Abstract
Full Text
UDC 517.54
MATHEMATICS
I. A. ALEKSANDROV, V. Ya. GUTLYANSKII
ON THE PROBLEM OF COEFFICIENTS
IN THE THEORY OF UNIVALENT FUNCTIONS
(Presented by Academician M. A. Lavrent’ev on 27 I 1969)
1°. In the complex plane (I), the set of values of the functional
[
I(f)=c_{n+1}\qquad (n=1,2,\ldots)
\tag{1}
]
on the class (S) of holomorphic univalent functions in the disk (E={z:\ |z|<1}),
[
f(z)=z+c_2z^2+\ldots+c_{n+1}z^{n+1}+\ldots
]
forms a closed disk (K_n) with center at the point (I=0). Finding the radius of this disk is one of the principal problems in the theory of univalent functions.
The boundary functions with respect to the functional (1) on the class (S) are solutions of the differential-functional equation
[
Q(f(z))\left(\frac{zf'(z)}{f(z)}\right)^2
=
N(z)+\overline{N}\left(\frac{1}{z}\right),
\tag{2}
]
where
[
Q(w)=x\sum_{m=1}^{n}{f^{m+1}}_{n+1}w^{-m},
]
[
N(z)=x\left[\frac{n}{2}{f}{n+1}+\sum\right].}^{n}m{f}_{m}z^{m-n-1
]
By ({\varphi}k) here and below we denote the coefficient of (z^k) in the expansion of the holomorphic function (\varphi(z)) in a series in a neighborhood of the origin; (x=e^{i\alpha}), (-\pi<\alpha\leq\pi), is the unit vector of the outward normal to (K_n) at the boundary point (I={f}).
The study of the geometric properties of the functions forming the set (\widetilde S(n,x)) of solutions of equation (2) on the class (S) is the subject of the works ((^{1-5})).
It is known that, for fixed (n) ((n=1,2,\ldots)) and (\alpha), (-\pi<\alpha\leq\pi), the set (\widetilde S(n,x)) contains those (2n) functions
[
f(z)=z/(1-\varepsilon z)^2
=
z+2\varepsilon z^2+\ldots+(n+1)\varepsilon^n z^{n+1}+\ldots,
\tag{3}
]
for which (\varepsilon=\varepsilon_l), (\varepsilon_l=e^{\alpha i/n}\varepsilon_l^{(2n)}), where (\varepsilon_l^{(2n)}=e^{\frac{\pi i}{n}l}) ((l=0,\ldots,2n-1)). Each of them contributes to (K_n) either the point ((n+1)x), or the point (-(n+1)x). Function (3) maps the disk (E) onto the plane with a ray removed, joining the point (w=-1/(4\overline{\varepsilon})) with (w=\infty) and, under continuation, passing through the point (w=0). (\widetilde S(1,x)) consists only of functions of the form (3) with (\varepsilon=\pm x).
2°. In this note we give new examples of solutions of equation (2) and find a lower estimate for the number of functions constituting (\widetilde S(n,x)). Satisfying the necessary condition of extremality in the coefficient problem, the proposed solutions contribute interior points to (K_n), which is easy to see by comparing their moduli with the moduli of the corresponding coefficient
functions (3). However, with respect to a sufficiently large part of their neighborhood in the class (S), regarded as a metric space with a metric convergence in which is equivalent to uniform convergence inside the disk, all known solutions of equation (2), including the new ones, are extremal.
(3^\circ). We denote, as usual, by (S_p) ((p=1,2,\ldots)) the subclass of the class (S) consisting of functions
[
f(z)=z+c_{p+1}^{(p)}z^{p+1}+\cdots+c_{kp+1}^{(p)}z^{kp+1}+\cdots,
]
possessing (p)-fold rotational symmetry. Obviously, (S_1\equiv S).
Taking (p) ((p=2,3,\ldots)) to be one of the divisors of the number (n) ((n=2,3,\ldots)) and putting (n=kp), consider the problem of determining the range of values of the functional
[
I(f)=c_{kp+1}^{(p)}
]
on the class (S_p). Every function contributing a boundary point to this domain satisfies the equation
[
Q_p(f(z))\,(zf'(z)/f(z))^2=N_p(z)+\overline{N}_p(1/z),
\tag{4}
]
where
[
Q_p(w)=x\sum_{s=1}^{k}{f^{sp+1}}_{kp+1}w^{-sp},\qquad
x=e^{i\alpha},\quad -\pi<\alpha\leq \pi,
]
[
N_p(z)=x\left[\frac{kp}{2}{f}{kp+1}+\sum\right].}^{k-1}(sp+1){f}_{sp+1}z^{p(s-k)
]
Theorem 1. Let (p) be a divisor of the number (n) ((n=2,3,\ldots;\ p=2,3,\ldots)), and let (k) be the quotient obtained by dividing (n) by (p). Then all solutions of equation (4) belonging to (S_p) are contained in (\widetilde S(n,x)).
The proof of the theorem can be carried out by directly verifying that every solution of equation (4) also satisfies equation (2).
Corollary 1. For any fixed (n) ((n=1,2,\ldots)), the functions
[
f(z)=\frac{z}{(1\pm xz^n)^{2/n}}
=z\mp \frac{2}{n}xz^{n+1}+\cdots \in \widetilde S(n,x).
]
Each of them maps the disk (E) onto the whole plane with slits removed, beginning at the points (w=\dfrac{1}{\sqrt[n]{4}}e^{-\alpha_i/n}e_l^{(n)}) (or (w=\dfrac{1}{\sqrt[n]{4}}e^{\frac{\pi-\alpha}{n}i}e_l^{(n)})),
[
e_l^{(n)}=e^{\frac{2\pi i}{n}}\qquad (l=0,\ldots,n-1),
]
and, upon continuation, passing through the point (w=0).
Corollary 2. Suppose that the number (n) admits a binary factorization (n=kp) with factors (k,p) different from unity. Then
[
f(z)=\frac{z}{(1-\varepsilon z^p)^{2/p}}
=z+\cdots+\binom{-2/p}{k}(-\varepsilon)^k z^{n+1}+\cdots \in \widetilde S(n,x)
]
for (\varepsilon=e^{\alpha i/k}e_l^{(2k)}) ((l=0,\ldots,2k-1)).
Corollary 3. In (\widetilde S(4,x)), in addition to the functions indicated in Corollaries 1 and 2, there are the functions (f(z)=\eta\varphi(\eta z)), (\eta=\pm e^{\frac{\alpha+k\pi}{2}i}) ((k=0,1)), where (w=\varphi(z)) is implicitly defined by the equation
[
\frac{a}{2}\ln\frac{1+\sqrt{1-aw}}{1-\sqrt{1-aw}}
-\frac{\sqrt{1-aw}}{w}
=z-\frac{1}{z}-\frac{a}{2}\ln z,\qquad a=4e^{-1}.
]
The function (\varphi(z)) maps the disk (E) onto the domain obtained by removing from the plane the ray (w_0\leq \operatorname{Re} z\leq +\infty), (w_0=1/a), and two analytic arcs symmetric with respect to the real axis, forming angles (2\pi/3) with this ray at the point (w_0).
The function (\varphi(z)) is a solution of equation (4) for (k=p=2) and (x=\pm1).
Theorem 2. If the canonical factorization of the number (n) has the form
[
n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}
\quad (p_1,p_2,\ldots,p_k \text{ are prime numbers}),
\tag{5}
]
then the set (\widetilde S(n,x)) consists of no fewer than (2\beta(n)) elements, where
[
\beta(n)=\prod_{r=1}^{k}\frac{p_r^{\alpha_r+1}-1}{p_r-1}.
]
In particular,
[
\beta(4)=7,\quad \beta(5)=6,\quad \beta(6)=12,\quad \beta(7)=8,\quad \beta(8)=15.
]
Upper estimates for the number of elements in (\widetilde S(n,x)) are unknown. The points obtained from (\widetilde S(n,x)) by functions in (K_n) form isolated circular orbits (\widetilde K(n)) with center at (I=0). Lower estimates for the number (\gamma(n)) of such orbits are given by
Theorem 3. If (n) has the canonical factorization (5), then
[
\gamma(n)\ge
1+(\alpha_1+\alpha_2+\cdots+\alpha_k)
+(\alpha_1\alpha_2+\alpha_1\alpha_3+\cdots+\alpha_{k-1}\alpha_k)+
]
[
+(\alpha_1\alpha_2\alpha_3+\alpha_1\alpha_2\alpha_4+\cdots+\alpha_{k-2}\alpha_{k-1}\alpha_k)
+\cdots+\alpha_1\alpha_2\cdots\alpha_k.
]
In particular,
[
\gamma(4)\ge 3,\quad \gamma(5)\ge 2,\quad \gamma(6)\ge 4,\quad \gamma(7)\ge 2,\quad \gamma(8)\ge 4.
]
Upper estimates for (\eta(n)) are unknown.
(4^\circ). Analogous propositions hold for solutions of the functional-differential equations characterizing, in the method of interior variations, the boundary functions in the problem on the range of values of the functional
[
I(F)=b_n
\tag{6}
]
on the class (\Sigma) of univalent functions in (|z|>1) of the form
(F(z)=z+b_0+b_1/z+\cdots+b_n/z^n+\cdots), holomorphic in
(1<|z|<\infty).
In view of the duality of the problem on the range of values of the functional (1), [(6)] on (S[\Sigma]) to a certain functional depending polynomially on the coefficients of functions of the class (\Sigma[S]), the non-uniqueness theorems indicated for the class (S[\Sigma]) carry over, with natural changes, to the class (\Sigma[S]).
In conclusion we note that the theorems on non-uniqueness of solutions of the equations constituting a necessary condition for extremality of a function in the method of interior variations also occur in more general problems connected with the determination of ranges of values of functionals analytically depending on the coefficients of a function (f(z)) of the class (S). We indicate only one of them, having first agreed to denote by (\widetilde S(n_1,\ldots,n_l;x)) the set of solutions of the differential-functional equation corresponding to the problem on the range of values of the functional
[
I(f)=J(c_{n_1+1},\ldots,c_{n_l+1})
\tag{7}
]
on the class (S); (J=J(w_1,\ldots,w_l)) is an analytic function in the domain of variation of the coefficients ((c_{n_1+1},\ldots,c_{n_l+1})), (f(z)\in S), in the space (C^l).
Theorem 4. Let (p) be a common divisor of the numbers (n_1,\ldots,n_l) ((l=2,3,\ldots)). Then all solutions of the equation characterizing boundary functions with respect to the functional (7) on the class (S_p) belong to
(\widetilde S(n_1,\ldots,n_l;x)).
Tomsk State University
named after V. V. Kuibyshev
Donetsk Computing Center
of the Academy of Sciences of the Ukrainian SSR
Received
17 I 1969
References
- M. Schiffer, Am. J. Math., 65, 341 (1943).
- M. Schiffer, appendix to the book by R. Courant, Dirichlet’s Principle, Conformal Mappings, and Minimal Surfaces, IL, 1953.
- G. M. Goluzin, Mat. sb., 19 (61), 203 (1946).
- A. C. Schaeffer, D. S. Spencer, Am. Math. Soc., Colloquium Publ., 35, N.Y. (1950).
- M. Schiffer, D. C. Spencer, Functionals on Finite Riemann Surfaces, IL, 1957.