Related papers: The Schwarzian derivative and the Wiman-Valiron pr…
For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are…
We show that if a meromorphic function has a direct singularity over infinity, then the escaping set has an unbounded component and the intersection of the escaping set with the Julia set contains continua. This intersection has an…
We prove a sharp Schwarz-type lemma for meromorphic functions with spherical derivative uniformly bounded away from zero. As a consequence we deduce an improved quantitative version of a recent normality criterion due to Grahl & Nevo and…
In this note it is shown that two key results on transcendental singularities for meromorphic functions of finite lower order have refinements which hold under the weaker hypothesis that the logarithmic derivative has finite lower order.
For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the…
A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these…
We study transcendental singularities of a Schr\"oder map arising from a rational function $f$, using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex…
This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are…
The Schwarzian derivative plays a fundamental role in complex analysis, differential equations, and modular forms. In this paper, we investigate its higher-order generalizations, known as higher Schwarzians, and their connections to…
Using the Schwarzian derivative we construct a sequence $\left(P_{\ell}\right)_{\ell \geqslant 2}$ of meromorphic differentials on every non-flat oriented minimal surface in Euclidean $3$-space. The differentials…
Quantitative estimates are obtained for the (finite) valence of functions analytic in the unit disk with Schwarzian derivative that is bounded or of slow growth. A harmonic mapping is shown to be uniformly locally univalent with respect to…
We show that the gradient of a multi-variable strongly differentiable function at a point is the limit of a single coordinate-free Clifford quotient between a multi-difference pseudo-vector and a pseudo-scalar, or of a sum of Clifford…
On Riemann surfaces $M$, there exists a canonical correspondence between a possibly multivalued function $\Psi_X$ whose differential is single valued ($i.e.$ an additively automorphic singular complex analytic function) and a vector field…
We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order,…
We argue relations between the Aharonov invariants and Tamanoi's Schwarzian derivatives of higher order and give a recursion formula for Tamanoi's Schwarzians. Then we propose a definition of invariant Schwarzian derivatives of a…
This paper establishes a sharp Schwarz-Pick type inequality for real-valued invariant harmonic functions defined on the complex unit ball $\mathbb B^n$. The proof of this main result simultaneously provides a solution to a natural extension…
Analogues of the key results of Wiman-Valiron theory are proved for a class of functions meromorphic in the unit disc, based on an approach developed by Bergweiler, Rippon and Stallard for the plane setting. The results give local…
In this paper, we investigate shared value problems for shifts and higher-order difference operators of meromorphic and entire functions in several complex variables. Using Nevanlinna theory in $\mathbb{C}^n$, we obtain new uniqueness…
We provide mathematical foundations for the Schwarzian Field Theory as a finite Borel measure on $\mathrm{Diff}^1(\mathbb{T})/\mathrm{PSL}(2,\mathbb{R})$, a quotient of the space of circle reparametrisations. The measure is defined by a…
The existence of subnormal solutions of following three difference equations with Schwarzian derivative $$\omega(z+1)-\omega(z-1)+a(z)(S(\omega,z))^n=R(z,\omega(z)),$$ $$\omega(z+1)\omega(z-1)+a(z)S(\omega,z)=R(z,\omega(z)),$$ and…