The Teichmüller and Grötzsch domains are defined by 𝒯(t)=ℂ∖( [-1,1]∪ [t,∞ )) and 𝒢(s)={z: |z|>1}∖[s,∞ ), where s,t>1, respectively. In 2011, Iwaniec et al. [9] raised the following two problems. In 2022, Bshouty et al. [4] addressed the Teichmüller-Nitsche problem and found a necessary condition for the existence of a harmonic homeomorphism f: 𝒯(s)→𝒯(t) symmetric with respect to the real axis, namely the inequality t≥ (s+3)/√(2(s+1))-1. In 2023, Bshouty and Lyzzaik [3] considered the same problem and found a necessary condition for the existence of a harmonic homeomorphism f: 𝒯(s)→𝒯(t), not necessarily symmetric, namely the inequality t≥√(2)(s+3)/(4√(s+1)). The purpose of this paper is to address the Grötzsch-Nitsche problem. We find a necessary condition for the existence of a symmetric harmonic homeomorphism f: 𝒢(s)→𝒢(t), namely the inequality t≥s^2 +6 s+1/2√(s)(s+1)-1.
Let G=ℝ^2 - { 0 } be equipped with the point pair function as a metric. We obtain several properties of equidistant circles and prove that the equilateral dimension of G is 4.
We study Schauder bases for spaces of holomorphic functions in the open unit disk 𝔻 . Given a non-Blaschke sequence (λ _n)_n≥ 1 in 𝔻 , we show that the associated sequence of finite Blaschke products (B_n)_n≥ 1 forms a Schauder basis for Hol(𝔻) when this space is endowed with a norm inherited from a Banach space X satisfying a set of natural structural assumptions. This abstract framework includes, in particular, the classical Hardy spaces H^p , 1≤ p≤∞ , the weighted Bergman spaces A_α ^p , 1≤ p≤∞ , α >-1 , and BMOA. We further prove that if the sequence (λ _n)_n≥ 1 is contained in a compact subset of the open unit disk, then (B_n)_n≥ 1 is a Schauder basis for Hol(𝔻) endowed with its natural topology. Moreover, in the case of Hardy spaces H^p , 1
Let G be a finite Jordan domain in the complex plane ℂ , bounded by a Carleson curve Γ and let ω be a weight function given on Γ . In this work some direct and inverse theorems of approximation theory in the weighted variable exponent Smirnov classes E_ω^p(· )(G) , under some restrictions on the weight ω and variable exponent p(· ) , are proved.
Honoring the 200th birthday of Bernhard Riemann on September 17, the cover of this volume shows a phase plot of the Riemann Zeta Function. In this essay we present some features of this function in an entertaining way. Using only basic tools, we try to make it clear that Zeta is, on the one hand, very "simple" and, on the other hand, incredibly complex. In particular, we give an interpretation of its universality and illustrate this property by phase plots. The last section is devoted to philosophical speculations on the Riemann Hypothesis.
