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

##### 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
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