Every projective Oka manifold is elliptic

In the present article, we discuss about the estimate of the pre-Schwarzian and Schwarzian norms for locally univalent harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:\, |z|<1\}$. First, we prove a general result for the estimate of the pre-Schwarzian norm which rectify few earlier flawed results. We also consider a new class $\mathcal{F}_0$ consisting of all harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that ${\rm Re\,}\left(1+z\frac{h''(z)}{h'(z)}\right)>0$ for $z\in\mathbb{D}$ with dilatation $ω_f(z)\in Aut(\mathbb{D})$ and obtain best possible estimates of the pre-Schwarzian and Schwarzian norms for functions in the class $\mathcal{F}_0$. Moreover, we obtain the distortion and coefficient estimates of the co-analytic function $g$ when $f=h+\overline{g}\in\mathcal{F}_0$.

math.CV↗

arXiv2026

The Reciprocal Problem on Weighted Bergman Spaces

The reciprocal problem on weighted Bergman spaces has been posed as an open problem. In this paper, we establish several sufficient conditions for the reciprocal property and clarify the parameter ranges in which the available methods are applicable. In particular, we prove that functions in $A_α^p\cap H^\infty$ enjoy the reciprocal property in the parameter ranges where the required analytic Besov composition theorem is available. In addition, using Hardy boundary estimates, we solve the reciprocal problem in the Drury--Arveson space $H_d^2$ when the dimension is $d=3$, and give an equivalent condition for the reciprocal problem in the four-dimensional Drury--Arveson space.

math.CV↗

arXiv2026

Möbius Maps, Reflections and Lipschitz Constants

We introduce the chordal isometric circle of a Möbius map, and use this to give a factorization of any Möbius map as the composition of a chordal isometry and either a reflection, or a rotary reflection, across a circle. We then use this to find the chordal, and spherical, Lipschitz constants of a Möbius map, and compare this with related results in the literature.

math.CV↗