Typically-real function
in a domain
A function , analytic in some domain
in the
-plane containing segments of the real axis, which is real on these segments and for which
whenever
. A fundamental class of typically-real functions is the class
of functions
![]() |
that are regular and typically real in the disc (cf. [1]). It follows from the definition of the class
that
is real for
. The class
contains the class
of functions
![]() |
with real coefficients , that are regular and univalent in
(cf. Univalent function). If
, then
![]() |
and, conversely, if , then
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where is the class of functions
![]() |
that are regular in with
in
and such that
is real for
.
Let be the class of non-decreasing functions
on
for which
. Functions of class
can be represented in
by Stieltjes integrals (cf. [2]):
![]() | (1) |
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in the sense that for each there exists an
such that (1) holds and, conversely, for any
formula (1) defines some function
. One has
for any fixed
. The largest domain in which every function in
is univalent is
. From the representation (1) for the class
, a number of rotation and distortion theorems have been obtained (cf. Distortion theorems; Rotation theorems). The following hold in the class
:
![]() | (2) |
![]() | (3) |
![]() |
with equality on the left in (2) only for and on the right only for
, on the left in (3) only for functions
for some
, and on the right only for
,
.
For , the coefficient regions for the systems
,
,
,
, have been found (cf. [3]).
References
[1] | W. Rogosinski, "Ueber positive harmonische Entwicklungen und typische-reelle Potenzreihen" Math. Z. , 35 (1932) pp. 93–121 |
[2] | G.M. Goluzin, "On typically real functions" Mat. Sb. , 27 : 2 (1950) pp. 201–218 (In Russian) |
[3] | G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian) |
Comments
References
[a1] | P.L. Duren, "Univalent functions" , Springer (1983) pp. Sect. 10.11 |
[a2] | A.W. Goodman, "Univalent functions" , 1 , Mariner (1983) |
Typically-real function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Typically-real_function&oldid=18776