Zero subsets, representation of meromorphic functions, and Nevanlinna characteristics in a disc

We introduce a new parametric representation of the class of holomorphic functions in the unit disk such that their Nevanlinna characteristic has a power growth near the boundary of the disk. The parameters of the obtained representation are determined explicitly by values of the function. In addition, the set of multipliers from the considered class to the Hardy and Bergman classes and the disk algebra is described completely. Bibliography: 10 titles.

Holomorphic functions differ fundamentally from real differentiable functions: they are infinitely often (real and complex) differentiable (II.3, II.7), they even admit power series expansions (II.4), their entire behaviour is determined by their values on arbitrarily small open sets (II.4, II.7), and they satisfy powerful convergence theorems and estimates (II.5). All of these properties are consequences of the Cauchy integral theorem and the integral representations that arise from it (II.1–3). Meromorphic functions extend the class of holomorphic functions (II.6); their study leads to the notion of isolated singularities and to generalizations of power series obtained by allowing negative powers (Laurent series). In addition to the phenomena that occur in the theory of functions of one complex variable, a fundamentally new phenomenon enters the picture in higher dimensions: the simultaneous holomorphic continuation of all holomorphic functions from a given domain to a larger one (II.7). Here the Cauchy integral formula (in one variable!) is again the decisive tool.

Recently J. Miao proved that if is a holomorphic function with Hadamard gaps on the open unit disc D then f ∈ Xp if and only if f ∈ Bp if and only if if and if only if where Xp, Bp and denote respectively the class of holomorphic functions on D which satisfy |f′(z)|p(1 − |z|2)p − 1dxdy is a finite measure, a Carleson measure and a little Carleson measure on D. In this paper we give a higher-dimensional version of Miao's result.

The relative rates of growth of a function F meromorphic in the complex plane and its q derivative F (q) are studied via the Nevanlinna Characteristics T(r.F)and T(r.F (q)) and It is shown that lim inf T(r.F)/T(r.F (q)) ≤ 3ethat for all meromorphic functions. A lower bound on the size of the set {r>1 T(r.F) T(r.F (q))for K 1 is obtained. The upper bounds established for T(r.F)/T(r.F′)justify in a weakened form an old conjecture of Nevanlinna.

By using the differential operator Dm f(z), z∈ U (Definition 1), we introduce a class of holomorphic functions, denoted by , and we obtain some inclusion relations.

Various BMO-type characteristics are given for some class of holomorphic functions with a mixed norm. Some criteria are established for Blaschke products to belong to analytic Besov spaces.

This article is concerned with the asymptotic behaviour of the Mahler measure (defined for the occasion) of functions of the form \(z^n + f(z)\), where f(z) is a holomorphic function on an open subset of \({\mathbb {C}}\) such that \(f(0) \ne 0\).

We introduce a generalized Cauchy integral formula for special classes of functions, holomorphic in several complex variables. These functions also satisfy a generalized Cachy-Riemann equation. The integral formula is then applied to demonstrate that these classes of holomorphic functions are Fréchet modules over complex Clifford algebras, and to provide maximal domains of convergence for Taylor expansions, with homogeneous terms grouped together, of real analytic solutions to Maxwell's equations in vacuo.

The class $S_{H}(p)$, $0 \\leq p \\lt 1$, of complex valued, meromorphic harmonic univalent sense-preserving functions in the unit disk $U \\backslash \\{p\\}$ is studied. The functions belong to $S_{H}(p)$ have the expansion $f(z) = \\frac{\\alpha}{z-p} + \\sum_{n=0}^{\\infty} c_{n} z^{n} + \\overline{\\sum_{n=1}^{\\infty} d_{n} z^{n}} + A\\log |z-p|$ and $\\lim_{z \\rightarrow p} f(z) = \\infty$. Some coefficient estimates, distortion and area theorems are obtained. Sufficient coefficient conditions for a class of meromorphic harmonic univalent sense-preserving functions that are starlike and convex are given.

In this paper we introduce and examine the differential subordination related to the geometric mean of the form [p(z)]1-γp(z)+zp′(z)φp(z),zp′(z)γ≺h(z),z∈D,where D:={z∈C:|z|<1} and h is a convex univalent function with 0∈∂h(D). In particular, the above differential subordination generalizes the well-known Briot-Bouquet differential subordination. At the same time new type of ordinary differential equation has been proposed for study. The main results allow us to construct non-obvious subclasses in the class of holomorphic functions on D.

On N-Widths of Holomorphic-Functions of Several Variables

In this article we investigate some classes of meromorphic or complex harmonic functions with a pole, which are generated either by analytic conditions or by “coefficient inequalities”. There are given theorems, which combine the geometrical properties of functions of the introduced classes. Some results broaden knowledge about the classes of functions, which were investigated in [

We study the uniqueness of meromorphic functions concerning nonlinear differential polynomials sharing a nonzero polynomial IM.Though the main concern of the paper is to improve a recent result of the present author [12], as a consequence of the main result we also generalize two recent results of X. M. Li and L. Gao [11].

ON SOME CLASSES OF BI-UNIVALENT FUNCTIONS

To consolidate or adapt to many studies on meromorphic functions, we define a new subclass of meromorphic functions of complex order involving a differential operator. The defined function class combines the concept of spiral-like functions with other studies pertaining to subclasses of multivalent meromorphic functions. Inclusion relations, integral representation, geometrical interpretation, coefficient estimates and solution to the Fekete-Szegö problem of the defined classes are the highlights of this present study. Further to keep up with the present direction of research, we extend the study using quantum calculus. Applications of our main results are given as corollaries.