a(2«+ a - 1) < 0, k = 0, 1,..., n - 1. Thus (2.8) follows easily from (2.9) (2.10). We also note that r- Vs - i (2.11) bn>bn_x, a>-, «=2,3,..., (2.12) b <b _x, 0<a<-, «= 2,3.

5 DERIVATIVES OF POLYNOMIALS 111 Using these ideas we can complete the proof of Theorem 1. For a > (VI - l)/2 we use (2.7), (2.8) (2.11) we obtain foi k = 2, 3,..., n, f >*'(*))V*x" dx - bk fœ(pk(x))2e-xxa dx (2.13) /.OO, /.OO - </ (Pk-i(x))2e-xx"dx-bk_xf (Pk_x(x))2e-Xxa dx. Adding all these equations we obtain (2 141 r(p x)fe-xx dx - bn[x(p (x))2e-xxa dx But a simple computation < (X(P'x(x)fe-xxa dx- bx (X(Px(x))2e-xxa dx. shows that if Px(x) = axx + a0, ax > 0, a0 > 0 then (2.15) C(P[(x)fe-xxa dx < bx C\px(x)fe-xxa dx, equality holds if Px(x) = axx, ax > 0. From (2.14) (2.15) we obtain (1.2) as desired for a > (VI - l)/2. For the proof of (1.3) for 0 < a < \ we use (2.7), (2.8) (2.12). From these results it follows that r(pi(x)fe-xxa dx - bk r(pk(x))2e-*xa dx (2.16) < /o (^-iw)2^^a dx - bk_xj\pk_x(x))2e-xx» dx Also from (2.4) it follows that + (V, - K) r{pk-x(x)fe-xxa dx. (2.17) fxp2_x(x)xae-xdx< (X(Pk(x))2xae-x dx. J<3 From (2.16) (2.17) we obtain for k = 2, 3,..., «1, «, r{pl(x))2e-xx» dx - bkr(pk(x))2e-xx dx (2.18) < f >;_,W)Vx dx - bk_x r(pk_x(x))2e-xx* dx + ( *-i - bk)r(pn(x))2e-xx" dx. We add all these equations for k = 2, 3,...,«r(P;(x)fe-xx dx - bnr(pn(x))2e-xx«dx 1, «we obtain < (X(P'x(x))2e-xxa dx- bx ÇX(Px(x))2e-xxa dx (2.19) ^ m J + (*. - b ) r{pn(x))2e-xxa dx Jn <(bx-bn)fœ(pn(x))2e-xx"dx. Jn

6 112 A. K. VARMA But clearly (2.18) is equivalent to (2.20) ( (P (x))2e-xxa dx < bx Ç\pn(x))2e-xxa dx. From (2.19) (2.5) follows (1.3). In order to show that (1.3) is best possible we consider f0(x) = x" + Xx, X > 0, 0 < a < \. A simple computation shows that where Ç{fo(x))2e-Xx dx r(f0(x))2e-xx j dx = («+ 2)(«+ 1) " Cn*' 2Af(«+ a)[a(a + 1) + «2 - n(a2 + a - I)] + d Q,A " (a + 2)(a + l)[a2r(a + 3) + 2Ar(«+ a + 2) + T(2«+ a + 1)] dn = T(2«+ a - 1)[«2(2-3a - a2) + 2n(2a - 1) + a(a - 1)]. Clearly CnJ< * 0 as X * oo from which our assertion follows. References 1. P. Erdos, Extremal properties of derivatives of polynomials, Ann. of Math. (2) 41 (1940), E. Hille, G. Szego J. D. Tamerkin, On some generalizations of a theorem of A. A. Markhoff, Duke Math. J. 3 (1937), G. G. Lorentz, 77ie degree of approximation by polynomials with positive coefficients, Math. Ann. 151 (1963), _, Derivatives of polynomials with positive coefficients, J. Approx. Theory 1 (1968), A. A. Markov, On aproblem of D. I. Mendeleev, Izv. Akad. Nauk 62 (1889), J. T. Scheick, Inequalities for derivatives of polynomials of special type, J. Approx. Theory 6 (1972), G. Szego, On some problems of approximations, Magyar Tud. Akad. Mat. Kutato Int. Dozl. 2 (1964), P. Turan, Remarks on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960), A. K. Vanna, Some inequalities of algebraic polynomials having real zeros, Proc. Amer. Math. Soc. 75 (1979), A. Zygmund, A remark on the conjugate series, Proc. London Math. Soc. 36 (1932), Department of Mathematics, University of Florida, Gainesville, Florida 32611