Open-problem statements, with their sources attached.

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$, $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii is $≤ 2$?

Let f be a transcendental entire function. What is the greatest possible value of liminf (fun r : ℝ => ratio r f) atTop?

Is it true that for all entire functions f = ∑ aₖzⁿₖ such that ∑' 1 / nₖ < ∞, limsup (fun r => ratio r f) atTop = 1?

If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?

Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense.

Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant.

In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value of the Landau constant.

Brennan's conjecture, part 1: $B(-2) = 1$.

Brennan's conjecture, part 2: $B_b(-2) = 1$.

Sendov's conjecture* states that for a polynomial $f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)$ with all roots $r_1, ..., r_n$ inside the closed unit disk $|z| ≤ 1$, each of the $n$ roots is at a distance no more than $1$ from at least one critical point.