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In 1984 J. Clunie and T. Sheil-Small proved ([2, Corollary 5.8]) that for any complex-valued and sense-preserving injective harmonic mapping F in the unit disk D, if F(D) is a convex domain, then the inequality |G(z2)− G(z1)| < |H(z2) − H(z1)| holds for all distinct points z1, z2∈ D. Here H and G are holomorphic mappings in D determined by F = H + Ḡ, up to a constant function. We extend this inequality by replacing the unit disk by an arbitrary nonempty domain Ω in ℂ and improve it provided F...
Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by the decreasing sequence of singular values of a matrix, we prove that the functions and are subharmonic on Ω for 1 ≤ k ≤ n.
The notion of a strong asymptotic tract for subharmonic functions is defined. Eremenko's value b(∞,u) for subharmonic functions is introduced and it is used to provide an exact upper estimate of the number of strong tracts of subharmonic functions of infinite lower order. It is also shown that b(∞,u) ≤ π for subharmonic functions of infinite lower order.
The article of J. Clunie and T. Sheil-Small [3], published in 1984, intensified the investigations of complex functions harmonic in the unit disc Δ. In particular, many papers about some classes of complex mappings with the coefficient conditions have been published. Consideration of this type was undertaken in the period 1998-2004 by Y. Avci and E. Złotkiewicz [2], A. Ganczar [5], Z. J. Jakubowski, G. Adamczyk, A. Łazińska and A. Sibelska [1], [8], [7], H. Silverman [12] and J. M. Jahangiri [6],...
In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.
We proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential...
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