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

Further, making the change , one obtains

Making the changes with and

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