Littlewood-type problems on \([0,1]\). (English) Zbl 1039.11046

Borwein, Peter; Erdélyi, Tamás; Kós, Géza

Proc. Lond. Math. Soc., III. Ser. 79, No. 1, 22-46 (1999).

Let \(p(x)=\sum_{j=0}^na_jx^j\) be a polynomial with complex coefficients \(a_j\) restricted in various ways. For such polynomials \(p(x)\) the authors discuss (i) the number of zeros at \(1\) in the sense of multiplicity, (ii) the Chebyshev problem on \([0,1]\), and (iii) the number of real zeros. Variants of these questions have begun with Bloch, Pólya, Schur, Szegő, Turán, Littlewood, Rudin-Shapiro, Erdős and others. The authors give a survey of the basic results and conjectures in (i)–(iii) and sharpen and generalize some of them.
To illustrate: In (i) they find that there is an absolute constant \(c>0\) such that every polynomial \(p(x)\) with \(| a_j| \leq1\), \(a_j\in\mathbb C\), has at most \(c\sqrt{n(1-\log | a_0| )}\) zeros at \(1\) (\(a_0\) can be replaced by \(a_n\)). This sharpens results of F. Amaroso [Ann. Inst. Fourier 40, 885–911 (1990; Zbl 0713.41004)] and E. Bombieri and J. Vaaler [in: Analytic number theory and diophantine problems, Proc. Conf., Stillwater/ Okla. 1984, Prog. Math. 70, 53–73 (1987; Zbl 0629.10024)]. In (ii) the authors prove that there are absolute constants \(c_1>0\) and \(c_2>0\) such that \(e^{-c_1\sqrt{n}}\leq\inf_{p(x)}\max_{x\in[0,1]}| p(x)| \leq e^{-c_2\sqrt{n}}\), where the infimum is taken over all polynomials \(p(x)\) with \(| a_j| \leq1\), \(a_j\in\mathbb C\), and the first non-zero coefficient of \(p(x)\) has the absolute value \(=1\). In (iii) they prove that there is an absolute constant \(c>1\) such that every polynomial \(p(x)\) with \(| a_j| \leq1\), \(| a_0| =1\), \(a_j\in\mathbb C\) has at most \(c\sqrt{n}\) zeros in \([-1,1]\). This improves the old bound \(c\sqrt{n\log n}\) given by I. Schur [Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl. 1933, No. 7/10, 403–428 (1933; Zbl 0007.00101)] and up to the constant \(c\) it is the best possible result.
The main tools used in proofs are Hadamard’s three-circles theorem and the authors’ theorem: For every analytic function \(f(z)\) on the open unit disk for which \(| f(z)| \leq1/(1-| z| )\) there are absolute constants \(c_1,c_2>0\) such that \(| f(0)| ^{c_1/t}\leq e^{c_2/t}\max_{x\in[1-t,1]}| f(x)| \) for every \(t\in(0,1]\). The authors provide a bibliography with 42 items.

Reviewer: Oto Strauch (Bratislava)

MSC:

11J54	Small fractional parts of polynomials and generalizations
12E10	Special polynomials in general fields
12D10	Polynomials in real and complex fields: location of zeros (algebraic theorems)
26C10	Real polynomials: location of zeros
30C80	Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination

Keywords:

polynomials; complex coefficients; uniform norm; number of zeros; Chebyshev problem

Citations:

Zbl 0713.41004; Zbl 0629.10024; Zbl 0007.00101