Mapping properties of analytic functions on the unit disk

Let be analytic on the unit disk with . In 1989, D. Marshall conjectured the existence of the universal constant such that whenever the area, counting multiplicity, of a portion of over is . Recently, P. Poggi-Corradini (2007) proved this conjecture with an unspecified constant by the method of extremal metrics. In this note we show that such a universal constant exists for a much larger class consisting of analytic functions omitting two values of a certain doubly-sheeted Riemann surface. We also find a numerical value, , which is sharp for the problem in this larger class but is not sharp for Marshall's problem.

     

Iteration of functions analytic on a disk

Aequationes mathematicae -     

On l p -multipliers of functions analytic in the disk

We consider bounded analytic functions in domains generated by sets with the Littlewood-Paley property. We show that each such function is an l ^p -multiplier.     

On analytic functions in the unit disk with univalent derivatives

We establish conditions on an increasing sequence (n_p) of nonnegative integers that guarantee that a functionf analytic in the unit disk {z ¦z¦< 1} can be analytically continued to a disk of larger radius provided all of its derivatives are univalent in the disk.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 58–65.     

Cyclic vectors of diagonal operators on the space of functions analytic on a disk

The purpose of this paper is to study cyclic vectors and invariant subspaces of operators on the space of functions analytic on an open disk in the complex plane having as eigenvectors the monomials zn.     

Bounds for higher order divided differences of analytic functions on a disk

We estimate the absolute value and real part of the order n divided difference of an analytic function on a disk. Particular attention in these estimates is paid to the question of equality.     

Further properties of discrete analytic functions

Analogs of the classical theorems of Liouville, Phragmén-Lindelöf, and Paley-Wiener are proved in the class of discrete analytic functions.     

Some properties of analytic in a disk functions with applications to the study of the behavior of singular integrals

We obtain representations for an analytic in a disc function such that its real part has a zero of an integer order at a fixed boundary point. We consider certain applications of these representations for studying properties of singular integrals with Hilbert kernel.     

Free boundary value problems for analytic functions in the closed unit disk

We shall prove (a slightly more general version of) the following theorem: let be analytic in the closed unit disk with , and let be a finite Blaschke product. Then there exists a function satisfying: i) analytic in the closed unit disk , ii) , iii) in , such that

satisfies

This completes a recent result of Kühnau for , , where this boundary value problem has a geometrical interpretation, namely that preserves hyperbolic arc length on for suitable
. For these important choices of we also prove that the corresponding functions are uniquely determined by , and that is univalent in . Our work is related to Beurling's and Avhadiev's on conformal mappings solving free boundary value conditions in the unit disk.

     

Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Denote by C _A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ? _n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R _n (f) (R _n (f) _A ) the best uniform approximation of a function f ∈ C _A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ? _n . The following equality is proved for any n ≥ 1: sup{R _n (f) _A /R _n (f): f ∈ C _A ? ? _n } = 2. We also consider a similar problem of comparing the best approximations of functions in C _A by polynomials and trigonometric polynomials.     

Mapping properties of the zero-balanced hypergeometric functions

In this paper, the order of convexity of $z_2{F}_1(a,b;c;z)$ is given under certain conditions on the positive real parameters a, b and c. We show also that the image domains of the unit disc $\mathbb{D}$ under some shifted zero-balanced hypergeometric functions $z_2{F}_1(a,b;a+b;z)$ are convex and bounded by two horizontal lines. This solves a problem raised by Ponnusamy and Vuorinen in [10].