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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.

MSC:
30C45Special classes of univalent and multivalent functions