Computational Methods and Function Theory

Ferguson, Timothy

2014 Computational Methods and Function Theory

doi: 10.1007/s40315-013-0046-7

In this paper, we discuss the explicit solution of certain extremal problems in Bergman spaces. In order to do this, we develop methods to calculate the Bergman projections of various functions. As a special case, we deal with canonical divisors for certain values of $$p$$ p .

Li, Xiao-Min; Yi, Hong-Xun; Shi, Yue

2014 Computational Methods and Function Theory

doi: 10.1007/s40315-014-0048-0

Using Zalcman’s lemma, we study a uniqueness question of meromorphic functions, where the meromorphic functions and their certain non-linear differential polynomials share a non-zero value with their shifts. The results in this paper extends Theorem 1 in Yang and Hua (Ann Acad Sci Fenn Math 22:395–406, 1997) and Theorem 1 in Fang (Comput Math Appl 44:823–831, 2002). Our reasoning in this paper also corrects the proof of Theorem 4 in Bhoosnurmath and Dyavanal (Comput Math Appl 53:1191–1205, 2007).

Betsakos, Dimitrios

2014 Computational Methods and Function Theory

doi: 10.1007/s40315-014-0049-z

Suppose that $$f$$ f is a holomorphic function in the unit disk. We provide bounds for the distance of $$f$$ f from its linearization $$f(0)+f^\prime (0)z$$ f ( 0 ) + f ′ ( 0 ) z ; the bounds involve the area, the diameter or the logarithmic capacity of the image $$f({\mathbb D})$$ f ( D ) of $$f$$ f . These results are motivated by a problem posed by Burckel, Marshall, Minda, Poggi-Corradini, and Ransford. We also prove that if, in addition, $$f(0)=0$$ f ( 0 ) = 0 , then the $$H^2$$ H 2 norm of $$f$$ f is bounded by a $$\mathrm{Area}f^{\circledcirc }({\mathbb D})/\pi $$ Area f ⊚ ( D ) / π , where $$f^{\circledcirc }$$ f ⊚ is a univalent function constructed via the symmetric decreasing rearrangement of the real part of $$f$$ f on the unit circle. The above estimate is stronger than a well-known inequality due to Alexander, Taylor, and Ullman. We give a description of the equality cases in Lindelöf’s principle for the Green function. We prove that the $$H^p$$ H p norm of $$f$$ f is smaller or equal to the $$H^p$$ H p norm of the universal covering map onto the circular ( $$0<p<\infty $$ 0 < p < ∞ ) or Steiner ( $$0<p\le 2$$ 0 < p ≤ 2 ) symmetrization of the range of $$f$$ f .

Marshall, Jonathan

2014 Computational Methods and Function Theory

doi: 10.1007/s40315-014-0055-1

In this paper, we construct an explicit parameterisation describing quadrature domains of finite area and arbitrary finite connectivity admitting quadrature identities with integrals along interior arcs. The formulation is in terms of conformal maps from circular domains, where these maps are expressed in terms of the associated Schottky–Klein prime function. An important role in the derivation is played by the Bergman kernel function. To illustrate the results, several examples are presented.

Böttcher, Albrecht

2014 Computational Methods and Function Theory

doi: 10.1007/s40315-014-0061-3

Let $$H(N)$$ H ( N ) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in $$H(N)$$ H ( N ) whose largest coefficients are as small as possible and also for polynomials in $$H(N)$$ H ( N ) with minimal sum of the coefficients. Let $$h(N)$$ h ( N ) and $$s(N)$$ s ( N ) denote these minimal values. Using Fekete’s subadditive lemma we show that the $$N$$ N th square roots of $$h(N)$$ h ( N ) and $$s(N)$$ s ( N ) have a limit as $$N$$ N goes to infinity and that these two limits coincide. We also derive tight bounds for the common value of the limits.