P. C. Fenton

Estimates for Singular Integral Operators

Estimates are obtained for

Phragmén-Lindelöf theorems

\int^\infty_0 K(r,t)\phi(t)dt

cos πρ theorems for δ-subharmonic functions

, where K(r,t) may have a singularity at t = −r.

Growth of functions in cercles de remplissage

Results of Phragmen-Lindelof type are obtained for subharmonic functions in sectorial domains of bounded angular extent.

On α-Polynomial Regular Functions, with Applications to Ordinary Differential Equations

For ϕ a δ-subharmonic function, sharp results are obtained that connectA(r, ϕ), B(r, ϕ) andT(r, ϕ), whereA(r, ϕ)=inf|z|=rϕ(z),B(r, ϕ)=sup|z|=rϕ(z), andT(r, ϕ) is the Nevanlinna characteristics.

A Reverse cos πρ Theorem and a Question of Fryntov

Suppose that f is meromorphic in the plane, and that there is a sequence zn !1 and a sequence of positive numbers n ! 0, such that njznj f # .zn/= log jzn j!1 . It is shown that if f is analytic and non-zero in the closed discs1n Df z Vj z zn jnjznjg, n D 1; 2; 3;:::, then, given any positive integer K , there are arbitrarily large values of n and there is a point z in 1n such that j f.z/j > jzj K . Examples are given to show that the hypotheses cannot be relaxed. 2000 Mathematics subject classification: primary 30D20.

Wiman-Valiron Theory in Simply Connected Domains

This research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman–Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

ODEs and Wiman–Valiron theory in the unit disc

The authors continue their work on reverse Denjoy theorems, proving a reverse cosπρ theorem. The theorem is connected to a question of Fryntov on entire functions with gaps.

Cercles De Remplissage for Entire Functions

We obtain asymptotic expressions for the derivatives of analytic functions in simply connected domains.