Let be the open unit disk in the complex plane . By it is denoted the set of all holomorphic functions in .
A univalent function on is called spirallike (resp., starlike) on if for some with (resp., with ) and for each , the element belongs to whenever .
It is clear that . Moreover,
(i) if , then is called spirallike (resp., starlike) with respect to an interior point;
(ii) if 0 it is not in , then is called spirallike (resp., starlike) with respect to a boundary point.
In this case, there is a boundary point (say, ) such that ; by symbol it is denoted the angular (non-tangential) limit of a function at a boundary point of .
The class of spirallike (starlike) functions with respect to a boundary point normalized by the conditions and will be denoted by Spiral (resp., Star).
It is known that for any function , the limit
exists with , where ,
For given , the class of spirallike functions which satisfy the above relation, will be denoted by . If the number is real, that is, , then the function is, in fact, starlike. In this case, we can write .
We can say that a univalent function belongs to the class if it is of the class and satisfies the conditions , . It is know that if is a real number (i.e. ), then is so and . Since in this case is starlike, we can write .
The authors study the angle bounds for starlike and spirallike functions with respect to a boundary point.
We say that a univalent function normalized by is convex in the positive direction of the real axis if for each and , , . 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.