Contour and Solid Structure Properties of Holomorphic Functions of a Complex Variable

Growth conditions for analytic functions in unbounded open sets

“…In [1], Aliev and Tamrazov obtained more refined bounds of the same kind, with C depending on the same quantity as in [16] measuring the thickness of C \ G. The new feature in their bounds for |f (z 1 ) − f (z 2 )| for z 2 ∈ G, for a fixed z 1 ∈ ∂G, is that they contain extra factors depending on the values of g D (z 2 , ζ j ) where the points ζ j lie in the same component D of G as z 2 and satisfy f (ζ j ) = f (z 1 ). Here g D (z, w) is the Green's function of the domain D with pole at w ∈ D.…”

“…Consider the problem of deducing (1.3) from (1.2). In [16], Theorem 3.1, p. 149 and Theorem 9.1, p. 166, Tamrazov obtained such results where G can be fairly arbitrary as long as ∂G is uniformly thick enough and f is bounded, with C in (1.3) depending on a quantity that measures the uniform lower density of the logarithmic capacity of the complement of G. Thus an absolute constant C = 108 can be obtained whenever G is a simply connected domain; the author improved this to C = 74 in [9], Theorem A, p. 309 by essentially the same method. These proofs only use the fact that log |f | is subharmonic and do not make use of the argument of f .…”

“…Tamrazov proved in [16], Theorem 9.3, p. 167, that the condition that µ is bilogarithmically concave is necessary for a result like Theorem 7 to hold (that is, for us to have the smallest possible constant C = 1 in (1.3)) even if we limit G to be an arbitrary bounded Jordan domain.…”

“…to be a subharmonic function of z in G for each fixed z 1 ∈ ∂G is precisely that log µ(e t ) must be a concave function of t for real t. Tamrazov [17] has called such functions µ bilogarithmically concave. Since it is not clear that such a function µ needs to satisfy the condition (1.1), which we have taken as the defining property of a majorant, we shall not call such functions µ "bilogarithmically concave majorants".…”

Volterra-Type Operators on Zygmund Spaces

“…To this end, we need two lemmas. Before formulating these lemmas, we quote the following result from [17]. …”