Related papers: On Becker's univalence criterion

In this note, we consider meromorphic univalent functions $f(z)$ in the unit disc with a simple pole at $z=p\in(0,1)$ which have a $k$-quasiconformal extension to the extended complex plane $\hat{\mathbb C},$ where $0\leq k < 1$. We denote…

In this note we consider some generalizations of the Schwarz lemma for harmonic functions on the unit disk, whereby values of such functions and the norms of their differentials at the point $z=0$ are given.

Let $\mathcal{A}$ denote the class of analytic functions $f$ such that $f(0)=0$ and $f'(0)=1$ in the unit disk $\mathbb{D}:=\{z \in \mathbb{C}: |z|<1\}.$ We examine the properties of the class $\mathcal{C}(\varphi)$ defined as…

The primary objective of this paper is to establish several sharp versions of Bohr inequalities for bounded analytic functions in the unit disk $\mathbb{D} := \{z\in\mathbb{C} : |z| < 1\}$ involving multiple Schwarz functions. Moreover, we…

The principal goal of this paper is to extend the classical problem of find the values of $\alpha\in \C$ for which the mappings, either $F_\alpha(z)=\int_0^z(f(\zeta)/\zeta)^\alpha d\zeta$ or $f_\alpha(z)=\int_0^z(f'(\zeta))^\alpha d\zeta$…

In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class $\mathcal{A}$ to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these…

Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general…

We obtain Schwarz-Pick lemma for $(\alpha, \beta)$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\Re \alpha + \Re \beta > -1$. We prove sharp estimate of derivative at the origin for such…

Functions, uniformly bounded in $BV$ norm in some bounded open set $U$ in $R^n$, are compact in $L_1(U)$. This result is known when $U$ has Lipschitz boundary [EG Th. 4 p. 176], [G 1.19 Th. p. 17], [Z 5.34 Cor. p. 227]; the proof for…

A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all…

We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be…

We consider functions of the type $f(z)=z+a_2z^2+a_3z^3+\cdots$ from a family of all analytic and univalent functions in the unit disk. Let $F$ be the inverse function of $f$, given by $F(z)=w+\sum_{n=2}^{\infty}A_nw^n$ defined on some…

Let $\mathcal{A}$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}: |z|<1\}$ such that $f(0)=f'(0)-1=0$. In this paper, we introduce a new subclass $\mathcal{C}_\theta(\gamma)$ of $\mathcal{A}$…

In this study, we establish certain Landau-type theorems for functions with logharmonic Laplacian of the form $F(z)=|z|^2L(z)+K(z)$, $|z|<1$, where $L$ is logharmonic and $K$ is harmonic, with $L$ and $K$ having bounded length distortion in…

We study conformal quantities at generic parameters with respect to the harmonic measure on the boundary of the connectedness loci ${\cal M}_d$ for unicritical polynomials $f_c(z)=z^d+c$. It is known that these parameters are structurally…

The main result shows a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighborhood consisting entirely of univalent functions. For the particular choice of a linear function in…

In this paper, we continue to discuss normality based on a single\linebreak holomorphic function. We obtain the following result. Let $\CF$ be a family of functions holomorphic on a domain $D\subset\mathbb C$. Let $k\ge2$ be an integer and…

A well-known theorem by J. Becker states that if a normalized univalent function $f$ in the unit disk $\mathbb{D}$ can be embedded as the initial element into a Loewner chain $(f_t)_{t\geqslant 0}$ such that the Herglotz function $p$ in the…

Let $\mathcal{A}$ be the family of analytic and normalized functions in the open unit disc $|z|<1$. In this article we consider the following classes \begin{equation*} \mathcal{R}(\alpha,\beta):=\left\{ f\in \mathcal{A}: {\rm…

Let $\mathcal{H}$ be the space of all functions that are analytic in $\mathbb{D}$. Let $\mathcal{A}$ denote the family of all functions $f\in\mathcal{H}$ and normalized by the conditions $f(0)=0=f'(0)-1$. Obradovi\'{c} and Ponnusamy have…