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For small $\epsilon>0$, the system $\dot x = \epsilon$, $\dot z = h(x,z,\epsilon)z$, with $h(x,0,0) 0$, admits solutions that approach the $x$-axis while $x 0$. The limiting attraction and repulsion points are given by the well-known entry-exit function. For $h(x,z,\epsilon)z$ replaced by $h(x,z,\epsilon)z^2$, we explain this phenomenon using geometric singular perturbation theory. We also show that the linear case can be reduced to the quadratic case, and we discuss the smoothness of the return map to the line $z=z_0$, $z_0>0$, in the limit $\epsilon\to0$.

In this paper, we shall study the geometric function theory for slice regular functions of a quaternionic variable. Specially, we give some coefficient estimates for slice regular functions among which a version of the Bieberbach theorem and the Fekete-Szeg\"{o} inequality for a subclass of slice regular functions are presented. The growth and distortion theorems for this class are also established. Finally, we give a Bloch theorem for slice regular functions with convex image.

We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding M\"obius invariant differential operators. In the doubly connected case the three Eisenstein series $E_2$, $E_4$, $E_6$ are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these. At the end we consider the projective properties of the prepotential associated to a second order differential operator depending canonically on the domain.

The theory of first-order differential subordination developed by Miller and Mocanu was recently extended to functions with fixed initial coefficient by R. M. Ali, S. Nagpal and V. Ravichandran [Second-order differential subordination for analytic functions with fixed initial coefficient, Bull. Malays. Math. Sci. Soc. (2) 34 (2011), 611--629] and applied to obtain several generalization of classical results in geometric function theory. In this paper, further applications of this subordination theory is given. In particular, several sufficient conditions related to starlikeness, convexity, close-to-convexity of normalized analytic functions are derived. Connections with previously known results are pointed out.

This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory.

This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory.

This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory.

In this paper, we develop the foundations of the theory of quasiregular mappings in general metric measure spaces. In particular, nine definitions of quasiregularity for a discrete open mapping with locally bounded multiplicity are proved to be quantitatively equivalent when the metric measure spaces have locally bounded geometry. We also demonstrate that some, though not all, of these implications remain true under fairly general hypotheses. The major new tool appeared in our approach is a powerful factorization, termed the pullback factorization, of a quasiregular mapping into the composition of a 1-BLD mapping and a quasiconformal mapping in locally compact complete metric measure spaces. This factorization also brings fundamental new point of view of the theory of quasiregular mappings in Euclidean spaces, in particular, a branched counterpart of quasisymmetric mappings is introduced and is shown to be locally equivalent with quasiregular mappings, quantitatively. As applications of our new techniques, we answer some well-known open problems in this field and characterize BLD mappings in metric spaces with locally bounded geometry. In particular, a conjecture of Heinonen--Rickman is shown to be false and an open problem of Heinonen--Rickman gets solved affirmatively.

This is a list of corrections for the book: J. Noguchi and T. Ochiai, Geometric Function Theory in Several Complex Variables, xi + 282 pp., Math.\ Monographs Vol.\ {\bf 80}, Amer.\ Math.\ Soc., Providence, 1990. The authors hope that this distribution will be helpful for readers to avoid unnecessary confusions.

Geometric models and Teichm\"uller structures have been introduced for the space of smooth expanding circle endomorphisms and for the space of uniformly symmetric circle endomorphisms. The latter one is the completion of the previous one under the Techm\"uller metric. Moreover, the spaces of geometric models as well as the Teichm\"uller spaces can be described as the space of H\"older continuous scaling functions and the space of continuous scaling functions on the dual symbolic space. The characterizations of these scaling functions have been also investigated. The Gibbs measure theory and the dual Gibbs measure theory for smooth expanding circle dynamics have been viewed from the geometric point of view. However, for uniformly symmetric circle dynamics, an appropriate Gibbs measure theory is unavailable, but a dual Gibbs type measure theory has been developed for the uniformly symmetric case. This development extends the dual Gibbs measure theory for the smooth case from the geometric point of view. In this survey article, We give a review of these developments which combines ideas and techniques from dynamical systems, quasiconformal mapping theory, and Teichm\"uller theory. There is a measure-theoretical version which is called $g$-measure theory and which corresponds to the dual geometric Gibbs type measure theory. We briefly review it too.

In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization $G$ of an analytic function $f$ in the unit disk $U$. We pose few problems we can not solve. An intriguing one is that of the inequality $$ \int_{0}^{2\pi} |f(re^{i\theta})|^{p}d\theta\le\int_{0}^{2\pi} |G(re^{i\theta})|^{p}d\theta,\,\,0

In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization $G$ of an analytic function $f$ in the unit disk $U$. We pose few problems we can not solve. An intriguing one is that of the inequality $$ \int_{0}^{2\pi} |f(re^{i\theta})|^{p}d\theta\le\int_{0}^{2\pi} |G(re^{i\theta})|^{p}d\theta,\,\,0