Class Of Meromorphic Functions Research Articles

The value regions of initial coefficients in a certain class of meromorphic functions

Let Mk,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+ao+a1z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition $$\mathop {\lim }\limits_{r \to 1 - } \int\limits_0^{2\pi } {\left| {\operatorname{Re} J_\lambda \left( {re^{i\theta } } \right)} \right|} d\theta \leqslant k\pi ,$$ where Jλ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class Mk,λ we consider sorne coefficient problems and problems concerning distortion theorems.

On the Size of the Exceptional Set in Nevanlinna's Second Fundamental Theorem for Certain Classes of Meromorphic Functions

AbstractSome conditions on the size of the exceptional set that arise in Nevanlinna's Second Fundamental Theorem are established, showing that previous sharp results can be improved by restricting the class of functions considered and suggesting a close relationship between the size of the exceptional set and the lower growth of the characteristic function. Examples of functions of rapid growth possessing exceptional sets are built, showing that these conditions are sharp for the class of functions considered.

Two problems in the value distribution theory

This paper includes two parts. In the first part, we deal with the relations between the orders, the lower orders and the number of Julia directions of entire functions. In the second part, we deal with the existence of the zeros of a class of meromorphic functions. This class of meromorphic functions has an interesting physical background.

Certain Classes of Meromorphic Functions with Positive and Missing Coefficients

The main object of the present paper is to introduce certain novel subclasses of meromorphic functions with positive and missing coefficients and investigate their basic properties including, for example, coefficient estimates, distortion theorem, and the radii of starlikeness and convexity. It is also proved that one of the classes (considered here) is closed under convex linear combinations. The various results obtained in this paper are shown to be sharp.

Newton's method and a class of meromorphic functions without wandering domains

AbstractLetNbe the class of meromorphic functionsfwith the following properties:fhas finitely many poles;f′ has finitely many multiple zeros; the superattracting fixed points offare zeros off′ and vice versa, with finitely many exceptions;fhas finite order. It is proved that iff∈N, thenfdoes not have wandering domains. Moreover, iff∈Nand if ∞ is among the limit functions offnin a cycle of periodic domains, then this cycle contains a singularity off−1. (Herefndenotes thenth iterate off) These results are applied to study Newton's method for entire functionsgof the formwherepandqare polynomials and wherecis a constant. In this case, the Newton iteration functionf(z) =z−g(z)/g′(z) is inN. It follows thatfn(z) converges to zeros ofgfor allzin the Fatou set off, if this is the case for all zeroszofg″. Some of the results can be extended to the relaxed Newton method.

A conjecture of L. Carleson and applications.

Let $T_\alpha$ be the class of functions meromorphic in the unit disk ${\bf D}$ such that $$\int_D{|f'(z)|^2\over (1+|f(z)|^2)^2}(1-|z|)^{1-\alpha}dx\,dy<\infty,\quad 0\leq\alpha<1.$$ It is known that $T_\alpha\subset N$, where $N$ denotes the Nevanlinna class of functions meromorphic in $\bold D$ and of bounded characteristic. Because every $f\in N$ is a quotient of two bounded functions, analytic in ${\bf D}$, it is natural to ask whether any $f\in T_\alpha$ may be represented as a quotient of two bounded analytic functions of the same class. This problem was posed for the first time by Carleson in his thesis [``On a class of meromorphic functions and its associated exceptional sets'', Univ. Uppsala, Uppsala, 1950; MR0033354 (11,427c)]. Carleson proved that any $f\in T_\alpha$ is a quotient of bounded functions $u,v\in T_\beta$, analytic in ${\bf D}$, for all $\beta<\alpha$. The considerable success of the author in the paper under review is in proving the following result. Theorem: For $0<\alpha\leq 1$, every function in $T_\alpha$ is a quotient of two bounded analytic functions belonging to $T_\alpha$. (Less)

Ranges of the first coefficients in the class of meromorphic functions with bounded boundary rotation

A description of the range of system (a1,a2, ⋯,an is given in class Λk(k≥2) of functions f(z)=z−1+∑n=0∞ anzn, regular in 0<∥z∥<1,f′(z)≠0 in∥z∥<1, mapping ∥z∥<1 onto a domain with boundary rotation α≤kπ. For this one makes use of the representation of functions of class λk in terms of functions of the Caratheodory class.

The numerical computation of the Voigt function by a corrected midpoint quadrature rule for (−∞,∞)

This paper presents a method for computing the Voigt function, through the application of a midpoint quadrature rule that has been corrected to accurately integrate a certain class of meromorphic functions.

A class of meromorphic functions with no wandering domains

A class of meromorphic functions with no wandering domains

BOUNDARY UNIQUENESS THEOREMS FOR MEROMORPHIC FUNCTIONS

A study is made of sets of uniqueness for the class of arbitrary meromorphic functions on the disk and for the limit values over -angles (domains with zero angle on the boundary and with form determined by a function ). The sets of uniqueness are characterized with the help of the concepts of -indecomposability of -regularity, introduced and studied in this article. These concepts turn out to be intermediate between measure and category. The concept of the porosity of a set served as a starting point for the definition of the property of -indecomposability. The central result in this paper is the following:Theorem. Let be the class of all meromorphic functions on the unit disk. A set on the boundary of the disk is a set of uniqueness for the class and for the limit values over -angles if and only if is -indecomposable.Bibliography: 13 titles.

The coefficients of a class of meromorphic functions and their inverses

We consider the coefficients in the expansion , where f(z) is in a class of functions meromorphic and univalent in . In particular, we obtain sharp bounds on for n = -1,0,1. We also obtain sharp bounds on the coefficients of functions which are inverses of functions in the class under consideration.

A Tauberian theorem for certain class of meromorphic functions

A Tauberian theorem for certain class of meromorphic functions

Solutions of a system of functional equations in the class of meromorphic functions

Solutions of a system of functional equations in the class of meromorphic functions

Coefficient Behavior of a Class of Meromorphic Functions

With , denote by Λk the class of functions ƒ of the formwhich are analytic in and which map y onto the complement of a domain with boundary rotation at most . It is known [2] that ƒ ∈ Λk if and only if there exist regular starlike functions s1 and s2, withsuch that

A note on the class of meromorphic functions, I

A note on the class of meromorphic functions, I

On the multivalence and order of starlikeness of a class of meromorphic functions

Basically this paper deals with the determination of the radius of starlikeness and radius of univalence of the class of meromorphic functions of the formg(z)=A/z−Φ(z) forA>0, and where Φ(z) is an analytic function defined in the unit disk whose modulus does not exceed unity. We estimate the radius ofp-valence of functions having the fromh(z)=Φ(z)+a/zp fora>1,p≧1, and also estimate the radius of starlikeness of certain Blaschke products which is also given as a function of the minimum modulus function. We discuss the question of sharpness of the results and mention some open problems.

A New Method in Arithmetical Functions and Contour Integration

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration(1.1)where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

Convergence of Padé approximants for a certain class of meromorphic functions

Convergence of Padé approximants for a certain class of meromorphic functions

On an extension of a theorem of Bernstein to meromorphic functions

A theorem of Bernstein [2; pp. 206-2111 asserts that if f(z) is an entire function of exponential type T and satisfies If(x) 1 < M uniformly for co < x < W, then If’(x) ( < Mr. In [3] A. J. Macintyre and S. M. Shah state sufficient conditions (same as (a), (c), and (d) below) to guarantee that meromorphic functions which are bounded on the real axis have derivatives which also are bounded there; moreover, they show by example that condition (d) is necessary. The theorem given here is proved by methods similar to those of Macintyre and Shah and extends their result to a class of meromorphic functions which are 0 (exp (/? ( x 17)) as x -+ f co. THEOREM. Iff(z) is a meromorphicfunction which satisjk:

On a Class of Meromorphic Functions

On a Class of Meromorphic Functions