A function
which is regular and univalent in the disc
,
,
and maps
onto a
star-like domain
with respect to
.
A function
,
in
,
,
,
regular in
,
is star-like in this disc if and only if it satisfies the condition
The family of star-like functions in
,
normalized so that
,
,
forms the
class
,
which admits a parametric representation by Stieltjes integrals:
where
is a non-decreasing function on
,
.
For the class
the
coefficient problem
has been solved; sharp estimates have been found for
,
,
,
(the argument of the function is the branch that vanishes at
).
The extremal functions for these estimates are
,
where
is real. The class
of functions
is related to the class of functions
,
,
,
that are regular and univalent in
and map
onto a convex domain, by the formula
.
A star-like function such that
is called a
star-like function of order
in
.
Attention has also been given to univalent star-like functions in an annulus (see
[1]),
-valent
star-like functions and weakly star-like functions in a disc (see
[2],
[4]),
-locally
star-like functions (see
[1]),
and functions which are star-like in the direction of the real axis (see
[3]).
For star-like functions of several complex variables, see
[5].