zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Angle distortion theorems for starlike and spirallike functions with respect to a boundary point. (English) Zbl 1137.30003

Let Δ be the open unit disk in the complex plane 𝒞. By Hol(Δ,𝒞) it is denoted the set of all holomorphic functions in Δ.

A univalent function h on Δ is called spirallike (resp., starlike) on Δ if for some μ𝒞 with Reμ>0 (resp., μ with μ>0) and for each t0, the element e -μt h(z) belongs to h(Δ) whenever zΔ.

It is clear that 0h(Δ) ¯. Moreover,

(i) if 0h(Δ), then h is called spirallike (resp., starlike) with respect to an interior point;

(ii) if 0 it is not in h(Δ), then h is called spirallike (resp., starlike) with respect to a boundary point.

In this case, there is a boundary point (say, z=1) such that h(1):=lim z1 h(z)=0; by symbol lim it is denoted the angular (non-tangential) limit of a function at a boundary point of Δ.

The class of spirallike (starlike) functions with respect to a boundary point normalized by the conditions h(1)=0 and h(0)=1 will be denoted by Spiral[1] (resp., Star[1]).

It is known that for any function hSpiral[1], the limit

lim z1 (z-1)h ' (z) h(z)=μ

exists with μΩ, where Ω={λ:λ-1|1, λ0}·

For given μΩ, the class of spirallike functions which satisfy the above relation, will be denoted by Spiral μ [1]. If the number μ is real, that is, 0<μ2, then the function h is, in fact, starlike. In this case, we can write hStar μ [1].

We can say that a univalent function h belongs to the class Spiral μ,ν [1,-1] if it is of the class Spiral μ [1] and satisfies the conditions lim z-1 h(z)=, lim z-1 h ' (z)(z+1) h(z)=ν0. It is know that if μ is a real number (i.e. 0<μ2), then ν is so and -μν<0. Since in this case h is starlike, we can write hStar μ,ν [1,-1].

The authors study the angle bounds for starlike and spirallike functions with respect to a boundary point.

We say that a univalent function hHol(Δ,𝒞) normalized by h(0)=0 is convex in the positive direction of the real axis if for each zΔ and t>0, h(z)+th(Δ), lim t h -1 (h(z)+t)=1. The authors find the maximal width size of the image for functions of this class using an angular limit characteristic of functions under consideration. In other words, given such a function, they find the minimal horizontal strip which contains its image. The following question is also natural but more complicated: characterize those functions convex in the positive direction of the real axis whose images contain a whole (two-sided) strip and find the size (width) of this strip. The authors solve this problem for functions having maximal horizontal strips of finite size. The problem is still open for the general case.

30C45Special classes of univalent and multivalent functions