Bazilevich functions
Let be the class of functions
that are analytic in the open unit disc
with
and
(cf. also Analytic function). Let
denote the subclass of
consisting of all univalent functions in
(cf. also Univalent function). Further, let
denote the subclass of
consisting of functions that are starlike with respect to the origin (cf. also Univalent function).
The Kufarev differential equation
![]() | (a1) |
![]() | (a2) |
where is a function regular in
, having positive real part and being piecewise continuous with respect to a parameter
, plays an important part in the theory of univalent functions. This differential equation can be generalized as the corresponding Loewner differential equation
![]() | (a3) |
![]() | (a4) |
where is a continuous complex-valued function with
(
).
Letting (
), (a1) can be written in the form
![]() | (a5) |
where ,
, and
is a function regular in
with
. Introducing a real parameter
, one sets
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Further, making the change , one obtains
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Making the changes with
and
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with , one obtains
![]() | (a6) |
which is the generalization of (a5).
Writing
![]() |
with
![]() |
and
![]() |
(a6) gives the Bernoulli equation
![]() | (a7) |
![]() |
If one takes
![]() |
in (a7), one obtains the integral
![]() | (a8) |
![]() |
Using
![]() |
and , one sees that
is uniformly convergent to a certain function
of the class
. This implies that
![]() | (a9) |
![]() |
Noting that (a9) implies
![]() | (a10) |
![]() |
I.E. Bazilevich [a1] proved that the function given by
![]() | (a11) |
![]() |
belongs to the class , where
![]() |
is regular in
with
,
is any real number, and
.
If one sets in (a11), then
![]() | (a12) |
Since in
, the function
given by (a12) satisfies
![]() | (a13) |
Therefore, the function satisfying (a13) with
is called a Bazilevich function of type
.
Denote by the class of functions
that are Bazilevich of type
in
.
1) If with
in
, then
![]() |
as (see [a3]).
2) Let be analytic in
. Then
if and only if
with
(see [a7]).
3) T. Sheil-Small [a12] has introduced the class of Bazilevich functions of type , given by
![]() |
4) If , then
is a close-to-convex
-valent function, where
is a rational number (see [a9]).
For other properties of Bazilevich functions, see [a4], [a8], [a10], [a2], [a6], [a11], and [a5].
References
[a1] | I.E. Bazilevich, "On a class of integrability by quadratures of the equation of Loewner–Kufarev" Mat. Sb. , 37 (1955) pp. 471–476 |
[a2] | R. Singh, "On Bazilevič functions" Proc. Amer. Math. Soc. , 38 (1973) pp. 261–271 |
[a3] | D.K. Thomas, "On Bazilevič functions" Trans. Amer. Math. Soc. , 132 (1968) pp. 353–361 |
[a4] | J. Zamorski, "On Bazilevič schlicht functions" Ann. Polon. Math. , 12 (1962) pp. 83–90 |
[a5] | P.L. Duren, "Univalent functions" , Grundl. Math. Wissenschaft. , 259 , Springer (1983) |
[a6] | P.J. Eenigenburg, S.S. Miller, P.T. Mocanu, M.O. Reade, "On a subclass of Bazilevič functions" Proc. Amer. Math. Soc. , 45 (1974) pp. 88–92 |
[a7] | F.R. Keogh, S.S. Miller, "On the coefficients of Bazilevič functions" Proc. Amer. Math. Soc. , 30 (1971) pp. 492–496 |
[a8] | S.S. Miller, "The Hardy class of a Bazilevič function and its derivative" Proc. Amer. Math. Soc. , 30 (1971) pp. 125–132 |
[a9] | P.T. Mocanu, M.O. Reade, E.J. Zlotkiewicz, "On Bazilevič functions" Proc. Amer. Math. Soc. , 39 (1973) pp. 173–174 |
[a10] | M. Nunokawa, "On the Bazilevič analytic functions" Sci. Rep. Fac. Edu. Gunma Univ. , 21 (1972) pp. 9–13 |
[a11] | Ch. Pommerenke, "Univalent functions" , Vandenhoeck&Ruprecht (1975) |
[a12] | T. Sheil-Small, "On Bazilevič functions" Quart. J. Math. , 23 (1972) pp. 135–142 |
Bazilevich functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bazilevich_functions&oldid=50402