In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separated. This separation is measured by the so-called separation constant, which is defined in terms of the pseudohyperbolic metric. Moreover, we study general properties of related extremal functions and associated vectors. Throughout, compressions of the shift serve as illustrating examples which also allow for refined results.

We some open questions linked to the notion of neighbourhood of a univalent function as introduced by Stephan Ruscheweyh in 1981.

We determine $$\displaystyle \sup \left\{ \frac{|(1-z)^\alpha -(1-w)^\alpha |}{|z-w|^\alpha }: |z|,|w|\le 1,z\not =w\right\} $$ where $$0<\alpha \le 1$$ .

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.

In this paper, we discuss the Polya–Szegő continuous symmetrization and its applications to isoperimetric inequalities. In particular, we survey results concerning monotonicity properties of certain characteristics, including torsional rigidity of cylindrical beams and principal frequency of a uniformly stretched elastic membrane of a drum, of triangles and other domains. Several remaining open problems on continuous symmetrization and relevant properties of domains and functions are also discussed.

Differential equations of the form f'' + A(z)f' + B(z)f = 0 (*) are considered, where A(z) and B(z) not equiv 0 are entire functions. The Lindelöf function is used to show that for any rho in (1/2, infty ), there exists an equation of the form (*) which possesses a solution f with a Nevanlinna deficient value at 0 satisfying rho =rho (f)ge rho (A)ge rho (B), where rho (h) denotes the order of an entire function h. It is known that such an example cannot exist when rho le 1/2. For smaller growth functions, a geometrical modification of an example of Anderson and Clunie is used to show that for any rho in (2, infty ), there exists an equation of the form (*) which possesses a solution f with a Valiron deficient value at 0 satisfying rho =rho _{log }(f)ge rho _{log }(A)ge rho _{log }(B), where rho _{log }(h) denotes the logarithmic order of an entire function h. This result is essentially sharp. In both proofs, the separation of the zeros of the indicated solution plays a key role. Observations on the deficient values of solutions of linear differential equations are also given, which include a discussion of Wittich’s theorem on Nevanlinna deficient values, a modified Wittich theorem for Valiron deficient values, consequences of Gol’dberg’s theorem, and examples to illustrate possibilities that can occur.

Exponential type functions are important subclasses of transcendental entire functions. In this paper, we will use some results given by Steinmetz (Manuscr Math 26:155–167, 1978) to consider the zeros of difference or differential-difference polynomials of exponential polynomials. In addition, we also consider the zeros of difference polynomials of exponential type functions with infinite order.

This paper gives a new approach to the calculation of the numerical radius of a restricted shift operator by linking it to the norm of a truncated Toeplitz operator (TTO), which can be calculated by various methods. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved.

The present paper is devoted to finding the classes of complex periodic functions whose iterations map lines into dense subsets of the complex plane. We distinguish a class of functions such that its second iterations map almost all oblique lines into dense sets. A class of complex periodic functions are obtained whose third iterations of any oblique line generate dense subsets of the complex plane.

For any finite Blaschke product B, there is an injective analytic map $$\varphi :{\mathbb {D}}\rightarrow {\mathbb {C}}$$ and a polynomial p of the same degree as B such that $$B=p\circ \varphi $$ on $${\mathbb {D}}$$ . Several proofs of this result have been given over the past several years, using fundamentally different methods. However, even for low-degree Blaschke products, no method has hitherto been developed to explicitly compute the polynomial p or the associated conformal map $$\varphi $$ . In this paper, we show how these functions may be computed for a Blaschke product of degree at most three, as well as for Blaschke products of arbitrary finite degree whose zeros are equally spaced on a circle centered at the origin.