Volume 28, Issue 3

This is an expository paper where we relate some aspects of the problem of looking for holomorphic functions with maximal cluster sets under the action of operators defined on spaces of holomorphic functions. Some functional generalizations of cluster sets, as well as special spaces of analytic functions, are also considered.

Using the notion of weighted sharing of sets we deal with the problem of uniqueness of meromorphic functions sharing three sets and obtain three results two of which will improve and the other will supplement a result of Lin-Yi [12].

We identify the extremal functions for a class of differential inequalities first introduced by Sanford S. Miller and P. T. Mocanu in 1978.

We prove the existence of an infinitesimal resistance metric on the Sierpinski gasket (SG) at boundary points, junction points and periodic points. This is a renormalized limit of the effective resistance metric as we zoom in on the point, and satisfies a self-similar identity. We obtain similar results on PCF fractals with three boundary points.

There are conformal maps f of open unit disc D onto a concave domain, i.e. a domain whose complement with respect to C is convex and unbounded. We say that f is a concave function if f (D) is a concave domain. We consider concave functions f with an opening angle of f (D) at ∞ is less than or equal to πα , α ∈ (1,2]. In this paper we discuss, for any fixed z 0 ∈ D, the region of variability for log f′ ( z 0 ) when f ranges over the class of concave univalent functions with a given opening angle and fixing f″ (0) suitably.

We study the smoothness properties of a complex-valued function f of several variables with absolutely convergent Fourier series. We prove sufficient conditions in terms of the Fourier coefficients of f in order that f belong to one of the multiplicative Zygmund classes Zyg( α 1 , α 2 ,…, α N ) and zyg( α 1 , α 2 ,…, α N ) for some α 1 , α 2 ,…, α N > 0. These Zygmund classes of functions are defined in terms of the multiple difference operator of second order in each variable. The conditions given by us are not only sufficient, but also necessary in the case when the Fourier coefficients are nonnegative.

For a sequence–to–sequence transformation defined by a matrix A denote by μ ( A ) the set of complex numbers μ such that I − μA is Mercerian, i.e., equivalent to convergence; in particular let D α = μ ( C α ) and E α = μ ( H α ), where C α and H α are the Cesàro and Hölder matrices of positive order α , respectively. We prove D α ⊂ E α for 0 < α < 1, and E α ⊂ D α for α > 1, as has been conjectured by B. Kuttner [3].

We show that if the series, ∑ a n e −λ n s , with Hadamard gaps, represents a function of Φ-bounded variation on (0, ∞), then ∑Φ(| a n |) converges.