UDC 517.54

MATHEMATICS

F. G. AVKHADIEV

ON SUFFICIENT CONDITIONS FOR THE UNIVALENCE OF SOLUTIONS OF INVERSE BOUNDARY-VALUE PROBLEMS

(Presented by Academician V. I. Smirnov on 14 IV 1969)

The present note continues the investigations of V. S. Rogozhin, L. A. Aksent'ev, and S. N. Kudryashov (see the bibliography in ((^1))). It gives sufficient conditions for the univalence of solutions of two inverse boundary-value problems (([^1],\ \text{Ch. }1;\ [^2],\ \S 33)) on the basis of the solution of certain problems in the theory of univalent functions.

§ 1. In Theorems 1–5 it is assumed that the function (f(z)) is regular in the disk (|z|<1), while the function (F(\zeta)) is regular in the domain (|\zeta|>1), except for the point (\zeta=\infty), where it has a simple pole.

Theorem 1. In order that the function (F(\zeta)) be univalent in (|\zeta|>1), it is necessary that

[
\left|\frac{F''(\zeta)}{F'(\zeta)}\right|\leq
\frac{6}{|\zeta|^3-|\zeta|},\qquad |\zeta|>1;
\tag{1}
]

sufficient that

[
\left|\frac{F''(\zeta)}{F'(\zeta)}\right|\leq
\frac{2(\sqrt5-2)}{|\zeta|^3-|\zeta|},\qquad |\zeta|>1.
\tag{2}
]

Inequality (1) follows from the theorem of G. M. Goluzin ((^3,\ \text{p. }139)). The proof of the sufficiency of condition (2) is carried out analogously to the proof of the corresponding result in ((^4)) for (f(z)).

Theorem 2. The function (F(\zeta)) is univalent in the domain (|\zeta|>1) if one of the following conditions is satisfied:

[
1^\circ.\quad
\frac{1}{2\pi}\int_{-\pi}^{\pi}
\left|\frac{F''(\zeta)}{F'(\zeta)}\right|^2\,d\theta
\leq \frac{8}{9},\qquad
\zeta=\rho e^{i\theta},\quad \rho>1.
\tag{3}
]

[
2^\circ.\quad
\left|\frac{F''(\zeta)}{F'(\zeta)}\right|\leq \frac{4}{3},\qquad |\zeta|>1.
\tag{4}
]

The proof is based on the following extension of V. V. Pokornyi’s sufficient condition for univalence ((^5)) to the domain (|\zeta|>1), by means of the change of variable (\zeta=1/z).

Lemma. In order that a meromorphic function (F(\zeta)) in (|\zeta|>1) be univalent in (|\zeta|>1), it is sufficient that, for some (\lambda) ((0\leq\lambda\leq2)), the inequality

[
|{F,\zeta}|\equiv
\left|
\left(\frac{F''(\zeta)}{F'(\zeta)}\right)'
-\frac{1}{2}
\left(\frac{F''(\zeta)}{F'(\zeta)}\right)^2
\right|
\leq
\frac{C(\lambda)}
{|\zeta|^4\left(1-|\zeta|^{-2}\right)^\lambda},
\qquad |\zeta|>1,
\tag{5}
]

holds, where

[
C(\lambda)=
\begin{cases}
2^{3\lambda-1}\pi^{2(1-\lambda)}, & 0\leq\lambda\leq1,\
2^{3-\lambda}, & 1\leq\lambda\leq2.
\end{cases}
]

By assumption, the function (F(\zeta)), satisfying the conditions of the theorem, is representable in (|\zeta|>1) by the series

[
F(\zeta)=a\zeta+a_0+a_1/\zeta+a_2/\zeta^2+\cdots,\qquad a\ne0.
]

Therefore, when (3) or (4) is fulfilled, the function regular in (|\zeta|>1),

(\varphi(\zeta)\equiv F''(\zeta)/F'(\zeta)), is representable by the series

[
\varphi(\zeta)=\sum_{k=3}^{\infty} c_k \zeta^{-k},\qquad |\zeta|>1 .
]

(1^\circ). Using the Cauchy–Bunyakovsky inequality, we have

[
|\varphi(\zeta)|^2
=
\left|\sum_{k=3}^{\infty} c_k \zeta^{-k}\right|^2
\le
\sum_{k=3}^{\infty}|c_k|^2\cdot
\sum_{k=3}^{\infty}|\zeta|^{-2k},
\qquad |\zeta|>1,
]

[
|\varphi'(\zeta)|^2
=
\left|\sum_{k=3}^{\infty} k c_k \zeta^{-k-1}\right|^2
\le
\sum_{k=3}^{\infty}|c_k|^2\cdot
\sum_{k=3}^{\infty} k^2|\zeta|^{-2k-2},
\qquad |\zeta|>1.
]

Condition (3), after passage to the limit as (|\zeta|\to 1), gives

(\sum_{k=3}^{\infty}|c_k|^2\le \dfrac{8}{9}), and therefore

[
|\varphi(\zeta)|^2\le
\frac{8}{9}\,
\frac{1}{|\zeta|^6(1-|\zeta|^{-2})},
\qquad
|\varphi'(\zeta)|^2\le
\frac{8}{9}\,
\frac{9-11|\zeta|^{-2}+4|\zeta|^{-4}}
{|\zeta|^8(1-|\zeta|^{-2})^3},
\qquad |\zeta|>1.
]

Noting that the Schwarzian

({F,\zeta}\equiv \varphi'(\zeta)-\tfrac12\varphi^2(\zeta)), we obtain the estimate

[
|{F,\zeta}|
\le
|\varphi'(\zeta)|+\tfrac12|\varphi(\zeta)|^2
\le
\frac{2\sqrt2}{3}\,S(t)\,
\frac{1}{|\zeta|^4(1-|\zeta|^{-2})^{3/2}},
]

where

[
S(t)=\sqrt{9-11t+4t^2}+\frac{\sqrt2}{3}\,t\sqrt{1-t},
\qquad
t=|\zeta|^{-2},\quad |\zeta|>1.
]

But (S'(t)<0,\ 0\le t\le 1); hence (S(t)\le S(0)=3). Then the Schwarzian ({F,\zeta}) satisfies inequality (5) of the lemma for (\lambda=3/2), which proves the univalence of the function (F(\zeta)).

(2^\circ). Condition (4) is equivalent to the inequality (|\varphi(\zeta)|\le 4/3). By the theorem of G. M. Goluzin (([3], p. 325)), applied to the function (\varphi(z^{-1})) in the disk (|z|<1), taking into account the order of the zero at the point (z=0), we shall have

[
\left|\varphi\left(\frac1z\right)\right|\le \frac43 |z|^3,
\qquad
\left|-\frac1{z^2}\varphi'\left(\frac1z\right)\right|
\le
\frac43\,\frac{3|z|^2}{1-|z|^6},
\qquad |z|<1,
]

or

[
|\varphi(\zeta)|\le \frac{4}{3|\zeta|^3},
\qquad
|\varphi'(\zeta)|\le
\frac{4}{|\zeta|^4(1-|\zeta|^{-6})},
\qquad |\zeta|>1.
]

The obtained inequalities allow us to estimate the Schwarzian

[
|{F,\zeta}|
\le
|\varphi'(\zeta)|+\tfrac12|\varphi(\zeta)|^2
\le
\frac{4}{|\zeta|^4(1-|\zeta|^{-2})},
\qquad |\zeta|>1,
]

i.e., the function (F(\zeta)) satisfies condition (5) of the lemma for (\lambda=1), and consequently is univalent.

§ 2. Let (f'(z)\ne 0) in (|z|<1) and (F'(\zeta)\ne 0) in (|\zeta|>1). Then the integrals

[
\int [f'(z)]^\alpha dz,\quad |z|<1;
\qquad
\int [F'(\zeta)]^\alpha d\zeta,\quad |\zeta|>1,
]

where (\alpha) is a complex constant, define certain analytic functions (f_\alpha(z)) and (F_\alpha(\zeta)) in the domains (|z|<1) and (|\zeta|>1), respectively. For the functions (f_\alpha(z)) and (F_\alpha(\zeta)) the following assertions are valid.

Theorem 3. If (f(z)) and (F(\zeta)) are univalent, then the functions (f_\alpha(z)) and (F_\alpha(\zeta)) are also univalent for every complex (\alpha) satisfying the inequality

[
3|\alpha|+9|\alpha-\alpha^2|\le 1.
]

We note that, for the function (f(z)), Theorem 3 includes the result from paper ((^{4})), coinciding with it for real and negative (\alpha). The following two theorems develop the ideas of the indicated paper ((^{4})) for certain subclasses of univalent functions.

Theorem 4. If the function (f(z)) is univalent and maps the disk (|z|<1) onto a convex domain, then the function (f_\alpha(z)) is univalent for any complex (\alpha) satisfying the inequality
[
|\alpha|+4|\alpha-\alpha^2|\leq 1;
]
for real (\alpha) it is sufficient to require only that the inequality
[
-\frac12 \leq \alpha \leq 1
]
hold.

Theorem 5. If the function (F(\zeta)) is univalent and maps (|\zeta|>1) onto a domain with convex complement, then the function (F_\alpha(\zeta)) is univalent for any complex (\alpha) satisfying the inequality
[
3|\alpha|+4|\alpha-\alpha^2|\leq 1.
]

The theorems formulated above are obtained using the known necessary conditions for univalence (\bigl((^{3}),) pp. 52, 139, 510, 582\bigr)) and Nehari’s results ((^{6})).

§ 3. Let
[
f(z)=\int \exp\left{\frac{1}{2\pi}\int_{-\pi}^{\pi} P(\theta)\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\theta\right}\,dz,\qquad |z|<1,
\tag{6}
]
where (P(\theta)) is a real (2\pi)-periodic function.

Theorem 6. The function (6) is univalent if one of the following conditions is fulfilled:

(1^\circ.) The function (P(\theta)) is continuous and, for any (\theta_1) and (\theta_2), (-\pi<\theta_1<\theta_2\leq \pi), the inequality
[
-\pi<(\theta_2-\theta_1)+\frac{P(\theta_1)-P(\theta_2)}{a}<3\pi
]
holds, or the stronger condition
[
|P(\theta_1)-P(\theta_2)|<\pi |a|,
]
where (a) is one of the real roots of the equation
[
3|a|+9|ai+a^2|=1.
]

(2^\circ.) There exists a continuous derivative (P'(\theta)) of the function (P(\theta)), essentially bounded on one side,
[
P'(\theta)>-|b|
]
or
[
P'(\theta)<|b|,
]
where (b) is a real root of the equation
[
|b|+4|bi+b^2|=1.
]

The proof is carried out by applying Theorems 3 and 4 to the auxiliary function (g(z)), defined by the equality ([g'(z)]^\alpha=f'(z)). For (\alpha=ai) ((\alpha=\pm bi)), it follows from condition (1^\circ) ((2^\circ)) that the function (g(z)) is almost convex (convex) in the disk (|z|<1).

Let
[
F(\zeta)=\int \exp\left{-\frac{1}{2\pi}\int_{-\pi}^{\pi} P(\theta)\frac{e^{i\theta}+\zeta}{e^{i\theta}-\zeta}\,d\theta\right}\,d\zeta,\qquad |\zeta|>1,
\tag{7}
]
where the real (2\pi)-periodic function (P(\theta)) satisfies the condition
[
\int_{-\pi}^{\pi} P(\theta)e^{i\theta}\,d\theta=0.
\tag{8}
]

Theorem 7. The function (F(\zeta)), defined by the integral representation (7) with the additional condition (8), is univalent if one of the following conditions is fulfilled:

(1^\circ). The function (P(\theta)) is continuous and has bounded oscillation

[
|P(\theta_1)-P(\theta_2)|<A,
]

where (A) is the real root of the equation (\operatorname{sh} A=(\sqrt{5}-2)/2).

(2^\circ). There exists a continuous derivative (P'(\theta)) of the function (P(\theta)), essentially bounded on one side, (P'(\theta)>-|c|) or (P'(\theta)<|c|), where (c) is the real root of the equation (3|c|+4|ci+c^2|=1).

(3^\circ). The function (P(\theta)) satisfies the Lipschitz condition

[
|P(\theta_1)-P(\theta_2)|\le K|\theta_1-\theta_2|,\quad \text{where } K\le {}^2!/_{3}.
]

From conditions (1^\circ) and (3^\circ) it follows that (F(\zeta)) satisfies inequalities (2) or (3), respectively, and therefore is univalent. If (2^\circ) is fulfilled, the function (\Phi(\zeta)) (([\Phi'(\zeta)]^{\pm ci}=F'(\zeta))) maps (|\zeta|>1) onto a domain with convex complement. The univalence of (F(\zeta)) in this case follows from Theorem 5 applied to (\Phi(\zeta)) with (a=\pm ci).

Let us draw some conclusions. Conditions (1^\circ) of Theorems 6 and 7 show that (P(\theta)) need not possess differentiability properties. Conditions (2^\circ) constitute a first attempt to restrict the interval of variation of (P'(\theta)) by half-infinite intervals, whereas in the known univalence conditions expressed in terms of (P'(\theta)) it was essentially assumed that (|P'(\theta)|\le M<\infty) (see, for example, (1), p. 56). Condition (3^\circ) of the last theorem was previously proved ((7), p. 16) only for (K\le 0,2,\ldots)

In conclusion, the author thanks N. A. Lebedev for valuable comments and his adviser L. A. Aksent’ev for constant attention to the work.

Kazan State University
named after V. I. Ulyanov-Lenin

Received
8 IV 1969

REFERENCES

(^{1}) G. G. Tumashev, M. T. Nuzhin, Inverse Boundary-Value Problems and Their Applications, Kazan, 1965.
(^{2}) F. D. Gakhov, Boundary-Value Problems, Moscow, 1963.
(^{3}) G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Moscow, 1966.
(^{4}) P. L. Duren, H. S. Shapiro, A. S. Shields, Duke Math. J., 33, No. 2, 247 (1966).
(^{5}) V. V. Pokornyi, DAN, 79, No. 5, 743 (1951).
(^{6}) Z. Nehari, Bull. Am. Math. Soc., 55, No. 6, 545 (1949).
(^{7}) L. A. Aksent’ev, Proceedings of the Seminar on Inverse Boundary-Value Problems, No. 2, 1964, p. 12.