Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let

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1 HOW FAR IS AN ULTRAFLAT SEQUENCE OF UNIMODULAR POLYNOMIALS FROM BEING CONJUGATE-RECIPROCAL? Tamás Erdélyi Abstract. In this paper we study ultraflat sequences (P n) of unimodular polynomials P n K n in general, not necessarily those produced by Kahane in his paper [Ka]. We examine how far is a sequence (P n) of unimodular polynomials P n K n from being conjugate reciprocal. Our main results include the following. Theorem. Given a sequence (ε n) of positive numbers tending to 0, assume that (P n) is a (ε n)-ultraflat sequence of unimodular polynomials P n K n. The coefficients of P n are denoted by a k,n, that is, Then P n(z) = nx nx a k,n z k,,k =0,1,...,n, n=1,2,.... k 2 a k,n a n k,n δn n 3, where (δ n) is a sequence of real numbers converging to Introduction Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by. Let { K n := p n : p n (z) = } a k z k, a k C, a k =1. The class K n is often called the collection of all (complex) unimodular polynomials of degree n. Let { L n := p n : p n (z) = } a k z k, a k { 1,1} Mathematics Subject Classification. 41A17. Key words and phrases. unimodular polynomials, ultraflat polynomials, angular derivatives. Research supported in part by the NSF of the USA under Grant No. DMS Typeset by AMS-TEX

2 2 TAMÁS ERDÉLYI The class L n is often called the collection of all (real) unimodular polynomials of degree n. By Parseval s formula, for all P n K n. Therefore 2π 0 P n (e it ) 2 dt =2π(n+1) (1.1) min z P n(z) < n +1< max z P n(z). An old problem (or rather an old theme) is the following. Problem 1.1 (Littlewood s Flatness Problem). How close can a unimodular polynomial P n K n or P n L n come to satisfying (1.2) P n (z) = n +1, z? Obviously (1.2) is impossible if n 1. So one must look for less than (1.2), but then there are various ways of seeking such an approximate situation. One way is the following. In his paper [Li1] Littlewood had suggested that, conceivably, there might exist a sequence (P n ) of polynomials P n K n (possibly even P n L n ) such that (n +1) 1/2 P n (e it ) converge to 1 uniformly in t R. We shall call such sequences of unimodular polynomials ultraflat. More precisely, we give the following definition. Definition 1.2. Given a positive number ε, we say that a polynomial P n K n is ε-flat if (1.3) (1 ε) n +1 P n (z) (1 + ε) n +1, z, or equivalently max Pn (z) n+1 ε n+1. z Definition 1.3. Given a sequence (ε nk ) of positive numbers tending to 0, wesay that a sequence (P nk ) of unimodular polynomials P nk K nk is (ε nk )-ultraflat if (1.4) (1 ε nk ) n k +1 P nk (z) (1 + ε nk ) n k +1, z, or equivalently max Pnk (z) n k +1 εnk nk +1. z The existence of an ultraflat sequence of unimodular polynomials seemed very unlikely, in view of a 1957 conjecture of P. Erdős (Problem 22 in [Er]) asserting that, for all P n K n with n 1, (1.4) max z P n(z) (1 + ε) n +1,

3 ULTRAFLAT POLYNOMIALS 3 where ε>0 is an absolute constant (independent of n). Yet, refining a method of Körner [Kö], Kahane [Ka] proved that there exists a sequence (P n )withp n K n which is (ε n )-ultraflat, where ( (1.5) ε n = O n 1/17 ) log n. Thus the Erdős conjecture (1.4) was disproved for the classes K n. For the more restricted class L n the analogous Erdős conjecture is unsettled to this date. It is a common belief that the analogous Erdős conjecture for L n is true, and consequently there is no ultraflat sequence of polynomials P n L n. An extension of Kahane s breakthrough is given in [Be]. For an account of some of the work done till the mid 1960 s, see Littlewood s book [Li2] and [QS]. 2. New Results In this paper we study ultraflat sequences (P n ) of unimodular polynomials P n K n in general, not necessarily those produced by Kahane in his paper [Ka]. With trivial modifications our results remain valid even if we study ultraflat sequences (P nk ) of unimodular polynomials P nk K nk. It is left to the reader to formulate these analogue results. We examine how far an ultraflat sequence (P n ) of unimodular polynomials P n K n is from being conjugate reciprocal. Our main results are formulated by the following theorems. In each of Theorems we assume that (ε n ) is a sequence of positive numbers tending to 0, and the sequence (P n )of unimodular polynomials P n K n is (ε n )-ultraflat. If Q n is a polynomial of degree n of the form Q n (z) = a k z k, a k C, then its conjugate polynomial is defined by Theorem 2.1. We have Q n (z) :=zn Q n (1/z) := a n k z k. ( P n (z) P n (z) )2 dz =2π where (γ n ) is a sequence of real numbers converging to γ n n 3, Theorem 2.2. If the coefficients of P n are denoted by a k,n, that is P n (z) = a k,n z k, k =0,1,...,n, n=1,2,..., then k 2 a k,n a n k,n δ n n 3, where (δ n ) is a sequence of real numbers converging to 0.

4 4 TAMÁS ERDÉLYI Theorem 2.3. We have P n (z) Pn(z) 2 dz 2π 3 +γ n n, where (γ n ) is a sequence of real numbers converging to 0. Using the notation of Theorem 2.2, in terms of the coefficients of P n, we have a k,n a n k,n δ n n, where (δ n ) is a sequence of real numbers converging to 0. Remark 2.4 Theorem 2.4 tells us much more than the non-existence of an ultraflat sequence of conjugate reciprocal unimodular polynomials. It measures how far such an ultraflat sequence is from being a sequence of conjugate reciprocal polynomials. 3. Lemmas To prove the theorems in Section 2, we need two lemmas. The first one can be checked by a simple calculation. Lemma 3.1. Let P n be an arbitrary polynomial of degree n with complex coefficients having no zeros on the unit circle. Let f n (z) := zp n(z) P n (z) and f n (z) := zp n (z) P n (z). Then f n (z)+f n (z)=n, z. Our next lemma may be found in [MMR] (page 676) and is due to Malik. Lemma 3.2. Let P n be an arbitrary polynomial of degree n with complex coefficients. We have max z ( P n (z) + P n (z) ) n max P n(z). z Lemma 3.3 (Bernstein s Inequality in L 2 ()). If Q n degree at most n with complex coefficients, then Q n (z) 2 dz n 2 Q n (z) 2 dz. is a polynomial of

5 ULTRAFLAT POLYNOMIALS 5 4. Proof of the Theorems Proof of Theorem 2.1. Lemma 3.2 combined with the ultraflatness of (P n ) implies that P n (z) + Pn (z) nmax P n(z) (1 + ε n )(n +1) 3/2 z for every z. Lemma 3.1 combined with the ultraflatness of P n imply P n (z) 1 (1 ε n ) n +1 + P n (z) 1 (1 ε n ) n +1 P n (z) P n (z) + P n (z) n, (z) that is for every z. We conclude that P n(z) + P n (z) (1 ε n )n 3/2 P n (1 ε n ) 2 n 3 ( P n(z) + P n (z) ) 2 (1 + ε n ) 2 (n +1) 3, z. Multiplying the expression in the middle out and integrating on with respect to dz, we obtain 2π(1 ε n ) 2 n 3 Note that (2.1) Hence P n (z) 2 dz + Pn (z) 2 dz +2 P n (z)p n (z) dz 2π(1 + ε n ) 2 n 3. P n (z) 2 dz = P n (z) 2 dz =2π k=1 k 2 n(n+ 1)(2n +1) =2π 2π 6 3 n3. ( ) P n 1 (z) Pn (z) dz =2π 6 +δ n n 3 with constants δ n converging to 0. Integrating the equation ( P n(z) P n (z) ) 2 = P n(z) 2 + P n (z) 2 2 P n(z)p n (z), and using observation (2.1) we obtain the theorem. Proof of Theorem 2.2. Parseval Formula and the triangle inequality give k 2 a k,n a n k,n 2 = P n(z) P n (z) 2 dz ( P n (z) P n (z) )2 dz, and the theorem follows from Theorem 2.2.

6 6 TAMÁS ERDÉLYI Proof of Theorem 2.3. Applying Theorem 2.1, the triangle inequality, and the Bernstein inequality in L 2 for P n Pn (see Lemma 3.3), we obtain ( ) 1 2π 3 + γ n n 3 = ( P n (z) P n (z) )2 dz P n (z) P n (z) 2 dz n 2 P n (z) Pn (z) 2 dz, where (γ n ) is a sequence of real numbers converging to 0. Now the first part of the theorem follows after dividing by n 2. To see the second part we proceed as in the proof of Theorem 2.2 by using Parseval s formula. 5. Acknowledgment. I thank Peter Borwein for many discussions related to the topic. References [Be] [BE] [DL] [Er] [Ka] J. Beck, Flat polynomials on the unit circle note on a problem of Littlewood, Bull. London Math. Soc. (1991), P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, R.A. DeVore and G.G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin, P. Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), Berlin. J.P. Kahane, Sur les polynomes a coefficient unimodulaires, Bull. London Math. Soc. 12 (1980), [Kö] T. Körner, On a polynomial of J.S. Byrnes, Bull. London Math. Soc. 12 (1980), [Li1] [Li2] [MMR] [Sa] [QS] J.E. Littlewood, On polynomials P ±z m, P exp(α m)z m,z = e iθ., J. London Math. Soc. 41, , yr J.E. Littlewood, Some Problems in Real and Complex Analysis, Heath Mathematical Monographs, Lexington, Massachusetts, Milovanović, G.V., D.S. Mitrinović, & Th.M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, B. Saffari, The phase behavior of ultraflat unimodular polynomials, in Probabilistic and Stochastic Methods in Analysis, with Applications (1992), Kluwer Academic Publishers, Printed in the Netherlands. H. Queffelee and B. Saffari, On Bernstein s inequality and Kahane s ultraflat polynomials, J.F.A.A. vol. 2 (1996), Department of Mathematics, Texas A&M University, College Station, Texas 77843

7 address: ULTRAFLAT POLYNOMIALS 7

THE RESOLUTION OF SAFFARI S PHASE PROBLEM. Soit K n := { p n : p n (z) = n

Comptes Rendus Acad. Sci. Paris Analyse Fonctionnelle Titre français: Solution du problème de la phase de Saffari. THE RESOLUTION OF SAFFARI S PHASE PROBLEM Tamás Erdélyi Abstract. We prove a conjecture