arXiv subjects

Sina Nadi

Publications and source records attributed to Sina Nadi.

3 recordsLinked to original sources

arXiv2026

Exceptional Sets of Positive Capacity in Gross's Star Theorem

In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. In 1977, Nagasaka asked whether this exceptional set must have logarithmic capacity zero. We prove that there exist a transcendental entire function and a regular local inverse for which the exceptional set contains a compact set of positive logarithmic capacity. In fact, for every $0<s<1$, such a function and inverse can be chosen so that this compact set has positive $s$-dimensional Hausdorff measure.

math.CV↗

arXiv2026

Sharp Deficiency Bounds for Meromorphic Functions in the Unit Disc

In 1986, Shea and Sons obtained the following bound for a meromorphic function $f$ of finite order $ρ$ in the unit disc, under the hypothesis $0<λ(f)\leq+\infty$, and for every positive integer $n$: \[ \sum_{a\ne\infty}δ(a,f) \leq δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n(ρ+1)}{λ(f)}. \] Under the condition $0<α(f)\leq+\infty$, they also obtained \[ \sum_{a\ne\infty}δ(a,f) \leq Δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{2n}{α(f)}. \] Shea and Sons asked whether the factor $2$ could be eliminated. We prove that it can. In fact, whenever $0<λ(f)\leq+\infty$, one has \[ \sum_{a\ne\infty}δ(a,f) \leq δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n(ρ+1)}{λ(f)}, \] and under the condition $0<α(f)\leq+\infty$ one has \[ \sum_{a\ne\infty}δ(a,f) \leq Δ(0,f^{(n)})\bigl(1+n k(f)\bigr) +\frac{n}{α(f)}. \] The coefficient $n$ in each of $n(ρ+1)/λ(f)$ and $n/α(f)$ is best possible for every $n$, and the dependence on $ρ$ and $α(f)$ in these terms is also essential. At the end, we extend both estimates to finite subsets $A$ of an $n$-dimensional complex vector space $V\subset\mathbb{C}(z)$, with $f^{(n)}$ replaced by the monic differential operator $D_Vf$ of order $n$ whose kernel is $V$. We prove that both extensions are sharp for every $V$.

math.CV↗

arXiv2026

Entire Functions Mapping Countable Dense Subsets of $\mathbb{R}$ onto Countable Dense Subsets of $\mathbb{C}$

In 1977, Karl F. Barth posed the following problem: given countable dense sets $A\subset\mathbb{R}$ and $B\subset\mathbb{C}$, does there exist a transcendental entire function $f$ such that $f(A)=B$ and $f(\mathbb{R}\setminus A)\subset\mathbb{C}\setminus B$? We review results related to this question and prove that there exist transcendental entire functions $f$ such that $f\restriction_A\colon A\to B$ is bijective, $f^{-1}(B)\cap\mathbb{R}=A$, and $f'(a)\neq0$ for every $a\in A$. In fact, the set of such functions has the cardinality of the continuum. At the end, we give two extensions of the result, one for countably many pairwise disjoint pairs of dense sets and one with $\mathbb{R}$ replaced by a closed unbounded subset of $\mathbb{C}$ of planar Lebesgue measure zero.

math.CV↗