Related papers: Meromorphic functions with small Schwarzian deriva…

The index of a meromorphic function $g$ on a compact Riemann surface is an invariant of $g$, which is defined as the number of negative eigenvalues of the differential operator $L:=-{\Delta}-|dG|^2$, where ${\Delta}$ is the Laplacian with…

In this work we consider a family of function classes constructed by means of the Gauss hypergeometric function $_2F_1(1,1;2;z) =-\frac{\log(1-z)}{z}$. We demonstrate that this family, in fact, constitutes classes of analytic functions…

We define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f.We then…

Let $f$ be a zero-free analytic function on $\Re(s) \geq 1$. We prove that there exists an entire zero-free function $g$ and a Helson zeta-function $\zeta_\chi(s)=\sum_{n=1}^\infty \chi(n) n^{-s}$, where $\chi(n)$ is a completely…

The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{\varphi…

Let $\mathcal{S}^*(\alpha_1,\alpha_2)$, where $ \alpha_1, \alpha_2 \in (0,1]$, represent the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$, normalized by $f(0) = f'(0) - 1=0$, and satisfying the following…

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is…

In 1984, Gehring and Pommerenke proved that if the Schwarzian derivative $S(f)$ of a locally univalent analytic function $f$ in the unit disk satisfies that $\limsup_{|z|\to 1} |S(f)(z)| (1-|z|^2)^2 < 2$, then there exists a positive…

Let $ \mathcal{S}(p) $ be the class of all meromorphic univalent functions defined in the unit disc $ \mathbb{D} $ of the complex plane with a simple pole at $ z=p $ and normalized by the conditions $ f(0)=0 $ and $ f^{\prime}(0)=1 $. In…

In this paper we prove the result: Let $\mathcal{F}$ be a family of meromorphic functions on a domain $\Omega$ such that every pair of members of $\mathcal{F}$ shares a set $S:=\left\{\psi_1(z), \psi_2(z), \psi_3(z) \right\}$ in $\Omega$,…

Let $M$\/ be a subharmonic function with Riesz measure $\mu_M$ on the unit disk $\mathbb D$ in the complex plane $\mathbb C$. Let $f$ be a nonzero holomorphic function on $\mathbb D$ such that $f$ vanishes on ${\sf Z}\subset \mathbb D$, and…

We prove that the Szeg\H{o} function, $D(z)$, of a measure on the unit circle is entire meromorphic if and only if the Verblunsky coefficients have an asymptotic expansion in exponentials. We relate the positions of the poles of $D(z)^{-1}$…

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the…

In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[0,+\infty)$, $f$ is twice differentiable on…

For $0<\lambda\le 1$, let $\mathcal{U}(\lambda)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<\lambda$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present…

Marx and Strohh\"acker showed around in 1933 that $f(z)/z$ is subordinate to $1/(1-z)$ for a normalized convex function $f$ on the unit disk $|z|<1.$ Brickman, Hallenbeck, MacGregor and Wilken proved in 1973 further that $f(z)/z$ is…

We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order $m_0(f)$ and their convergence exponent $d(f)$. When $m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the…

Let F and G be two families of meromorphic functions on a domain D, and let a, b and c be three distinct points in the extended complex plane. Let G be a normal family in D such that all limit functions of G are non-constant. If for each f…

It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{\Delta_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $\Delta_c f=f(z+c)-f(z)$ and…

The logarithmic coefficients $\gamma_n$ of an analytic and univalent function $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…