Related papers: The Schwarzian derivative and polynomial iteration
We relate in this note the classical Schwarzian derivative to the curvature of time-like curves in Lorentz surfaces of constant curvature.
In this paper, we study delay differential equations involving the Schwarzian derivative $S(f,z)$, expressed in the form \begin{equation*} f(z+1)f(z-1) + a(z)S(f,z) =R(z,f(z))= \frac{P(z,f(z))}{Q(z,f(z))} \end{equation*} where $a(z)$ is…
The Schwarzian derivative of a function f(x) which is defined in the interval (a, b) having higher order derivatives is given by Sf(x)=(f''(x)/f'(x))'-1/2(f''(x)/f'(x))^2 . A sufficient condition for a function to behave chaotically is that…
Motivated by the effective impact of the Pascal functional and the Wronskian matrices, we investigate several identities and differential equation for the Sheffer-Appell polynomial sequence by using matrix algebra. The matrix approach,…
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…
In this note we present an application of the Schwarzian derivative. By exploiting some properties of the Schwarzian derivative, we solve the equation appearing in the gravity-dilaton-antisymmetric tensor system. We also mention that this…
We gain further insight into the use of the Schwarzian derivative to obtain new results for a family of functional differential equations including the famous Wright's equation and the Mackey-Glass type delay differential equations. We…
We give a simple formula for some determinants, and an analogous formula for pfaffians, both of which are polynomial identities. The second involve some expressions that interpolate between determinants and pfaffians. We give several…
In this paper we study a multiplier operator which is induced by the Schwarzian derivative of univalent functions with a quasiconformal extension to the extended complex plane. As applications, we show that the Brennan conjecture is…
We sharpen the bound $n^{2k}$ on the maximum modulus of the $k^{{\rm th}}$ radial derivative of the Zernike circle polynomials (disk polynomials) of degree $n$ to $n^2(n^2-1^2)\cdot ... \cdot(n^2-(k-1)^2)/2^k(1/2)_k$. This bound is obtained…
In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=sE_4(\tau)$ where $E_4(\tau)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit…
In this paper we introduce a definition of the pre-Schwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping $f$ in the complex plane without assuming any additional condition on the (second complex) dilatation…
Let $(M,F)$ be a Finsler manifold. We construct a 1-cocycle on $\Diff(M)$ with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function $F.$ As an operator, it has several…
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…
We obtain exact expressions for a general class of correlation functions in the 1D quantum mechanical model described by the Schwarzian action, that arises as the low energy limit of the SYK model. The answer takes the form of an integral…
Consider a sequence of meromorphic functions $(f_n)_n$. This paper presents a technique that enables the transfer of convergence properties from $(f_n^{(m+1)}/f_n^{(m)})_n$ to subsequences of $(f_n^{(m)}/f_n^{(m-1)})_n$. As an application,…
In the study of one dimensional dynamical systems one often assumes that the functions involved have a negative Schwarzian derivative. In this paper we consider a generalization of this condition. Specifically, we consider the interval…
Given a sheaf on a projective space P^n we define a sequence of canonical and easily computable Chow complexes on the Grassmannians of planes in P^n, generalizing the Beilinson monad on P^n. If the sheaf has dimension k, then the Chow form…
Because of all the known integrable models possess Schwarzian forms with M\"obious transformation invariance, it may be one of the best way to find new integrable models starting from some suitable M\"obious transformation invariant…
In this article, a new approach based on linear algebra is adopted to study a hybrid Sheffer polynomial sequences. The recurrence relations and differential equation for these polynomials are derived by using the properties and…