UDC 517.54

MATHEMATICS

Yu. E. ALENITSYN

ON UNIVALENT FUNCTIONS WITHOUT COMMON VALUES IN A MULTIPLY CONNECTED DOMAIN

(Presented by Academician V. I. Smirnov on 14 VI 1965)

In the present note some new results are given for univalent functions without common values in a finitely connected domain, obtained from the consideration of suitable Dirichlet integrals.

Let (B) be a finite and finitely connected domain of the (z)-plane, bounded by closed analytic Jordan curves; (\mathscr L^2(B)) the class of all functions regular and with integrable squared modulus in the domain (B); (\mathscr L_0^2(B)) the subclass of functions from (\mathscr L^2(B)) with single-valued integrals in the domain (B); (K(z,\zeta)) and (l(z,\zeta)) the Bergman kernels ((^1)) of the first and second kind of the domain (B) of the class (\mathscr L^2(B)); (K_0(z,\zeta)) and (l_0(z,\zeta)) those of the class (\mathscr L_0^2(B));

[
U_k(z,\zeta)=\frac{1}{\pi}\left[\frac{f_k'(z)f_k'(\zeta)}
{(f_k(z)-f_k(\zeta))^2}-\frac{1}{(z-\zeta)^2}\right],\quad z,\zeta\in B,
]

where the function (f_k(z)) is regular* and univalent in the domain (B).

Theorem 1. If the functions (f_k(z)), (k=1,\ldots,n), are regular, univalent, and without common values in the domain (B), then for any points (\zeta_{k\mu}) of this domain and any constants (\alpha_{k\mu}), (\mu=1,\ldots,N), (k=1,\ldots,n), the inequality holds:

[
\left|
\sum_{\mu,\nu=1}^{N}
\left{
\sum_{k=1}^{n}\alpha_{k\mu}\alpha_{k\nu}
\bigl[U_k(\zeta_{k\mu},\zeta_{k\nu})+l_0(\zeta_{k\mu},\zeta_{k\nu})\bigr]
+
\frac{2}{\pi}
\sum_{1\le j<k\le n}
\alpha_{j\mu}\alpha_{k\nu}
\frac{f_j'(\zeta_{j\mu})f_k'(\zeta_{k\nu})}
{(f_j(\zeta_{j\mu})-f_k(\zeta_{k\nu}))^2}
\right}
\right|
\le
\sum_{\mu,\nu=1}^{N}\sum_{k=1}^{n}
\alpha_{k\mu}\overline{\alpha}{k\nu}
K_0(\zeta).},\overline{\zeta}_{k\nu
\tag{1(^0)}
]

This inequality remains valid also after the simultaneous replacement in it of the functions (l_0) and (K_0), respectively, by the functions (l) and (K) (this second inequality will henceforth be cited as inequality (1)).

One can indicate necessary and sufficient conditions under which equality holds in the inequalities ((1^0)) and (1)).

Each of the inequalities ((1^0)) and (1) determines the corresponding disk in which lie the possible values of the functional

[
\sum_{\mu,\nu=1}^{N}
\left[
\sum_{k=1}^{n}\alpha_{k\mu}\alpha_{k\nu}
U_k(\zeta_{k\mu},\zeta_{k\nu})
+
\frac{2}{\pi}
\sum_{1\le j<k\le n}
\alpha_{j\mu}\alpha_{k\nu}
\frac{f_j'(\zeta_{j\mu})f_k'(\zeta_{k\nu})}
{(f_j(\zeta_{j\mu})-f_k(\zeta_{k\nu}))^2}
\right]
]

for fixed (\zeta_{k\mu}) and (\alpha_{k\mu}), (\mu=1,\ldots,N); (k=1,\ldots,n).

The disk determined by inequality ((1^0)) lies in the disk determined by inequality (1).

For (n=1), inequalities ((1^0)) and (1) give inequalities previously obtained by another method by Bergman and Schiffer ((^1)).

Corollary 1. If the function (f(z)) is regular, univalent, and bounded in the domain (B): (|f(z)|<1), (z\in B), then for any points (\zeta_\mu) of this domain

* Here and below—including single-valuedness.

and any constants (\alpha_\mu,\ \mu=1,\ldots,N), the inequality holds:
[
\left|
\sum_{\mu,\nu=1}^{N}\alpha_\mu\alpha_\nu
\left[U(\xi_\mu,\xi_\nu)+l_0(\xi_\mu,\xi_\nu)\right]
\right|
\le
\sum_{\mu,\nu=1}^{N}\alpha_\mu\bar\alpha_\nu
\left[
K_0(\xi_\mu,\bar\xi_\nu)-
\frac{f'(\xi_\mu)\overline{f'(\xi_\nu)}}{\pi(1-f(\xi_\mu)\overline{f(\xi_\nu)})^2}
\right].
\tag{2}
]

In particular:
[
\left|
\frac{1}{6}{f,z}+\pi l_0(z,z)
\right|
+
\frac{|f'(z)|^2}{(1-|f(z)|^2)^2}
\le
\pi K_0(z,\bar z),\qquad z\in B,
]
where ({f,z}) is the Schwarzian invariant.

Inequality (2) strengthens Singh’s inequality ({}^{(2)}) for bounded univalent functions, in which (l_0) and (K_0) are replaced by (l) and (K).

Let (\widetilde C(B)) denote the class of all functions (f(z)), regular and univalent in the domain (B), satisfying the condition: (f(z_1)f(z_2)\ne 1) for any points (z_1) and (z_2) of the domain (B).

Corollary 2. If (f(z)\in\widetilde C(B)), then for any points (\xi_\mu) of the domain (B) and any constants (\alpha_\mu,\ \mu=1,\ldots,N), the inequality holds:
[
\left|
\sum_{\mu,\nu=1}^{N}\alpha_\mu\alpha_\nu
\left[
U(\xi_\mu,\xi_\nu)\pm
\frac{f'(\xi_\mu)f'(\xi_\nu)}
{\pi(1-f(\xi_\mu)f(\xi_\nu))^2}
+l_0(\xi_\mu,\xi_\nu)
\right]
\right|
\le
]
[
\le
\sum_{\mu,\nu=1}^{N}\alpha_\mu\bar\alpha_\nu K_0(\xi_\mu,\bar\xi_\nu),
]
In particular:
[
\left|
\frac{1}{6}{f,z}\pm
\left(\frac{f'(z)}{1-f^2(z)}\right)^2
+\pi l_0(z,z)
\right|
\le
\pi K_0(z,\bar z),\quad z\in B.
]

Each of these inequalities remains valid after replacing (l_0) and (K_0) in it respectively by (l) and (K).

Theorem 2. Let the functions (f_k(z)), (k=1,\ldots,n), be regular in a domain (B) containing the origin and satisfy the conditions: (f_j(0)\ne f_k(0)), (j\ne k), (f'k(0)\ne 0). In order that these functions be univalent functions without common values in the domain (B), it is necessary and sufficient that, for any (n) complex vectors ({\alpha}), (k=1,\ldots,n), with arbitrary (N=0,1,2,\ldots), the conditions},\ldots,\alpha_{kN
[
\sum_{\mu,\nu=0}^{N}
\left[
\sum_{k=1}^{n}\alpha_{k\mu}\alpha_{k\nu}(c_{\mu\nu}^{[k]}+\lambda_{\mu\nu})
+
\sum_{1\le j<k\le n}\alpha_{j\mu}\alpha_{k\nu}d_{\mu\nu}^{[jk]}
\right]
\le
\sum_{\mu,\nu=0}^{N}\varkappa_{\mu\nu}
\sum_{k=1}^{n}\alpha_{k\mu}\bar\alpha_{k\nu},
]
be satisfied, where (c_{\mu\nu}^{[k]}), (d_{\mu\nu}^{[jk]}), (\lambda_{\mu\nu}), and (\varkappa_{\mu\nu}) are determined by the following expansions into double series in a neighborhood of the origin:
[
U_k(z,\zeta)=
\sum_{\mu,\nu=0}^{\infty}c_{\mu\nu}^{[k]}z^\mu\zeta^\nu,
\qquad k=1,\ldots,n;
]
[
\frac{2f'j(z)f'_k(\zeta)}
{\pi[f_j(z)-f_k(\zeta)]^2}
=
\sumz^\mu\zeta^\nu,}^{\infty}d_{\mu\nu}^{[jk]
\qquad 1\le j<k\le n;
]
[
l(z,\zeta)=
\sum_{\mu,\nu=0}^{\infty}\lambda_{\mu\nu}z^\mu\zeta^\nu,\qquad
K(z,\bar\zeta)=
\sum_{\mu,\nu=0}^{\infty}\varkappa_{\mu\nu}z^\mu\bar\zeta^\nu.
]

For (n=1) this theorem gives the known ({}^{(1)}) necessary and sufficient conditions for univalence of a function in a multiply connected domain.

Corollary. Let the function (f(z)) be regular in the domain (B), containing the origin, and satisfy the conditions (f(0)\ne \pm 1), (f'(0)\ne 0). In order that the function (f(z)) belong to the class (\widetilde C(B)), it is necessary and sufficient that, for any two complex vectors ({a_{k0}, \ldots, a_{kN}}), (k=1,2), with arbitrary (N=0,1,2,\ldots), the conditions

[
\left|
\sum_{\mu,\nu=0}^{N}
\left[
(\alpha_{1\mu}\alpha_{1\nu}+\alpha_{2\mu}\alpha_{2\nu})(c_{\mu\nu}+\lambda_{\mu\nu})
-\alpha_{1\mu}\alpha_{2\nu}d_{\mu\nu}
\right]
\right|
\le
\sum_{\mu,\nu=0}^{N}
(\alpha_{1\mu}\overline{\alpha}{1\nu}
+\alpha}\overline{\alpha{2\nu})\,\chi,
]

hold, where the coefficients (c_{\mu\nu}), (\lambda_{\mu\nu}), and (\chi_{\mu\nu}) are defined in Theorem 2, while (d_{\mu\nu}) are determined by the expansion in a neighborhood of the origin

[
\frac{2f'(z)f'(\zeta)}
{\pi[1-f(z)f(\zeta)]^{2}}
=
\sum_{\mu,\nu=0}^{\infty} d_{\mu\nu}z^\mu \zeta^\nu .
]

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
10 VI 1965

REFERENCES

({}^{1}) S. Bergman, M. Schiffer, Comp. Math., 8 (1951). ({}^{2}) V. Singh, Ann. Acad. Sci. Fenn., Ser. A 1, 310 (1962).