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Shapiro's problem on polynomials with large partial sums of coefficients

Marc Technau

Abstract

Given a polynomial $\sum_νa_νX^ν$ of degree $<d$, bounded by one on the unit disk, how large can $\lvert a_0+a_1+\ldots+a_n \rvert$ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on this theme, obtaining exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that $n$ is not too large in terms of $d$. The latter is achieved via a quantitative Eneström--Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $\lvert t_0 a_0 + \ldots + t_{d-1} a_{d-1} \rvert$ for arbitrary complex numbers $t_0,\ldots,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro's problem for some choices of $n$ and $d$.

Shapiro's problem on polynomials with large partial sums of coefficients

Abstract

Given a polynomial of degree , bounded by one on the unit disk, how large can () get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for , but there does not seem to have been further progress on the question since. We study variations on this theme, obtaining exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that is not too large in terms of . The latter is achieved via a quantitative Eneström--Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums for arbitrary complex numbers . This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro's problem for some choices of and .

Paper Structure

This paper contains 19 sections, 24 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Approximate formulæ
  4. Exact formulæ
  5. A general bound
  6. Plan of the paper
  7. Polynomials and Landau's bound
  8. Landau's argument
  9. Reflections on Landau's argument
  10. Asymptotic estimates
  11. Proof of Theorem \ref{['thm:Quantitative']}
  12. A quantitative Eneström--Kakeya theorem
  13. Growth of the Taylor coefficients of Landau's extremal functions
  14. Remark on the zeros of Pn
  15. Interpolation with roots of unity
  16. ...and 4 more sections

Key Result

Theorem 1.2

Let $n$ be a non-negative integer. Then, for any holomorphic function $f$ on $\mathbb{D}$ with $\lVert f\rVert_\infty \leq 1$ and power series expansion $f(z) = a_0 + a_1 z + a_2 z^2 + \ldots$, one has where Moreover, '$\leq$' in eq:LandausBound is attained with equality precisely for the rational function $f_n$ given by and the functions $\eta f_n$, where $\eta$ is any unimodular constant.

Figures (2)

  • Figure 1: A numerical experiment showing when cor:NewmanGeneralised:Combined produces stronger upper bounds on the maximum $\mathscr{M}_{n,d}$ from prob:Shapiro than previously known (cf. example:NumericalExperiment). The plot shows the points $(d,n)$, $0\leq n<d\leq75$, for which the upper bound eq:NewmanGeneralised:Combined furnished by cor:NewmanGeneralised:Combined for $\boldsymbol{t} = (1_{\times (n+1)},0_{\times(d-n-1)}) \in \mathbb{C}^d$ is strictly smaller than the bound $\mathscr{L}_n$ for $\mathscr{M}_{n,d}$ coming from Landau's theorem (see eq:PolHolComparison)
  • Figure 2: Zeros (black dots) and poles (white dots) of $f_n$ from eq:LandausBound:Extremizer for $n\in \lbrace4,7,17\rbrace$. In each picture the two circles are centred about $0$ and have radii $1$ and $1 + 1/(2n+1)$, respectively

Theorems & Definitions (50)

  • Theorem 1.2: Landau, 1913
  • Remark
  • Theorem 1.3: Newman, 1978
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Remark 2.4: Improvements to the error terms
  • Theorem 2.5
  • Remark
  • Theorem 2.6
  • ...and 40 more