On a certain classes of meromorphic functions with positive coefficients

In this paper certain classes of meromorphic functions in punctured unit disk are defined. Some properties including coefficient inequalities, convolution and other results are investigated.     

Iteration of certain exponential-like meromorphic functions

The dynamics of functions \(f_\lambda (z)= \lambda \frac{\mathrm{e}^{z}}{z+1}\ \text{ for }\ z\in \mathbb {C}, \lambda >0\) is studied showing that there exists \(\lambda ^* > 0\) such that the Julia set of \(f_\lambda \) is disconnected for \(0< \lambda < \lambda ^*\) whereas it is the whole Riemann sphere for \(\lambda > \lambda ^*\). Further, for \(0< \lambda < \lambda ^*\), the Julia set is a disjoint union of two topologically and dynamically distinct completely invariant subsets, one of which is totally disconnected. The union of the escaping set and the backward orbit of \(\infty \) is shown to be disconnected for \(0<\lambda < \lambda ^*\) whereas it is connected for \(\lambda > \lambda ^*\). For complex \(\lambda \), it is proved that either all multiply connected Fatou components ultimately land on an attracting or parabolic domain containing the omitted value of the function or the Julia set is connected. In the latter case, the Fatou set can be empty or consists of Siegel disks. All these possibilities are shown to occur for suitable parameters. Meromorphic functions \(E_n(z) =\mathrm{e}^{z}(1+z+\frac{z^2}{2!}+\cdots +\frac{z^n}{n!})^{-1}\), which we call exponential-like, are studied as a generalization of \(f(z)=\frac{\mathrm{e}^{z}}{z+1}\) which is nothing but \(E_1(z)\). This name is justified by showing that \(E_n\) has an omitted value 0 and there are no other finite singular value. In fact, it is shown that there is only one singularity over 0 as well as over \(\infty \) and both are direct. Non-existence of Herman rings are proved for \(\lambda E_n \).     

Distribution of zeros of certain meromorphic functions

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1428–1430, October, 1989.     

Iteration of a class of hyperbolic meromorphic functions

We look at the class which contains those transcendental meromorphic functions for which the finite singularities of are in a bounded set and prove that, if belongs to , then there are no components of the set of normality in which as . We then consider the class which contains those functions in for which the forward orbits of the singularities of stay away from the Julia set and show (a) that there is a bounded set containing the finite singularities of all the functions and (b) that, for points in the Julia set of , the derivatives have exponential-type growth. This justifies the assertion that is a class of hyperbolic functions.

     

Some properties of certain meromorphic close-to-convex functions

In the present paper, we introduce and investigate a certain subclass of meromorphic close-to-convex functions. Such results as coefficient inequalities, convolution property, distortion property and radius of meromorphic convexity are derived.     

The boundary behavior of functions meromorphic in bounded plane regions

The object of this note is to extend the classical theorems of Fatou-Nevanlinna, Riesz-Nevanlinna, Riesz-Tsuji, Lusin-Privalov and Lusin-Privalov-Tsuji to bounded plane regions.     

具1 IM公共值的亚纯函数的唯一性

主要在涉及重值的情况下,得到一个具１ＩＭ公共值的亚纯函数的唯一性定理,推广并改进了Ｌａｈｉｒｉ的一个结果。     

On the order of a class of meromorphic functions

This paper proves the following result: Letf(z) be a meromorphic function in thez-plane with a deficient value, and δ(θ _k )(k=1,2, ...,q;0≤θ ₁<θ₂<...<θ _q<θ _q+1=θ ₁+2π) beq rays (1≤q<∞) starting at the origin, and letn≥3 be an integer such that for any given positive numberε,0<ε<π/2, $$\overline {\mathop {\lim }\limits_{r \to \infty } } \frac{{\log ^ + n\left\{ { \cup _{k = 1}^q \Omega \left( {\theta _k + \varepsilon ,\theta _{k + 1} - \varepsilon ,r} \right),f\prime f^n = 1} \right\}}}{{\log r}} \leqslant v< \infty ,$$ whereΝ is a constant independent ofε. IfΜ<∞, then we have $$\lambda \leqslant \frac{\pi }{\omega } + v,$$ whereΜ andλ denote the lower order and order off(z), respectively,Ω=minθ _k+1 ?θ _k ;1≤k≤q, andn(E, f=a) is the number of zeros off(z)?a inE with multiple zeros being counted with their multiplicities.     

Schlicht regions for entire and meromorphic functions

Let f:C →C be a meromorpMc function. We study the size of the maximal disc inC, with respect to the spherical metric, in which a single-valued branch of f^-1 exists. This problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for the class of rational functions is also given which is best possible for rational functions of degree 7. For algebraic functions of given genus we obtain an estimate which is precise for genera 2 and 5 and asymptotically best possible when the genus tends to infinity. Supported by a Heisenberg fellowship of the DFG. Partially supported by NSF grant DMS-950036 and by the Lady Davis Foundation.     

Coefficient estimates for a class of meromorphic bi-univalent functions

Applying the Faber polynomial coefficient expansions to a class of meromorphic bi-univalent functions, we obtain the general coefficient estimates for such functions and also examine their early coefficient bounds. A function univalent in the open unit disk is said to be bi-univalent if its inverse map is also univalent there. Both the technique and the coefficient bounds presented here are new on their own kind. We hope that this article will generate future interest in applying our approach to other related problems.     

On certain subclasses of meromorphic functions associated with certain differential operators

In this work, we study some subordination and convolution properties of certain subclasses of meromorphic functions which are defined by a previously mentioned differential operator.     

On deficient values of meromorphic functions satisfying a certain functional equation

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