Function Theory 2
Something Like Picard for 1-Forms
Picard’s great theorem is a statement about how wildly a holomorphic function can behave near an essential singularity. The conjecture below asks whether injectivity of local primitives of a 1-form is enough to rule out such wild behaviour at the origin, forcing the 1-form to extend meromorphically across the puncture. Conjecture (Elsner, 2010) Let $D$ be the open unit disk and let $U_1,\dots,U_n$ be open sets with $\bigcup_{j=1}^n U_j = D\setminus{0}$. Suppose there are injective holomorphic functions $f_j : U_j \to \mathbb{C}$ such that $$\mathrm{d}f_j = \mathrm{d}f_k \quad \text{on every connected component of } U_j \cap U_k.$$ Then the $\mathrm{d}f_j$ glue together to a meromorphic 1-form on $D$.
Criterion for Boundedness of Power Series
Introduction & Problem Statement # Power series constitute one of the most ubiquitous objects in analysis. A power series $\sum_{n=0}^{\infty}a_n x^n$ with infinite radius of convergence defines a real-entire function $f:\mathbb{R}\to\mathbb{R}$. Whereas the question of convergence is completely settled by Cauchy–Hadamard theory, the question of boundedness of the sum function is far subtler and, as of this writing, remains open. Question 1 (Rüdinger, 2009) Let $(a_n) _{n\ge 0}$ be a sequence of real numbers such that the power series $\sum _{n=0}^{\infty}a_n x^n$ converges for every $x\in\mathbb{R}$, thereby defining a smooth function $f:\mathbb{R}\to\mathbb{R}$. Give a necessary and sufficient criterion on $(a_n)$ for $f$ to be bounded on $\mathbb{R}$.