A note on the growth of entire functions.
A note on the number of zeros of polynomials in an annulus
Let p(z) be a polynomial of the form , . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.
A note on the oscillation theory of certain second order differential equations.
A Note on the Pompeiu Problem for Convex Domains.
A note on the separated maximum modulus points of meromorphic functions
We give an upper estimate of Petrenko's deviation for a meromorphic function of finite lower order in terms of Valiron's defect and the number p(∞,f) of separated maximum modulus points of the function. We also present examples showing that this estimate is sharp.
A note on the solvability of homogeneous Riemann boundary problem with infinity index
In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points.
A note on the structure of quadratic Julia sets
In a series of papers, Bandt and the author have given a symbolic and topological description of locally connected quadratic Julia sets by use of special closed equivalence relations on the circle called Julia equivalences. These equivalence relations reflect the landing behaviour of external rays in the case of local connectivity, and do not apply completely if a Julia set is connected but fails to be locally connected. However, rational external rays land also in the general case. The present...
A note on the three-segment problem
We improve a theorem of C. L. Belna (1972) which concerns boundary behaviour of complex-valued functions in the open upper half-plane and gives a partial answer to the (still open) three-segment problem.
A note on the zeros of exponential polynomials
A Note on Univalent Functions with Finitely many Coefficients
MSC 2010: 30C45The main object of this article is to introduce sufficient conditions of univalency for a class of analytic functions with finitely many coefficients defined by approximate functions due to Suffridge on the unit disk of the complex plane whose image is saddle-shaped. Sandwich theorem is also discussed.
A note to a certain pair of parametric integrals
A notion of extremal analytic discs related to interpolation in the ball.
A numerical algorithm for the Stokes problem based on an integral equation for the pressure via conformal mappings.
A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix
It is shown that if A is a bounded linear operator on a complex Hilbert space, then , where w(A) and ||A|| are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.
A inequality for Kleinian groups.
A parabolic Pommerenke-Levin-Yoccoz inequality
In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only...
A percolation formula.
A Phragmén-Lindelöf type quasi-analyticity principle
Quasi-analyticity theorems of Phragmén-Lindelöf type for holomorphic functions of exponential type on a half plane are stated and proved. Spaces of Laplace distributions (ultradistributions) on ℝ are studied and their boundary value representation is given. A generalization of the Painlevé theorem is proved.
A Picard type theorem for quasiregular mappings of Rn into n-manifolds with many ends.
A Pick function related to an inequality for the entropy function.