Hayman lingham 2018 research problems function theory

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problem_1_25: Erdős's question, in Hayman's collection, whether some meromorphic function has limsup n(r,a)/n(r,b) = infinity and liminf n(r,a)/n(r,b) = 0 for every pair of distinct values a, b, with the 2018 update crediting Gol'dberg and Toppila with entire examples and Toppila with a meromorphic one.

problem_2_16: Erdős's question, in Hayman's collection, whether the number nu(r) of points on |z| = r where an entire function attains its maximum modulus can have limsup or liminf infinite, with the 2018 update crediting Herzog and Piranian with the limsup and recording the liminf as unknown.

problem_2_40: Erdős's question, in Hayman's collection, on the minimum growth of a non-constant entire function with {|f(z)| > c} of finite plane measure, with Hayman's conjecture and the 2018 update crediting Camera, Hansen and Gol'dberg.

problem_2_41: Erdős's question, in Hayman's collection, how slowly the length l(r) of an asymptotic path to infinity in |z| < r can grow for an entire function of finite order, and whether l(r) = O(r) is possible, with the 2018 update recording the negative answer of Gol'dberg and Eremenko.

problem_2_7: A question in Hayman's collection asking what can be said about the length of an asymptotic path to infinity of an entire function of finite order, with the 2018 update reporting Hayman's ray theorem, the Gol'dberg–Eremenko and Toppila counterexamples to linear length, and Chang's bound O(r^{1+rho/2+epsilon}).

problem_4_13: A question in Hayman's collection asking whether a polynomial with coefficients -1 or 1 can satisfy max_{|z|=1} |P(z)| < C_1 sqrt(n) only with C_1 > 1 + A for an absolute constant A > 0, with the 2018 update reporting no progress.

problem_4_14: A question in Hayman's collection asking whether some polynomial with coefficients -1 or 1 has minimum modulus on the unit circle above C_2 sqrt(n) for every n, and whether one can also keep the maximum below C_1 sqrt(n), with the 2018 update recording the coefficient case -1 or 1 as open.

problem_4_15: A question in Hayman's collection asking whether, for large n, all but o(2^n) of the polynomials P(z) = sum_{k=1}^n epsilon_k z^k with epsilon_k = -1 or 1 have n/2 + o(n) roots in the unit disc, with the 2018 update reporting no progress.

problem_4_17: A question in Hayman's collection asking whether the Salem–Zygmund bounds (C_3-epsilon)(n log n)^{1/2} < max_{|z|=1} |P(z)| < (C_4+epsilon)(n log n)^{1/2}, valid apart from o(2^n) polynomials, hold with C_3 = C_4, with the 2018 update crediting Halász with the common value 1.

problem_4_8: A question in Hayman's collection asking whether max |f'(z)| over a connected set {|f(z)| <= 1} is at most n^2/2 for a monic polynomial f of degree n, with the 2018 update recording that Chebyshev polynomials violate it and that Eremenko and Lempert proved the sharp bound 2^{1/n-1} n^2.

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W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (New Edition), arXiv:1809.07200v2 (21 September 2018), 256 pp.; its title page marks it a draft copy, with replies due by 6 January 2019. It expands Hayman, W. K., Research problems in function theory (Athlone Press, London, 1967), vii+56, the edition that Problem 522 cites as [Ha67]. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1809.07200), every other right reserved.

This is a 256-page problem collection in complex analysis organized into nine chapters (meromorphic functions, entire functions, subharmonic and harmonic functions, polynomials, functions in the unit disc, univalent and multivalent functions, miscellaneous, spaces of functions, interpolation and approximation), each problem followed by an 'Update' recording progress since it was first posed. The polynomial chapter collects questions about plus-minus-one polynomials P(z) = sum_{k=1}^n epsilon_k z^k: Problem 4.13 asks whether the constant in max_{|z|=1} |P| < C_1 sqrt(n) must exceed 1 + A; Problem 4.14, on min_{|z|=1} |P| > C_2 sqrt(n), has an Update crediting affirmative answers to Beller and Newman for |epsilon_k| <= 1 and to Körner for |epsilon_k| = 1 and recording the plus-minus-one case as open; Problem 4.17, on the Salem-Zygmund constants for max_{|z|=1} |P|, has an Update crediting Halász with C_3 = C_4 = 1. Problem 4.15 asks whether, for large n, all but o(2^n) of the plus-minus-one polynomials of degree n have just n/2 + o(n) roots in the unit disc D, and the Update records that no progress has been reported. It is a weaker, in-probability form of problem [522], which asks for almost-sure convergence of the root count, with indices from 0 and the closed disc. The copy read for this card is the 2018 Hayman-Lingham expanded draft edition, not the cited 1967 Hayman booklet, so its pages differ from the 1967 original; its Table 2 (p. 253) lists Problems 4.1--4.21 as those of the 1967 booklet, so the polynomial problem numbers above are the 1967 ones.

Source: https://arxiv.org/abs/1809.07200.

Page references are the book's printed page numbers.

Read status: claims checked for Problems 1.25, 2.7, 2.16, 2.40, 2.41, 4.8, 4.13--4.17 and their updates, and for the chapter notation and reference entries they use (statements read clause by clause on the printed pages). The book proves nothing; it poses problems and reports progress. No other problem was read for this card. Nothing here is independently reviewed.

Bears on. #522, as Problem 4.15 (p. 76), the in-probability form of the problem's almost-sure question, for the sum from $k=1$ and roots in $\mathbb{D}$; an affirmative answer to #522 gives one to Problem 4.15, as its page explains. #228, as the "more generally" question of Problem 4.14 (pp. 75--76) read with coefficients $\pm1$, which Update 4.14 records as open in 2018. #230, as Problem 4.13 (p. 75), the same lower-bound question for coefficients $\pm1$ in place of complex coefficients of modulus one; an affirmative answer to #230 would answer it. #523, as Problem 4.17 (p. 76), the same question in the form "apart from $o(2^n)$ polynomials" rather than almost surely; Update 4.17 credits Halász with the common value $1$. #115, as Problem 4.8 (p. 74), the question with the bound $\frac12n^2$, in the setting where Problem 4.7 takes $f$ monic; its update credits Eremenko and Lempert with the sharp bound $2^{1/n-1}n^2$; the problem page cites it for the monic normalization. #1115, as Problem 2.41 (p. 38), whose first paragraph the problem's statement follows and whose update records the Gol'dberg–Eremenko negative answer, and as Problem 2.7 (p. 25), its earlier form, whose update the problem page cites for Toppila's proof and Chang's bound. #1116, as Problem 1.25 (p. 14), whose wording the problem follows; the update credits entire examples to Gol'dberg and Toppila and a meromorphic one to Toppila. #1117, as Problem 2.16 (p. 29), the same two questions; the update credits Herzog and Piranian with the first and records the second as unknown in 2018. #1118, as Problem 2.40 (pp. 37--38), the same two questions, except that the book asks whether $E(c')$ has finite measure for $c'<c$ and the problem asks whether some $c'<c$ has it; the update credits Camera, Hansen and Gol'dberg.

Results.

• Problem 1.25 (p. 14): asks for a meromorphic function with $\limsup n(r,a)/n(r,b)=\infty$ and $\liminf n(r,a)/n(r,b)=0$ for every pair of distinct values; the update credits entire examples to Gol'dberg and Toppila and a meromorphic one to Toppila.
• Problem 2.7 (p. 25): asks what can be said about the length of an asymptotic path to $\infty$ of an entire function of finite order; the update reports Hayman's ray theorem, the Gol'dberg–Eremenko and Toppila counterexamples to $\ell(r)=O(r)$, and Chang's bound $O(r^{1+\rho/2+\varepsilon})$ for the length to the first intersection with $|z|=r$.
• Problem 2.16 (p. 29): asks whether the number $\nu(r)$ of points of maximum modulus on $|z|=r$ can have $\limsup$, or $\liminf$, infinite; the update credits Herzog and Piranian with the $\limsup$ and records the $\liminf$ as unknown.
• Problem 2.40 (pp. 37--38): asks for the minimum growth of a non-constant entire function with $\{|f|>c\}$ of finite measure, with Hayman's conjecture $\int_0^\infty r\,dr/\log\log M(r,f)<\infty$, and whether the measure stays finite for $c'<c$; the update credits Camera, Hansen and Gol'dberg, with a misprinted sharpness condition.
• Problem 2.41 (p. 38): asks how slowly the length $\ell(r)$ in $|z|<r$ of an asymptotic path to $\infty$ can grow for an entire function of finite order, and whether $\ell(r)=O(r)$ is possible; the update records the Gol'dberg–Eremenko negative answer.
• Problem 4.8 (p. 74): asks whether $\max_{E_f^{(n)}}|f'|\le\frac12n^2$ when $E_f^{(n)}=\{|f|\le1\}$ is connected; the update records Eremenko's remark that Chebyshev polynomials violate it and the sharp bound $2^{1/n-1}n^2$ of Eremenko and Lempert.
• Problem 4.13 (p. 75): given that some $\pm1$ polynomial satisfies $\max_{|z|=1}|P(z)|<C_1\sqrt n$ (Clunie), asks whether necessarily $C_1>1+A$ for a positive absolute constant $A$; no progress reported.
• Problem 4.14 (pp. 75--76): asks for a $\pm1$ polynomial with $\min_{|z|=1}|P(z)|>C_2\sqrt n$ for every $n$, and more generally for one that also satisfies the upper bound of Problem 4.13; the update credits affirmative answers to Beller and Newman for $|\varepsilon_k|\le1$ and to Körner for $|\varepsilon_k|=1$, and records the case $\varepsilon_k=\pm1$ as open.
• Problem 4.15 (p. 76): asks whether, for large $n$, all but $o(2^n)$ of the polynomials $\sum_{k=1}^n\varepsilon_kz^k$ with $\varepsilon_k=\mp1$ have $n/2+o(n)$ roots in the unit disc; the update reports no progress.
• Problem 4.17 (p. 76): asks whether the Salem–Zygmund bounds $(C_3-\varepsilon)(n\log n)^{1/2}<\max_{|z|=1}|P(z)|<(C_4+\varepsilon)(n\log n)^{1/2}$, valid apart from $o(2^n)$ polynomials, hold with $C_3=C_4$, and for the common value; the update credits Halász with $C_3=C_4=1$, together with the analogue for trigonometric polynomials.
• Problem 4.16 (p. 76), not paged: asks whether all but $o(2^n)$ of the $\pm1$ polynomials satisfy $\min_{|z|=1}|P(z)|<1$, or, if not, what the correct result is; no progress reported.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.