A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformal metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.
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 related by the "Schwarzian derivative" of the metrics of the surfaces (at least for the case of virtual orders 2 and 3). As an application, we give univalence criteria for a meromorphic function on the unit disk in terms of the projective Schwarzian derivative of virtual order 3.
In the present paper, we obtain a more general conditions for univalence of analytic functions in the open unit disk U. Also, we obtain a refinement to a quasiconformal extension criterion of the main result.
In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields new a criterion for a polyharmonic functions to be fully $α$--accessible. Several examples are presented to illustrate the use of these criteria.
The article discusses criteria for univalence of analytic functions in the unit disc. Various families of analytic functions depending on real parameters are considered. A unified method for creating new sets of conditions ensuring univalence is presented. Applying this method we are able to find several families of new sharp criteria for univalence.
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
Let $p(z)=zf'(z)/f(z)$ for a function $f(z)$ analytic on the unit disk $|z|<1$ in the complex plane and normalized by $f(0)=0, f'(0)=1.$ We will provide lower and upper bounds for the best constants $δ_0$ and $δ_1$ such that the conditions $e^{-δ_0/2}<|p(z)|<e^{δ_0/2}$ and $|p(w)/p(z)|<e^{δ_1}$ for $|z|,|w|<1$ respectively imply univalence of $f$ on the unit disk.
In this paper our aim is to extend and improve the sufficient conditions for integral operators involving the normalized forms of the generalized Bessel functions of the first kind to be univalent in the open unit disk as investigated recently by (Erhan, E. Orhan, H. and Srivastava, H. (2011). Some sufficient conditions for univalence of certain families of integral operators involving generalized Bessel functions. Taiwanese Journal of Mathematics, 15 (2), pp.883-917) and (Baricz, Á. and Frasin, B. (2010). Univalence of integral operators involving Bessel functions. Applied Mathematics Letters…
In this paper we obtain, by the method of Loewner chains, some sufficient conditions for the analyticity and the univalence of the functions defined by an integral operator. These conditions in- volves Ruscheweyh and Salagean derivative operator in the open unit disk. In particular cases, we find the well-known conditions for univalency established by Becker [3], Ahlfors [2], Kanas and Srivastava [8] and others for analytic mappings f : U ! C: Also, we obtain the corresponding new, useful and simpler conditions for this integral operator.
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …
In this paper we study sharp estimates of pre-Schwarzian derivatives of functions belonging to the Nehari-type classes by using techniques from differential equations. In the sequel, we also see that a solution of a complex differential equation has a special form in terms of ratio of hypergeometric functions resulting to an integral representation. Finally, we attempt to study those univalent functions in the unit disk for which the image domain is an unbounded John domain.
We introduce a class of Weierstrass-Enneper lifts of harmonic mappings which satisfy a criterion for univalence introduced by Duren, Osgood and the first author in [J. Geom. Anal. (2007)].
Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic in $\ID\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$. In this article, we first prove a sufficient condition for univalency for functions in $\mathcal{A}(p)$. Thereafter, we consider the class denoted by $Σ(p)$ that consists of functions $f \in \mathcal{A}(p)$ that are univalent in $\ID$. We obtain the exact value for $\ds\max_ {f\in Σ(p)}Δ(r,z/f)$, where the Dirichlet integral $Δ(r,z/f)$ is given by $$ Δ(r,z/f)=\d…
Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $ϕ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a generalization of a classical condition for univalence of Nehari.
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 Weierstrass--Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of harmonic mappings are shown to have finite Schwarzian norm.
In earlier work the authors have extended Nehari's well-known Schwarzian derivative criterion for univalence of analytic functions to a univalence criterion for canonical lifts of harmonic mappings to minimal surfaces. The present paper develops some quantitative versions of that result in the form of two-point distortion theorems. Along the way some distortion theorems for curves in ${\Bbb R}^n$ are given, thereby recasting a recent injectivity criterion of Chuaqui and Gevirtz in quantitative form.
We establish two-point distortion theorems for sense-preserving planar harmonic mappings $f=h+\overline{g}$ which satisfies the univalence criteria in the unit disc such that, Becker's and Nehari`s harmonic version. In addition, we find the sharp two-point distortion theorem when $h$ is a convex function, and normalized mappings such that $h(\D)$ is a $c$-linearly connected domain. To do this, we use the order of this family.
Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible $r\in (0,1)$ for which the pre-Schwarzian norm of the dilation $r^{-1}f(rz)$ is not greater than a prescribed number for normalized univalent functions $f(z)$ in the unit disk. Similar results concerning the Schwarzian derivative are also obtained
