Reports of the Academy of Sciences of the USSR

1957, Volume 117, No. 5

MATHEMATICS

G. V. KUZ’MINA

DETERMINATION OF THE SMALLEST RADIUS OF UNIVALENCE FOR A CLASS OF ANALYTIC FUNCTIONS

(Presented by Academician V. I. Smirnov on 21 VI 1957)

Let \(H_1^*(a)\) denote the class of functions of the form

\[ f(z)=\sum_{k=0}^{\infty} c_k z^k=\left(\sum_{k=0}^{\infty} d_k z^k\right)^2, \tag{1} \]

where the functions \(F(z)=\sum_{k=0}^{\infty} d_k z^k\) are regular in \(|z|<1\),

\[ \sum_{k=0}^{\infty} |d_k|^2=1, \tag{2} \]

\[ |c_1|=2|d_0|\,|d_1|=a \quad (0<a\le 1). \tag{3} \]

The class \(H_1^*(a)\) is a subclass of the functions of the class \(H_1\), investigated by G. M. Goluzin \((^1)\).

We shall find the smallest radius of univalence \(R\) for all functions of the class \(H_1^*(a)\). For this purpose, for any fixed \(n\) we determine the smallest radius of univalence \(R_{2n}\) of all polynomials of degree \(\le 2n\) from this class, i.e., of all polynomials representable in the form

\[ f_{2n}(z)=\sum_{k=0}^{2n} c_k z^k=\left(\sum_{k=0}^{n} d_k z^k\right)^2, \tag{1_n} \]

where

\[ \sum_{k=0}^{n} |d_k|^2=1, \tag{2_n} \]

\[ |c_1|=2|d_0||d_1|=a, \tag{3} \]

and then let \(n\to\infty\).

Let \(f_{2n}(z)\in H_1^*(a)\). Form for it the divided difference

\[ K_{2n}(z_1,z_2)=\frac{f_{2n}(z_1)-f_{2n}(z_2)}{z_1-z_2} =\frac{(d_0+d_1z_1+\ldots+d_nz_1^n)^2-(d_0+d_1z_2+\ldots+d_nz_2^n)^2}{z_1-z_2} = \]

\[ =\bigl[2d_0+d_1(z_1+z_2)+\ldots+d_n(z_1^n+z_2^n)\bigr] \bigl[d_1+d_2(z_1+z_2)+\ldots \]

\[ \ldots+d_n(z_1^{\,n-1}+z_1^{\,n-2}z_2+\ldots+z_2^{\,n-1})\bigr]. \tag{4_n} \]

It is obvious that, for \(|z_1|=|z_2|=r\),

\[ \left|2d_0+d_1(z_1+z_2)+\ldots+d_n(z_1^n+z_2^n)\right| \ge 2\bigl(|d_0|-r|d_1|-\ldots-r^n|d_n|\bigr), \tag{5_n} \]

\[ \left|d_1+d_2(z_1+z_2)+\ldots+d_n(z_1^{\,n-1}+z_1^{\,n-2}z_2+\ldots+z_2^{\,n-1})\right| \ge \]

\[ \ge |d_1|-2r|d_2|-\ldots-nr^{\,n-1}|d_n|, \tag{6_n} \]

and the equality signs are attained.

From the representation of the divided difference \(K_{2n}(z_1,z_2)\) in the form \((4_n)\) and from inequalities \((5_n)\) and \((6_n)\), it follows that

\[ R_{2n}=\min\{r_n^*,r_n^{**}\}, \tag{7_n} \]

where \(r_n^*\) and \(r_n^{**}\) are, respectively, the least positive roots of the equations

\[ \Phi_n^*(r)=|d_0|-r|d_1|-\ldots-r^n|d_n|=0, \tag{8_n} \]

\[ \Phi_n^{**}(r)=|d_1|-2r|d_2|-\ldots-nr^{n-1}|d_n|=0 \tag{9_n} \]

with respect to all \(|d_0|, |d_1|,\ldots, |d_n|\) satisfying the conditions \((2_n)\) and \((3)\).

Let first \(n=1\). It is easy to observe that the set of admissible values of the parameters \(|d_0|, |d_1|\) consists in this case only of two pairs of values

\[ 1)\ |d_0|=\sqrt{\frac{1-\sqrt{1-a^2}}{2}},\qquad |d_1|=\sqrt{\frac{1+\sqrt{1-a^2}}{2}}; \]

\[ 2)\ |d_0|=\sqrt{\frac{1+\sqrt{1-a^2}}{2}},\qquad |d_1|=\sqrt{\frac{1-\sqrt{1-a^2}}{2}} \]

(for \(a=1\) we have only one pair of values). Then from the equation

\[ \Phi_1^*(r)=|d_0|-r|d_1|=0 \tag{8_1} \]

it follows that either

\[ 1)\ r=\frac{1-\sqrt{1-a^2}}{a} \]

or

\[ 2)\ r=\frac{1+\sqrt{1-a^2}}{a}. \]

Hence it is clear that

\[ R_2=r_1^*=\frac{1-\sqrt{1-a^2}}{a}. \]

Let \(n\ge 2\). To find \(r_n^*\), we first note that the least value of the function \(\Phi_n^*(r)\), for any fixed \(r\), with respect to all \(|d_k|\), \(k=0,1,\ldots,n\), satisfying the conditions \((2_n)\) and \((3)\), is attained only when the equalities

\[ |d_{k+1}|=r|d_k|,\quad k=2,3,\ldots,n. \tag{10_n} \]

hold. For such \(|d_k|\), \(k=2,3,\ldots,n\), equation \((8_n)\) and conditions \((2_n)\), \((3)\) can be rewritten in the following form:

\[ |d_0|-r|d_1|-r^2\frac{1-r^{2n-2}}{1-r^2}|d_2|=0, \tag{11_n} \]

\[ |d_0|^2+|d_1|^2+\frac{1-r^{2n-2}}{1-r^2}|d_2|^2=1, \tag{12_n} \]

\[ |d_1|=\frac{a}{2|d_0|}. \tag{3} \]

Eliminating \(|d_1|\) and \(|d_2|\) from equation \((11_n)\) by means of equalities \((12_n)\) and \((3)\), we obtain a biquadratic equation with respect to \(|d_0|\):

\[ 4\left(1+r^4\frac{1-r^{2n-2}}{1-r^2}\right)|d_0|^4 -4\left(ar+r^4\frac{1-r^{2n-2}}{1-r^2}\right)|d_0|^2 +a^2\left(r^2-r^4\frac{1-r^{2n-2}}{1-r^2}\right)=0. \tag{13_n} \]

The problem has been reduced to determining the smallest positive root \(r_n^*\) of equation \((13_n)\) for all possible admissible values of \(|d_0|\)

\[ \frac{1-\sqrt{1-a^2}}{2}\leq |d_0|^2\leq \frac{1+\sqrt{1-a^2}}{2}. \tag{14} \]

The smallest positive root \(r_n^*\) of equation \((13_n)\) with respect to all admissible values of \(|d_0|\) must be either 1) a root of equation \((13_n)\) for

\[ |d_0|^2=\frac{1-\sqrt{1-a^2}}{2}, \]

or 2) a root of equation \((13_n)\) for

\[ |d_0|^2=\frac{1+\sqrt{1-a^2}}{2}, \]

or 3) the smallest positive value \(r=\widetilde r_n\) at which the discriminant of equation \((13_n)\) with respect to \(|d_0|^2\) becomes zero.

In the first case we obtain that

\[ r=\frac{1-\sqrt{1-a^2}}{a}. \]

In the second case we find that

\[ r=\frac{1+\sqrt{1-a^2}}{a}. \]

In the third case we arrive at the equation

\[ a^2-2ar+a^2r^2-(1-a^2)r^4\frac{1-r^{2n-2}}{1-r^2}=0. \]

This equation has the smallest positive root

\[ \widetilde r_n<\frac{1-\sqrt{1-a^2}}{a}. \]

It is not difficult to verify that, in this case, the corresponding value

\[ |\widetilde d_0|^2= \frac{a\widetilde r_n+\widetilde r_n^4\dfrac{1-\widetilde r_n^{2n-2}}{1-\widetilde r_n^2}} {2\left(1+\widetilde r_n^4\dfrac{1-\widetilde r_n^{2n-2}}{1-\widetilde r_n^2}\right)} = \frac12\,\frac{a(a-\widetilde r_n)}{1-a\widetilde r_n} \]

satisfies condition (14) for all \(a\), \(0<a\leq 1\). This means that \(r_n^*=\widetilde r_n\).

Thus, \(r_n^*\) is the smallest positive root of the equation

\[ \varphi_n^*(r)=a^2-2ar+a^2r^2-(1-a^2)r^4\frac{1-r^{2n-2}}{1-r^2}=0. \tag{15_n} \]

To determine \(r_n^{**}\), we note that, for any fixed \(r\), the smallest value of the function \(\Phi_n^{**}(r)\) with respect to all admissible values \(|d_k|\), \(k=0,1,\ldots,n\), is attained only when the condition

\[ |d_{k+1}|=\frac{k+1}{k}\,r\,|d_k|,\qquad k=2,3,\ldots,n. \tag{16_n} \]

is fulfilled. For such values \(|d_k|\), \(k=2,3,\ldots,n\), equation \((9_n)\) and the equalities \((2_n)\), (3) take the following form:

\[ |d_1|-\frac{|d_2|}{2}\,r\,(4+9r^2+\cdots+n^2r^{2n-4})=0, \tag{17_n} \]

\[ |d_0|^2+|d_1|^2+\frac{|d_2|^2}{4}\,(4+9r^2+\cdots+n^2r^{2n-4})=1, \tag{18_n} \]

\[ |d_1|=\frac{a}{2|d_0|}. \tag{3} \]

Denoting

\[ 4+9r^2+\cdots+n^2r^{2n-4}=S_n(r) \]

and, eliminating from \((17_n)\) \(|d_1|\) and \(|d_2|\) with the aid of \((18_n)\) and (3), we obtain a biquadratic equation with respect to \(|d_0|\):

\[ 4r^2 S_n(r)|d_0|^4-4r^2 S_n(r)|d_0|^2+a^2(1+r^2S_n(r))=0, \tag{19_n} \]

and the problem reduces to finding the least positive root \(r_n^{**}\) of equation \((19_n)\) for all admissible values of \(|d_0|\).

Repeating the arguments given above, we find that \(r_n^{**}\) is the least positive root of the equation

\[ \varphi_n^{**}(r)=a^2-(1-a^2)r^2S_n(r)=0. \tag{20_n} \]

Passing in \((15_n)\) and \((20_n)\) to the limit as \(n\to\infty\), we obtain that the least radius of univalence \(R\) of the functions of the class \(H_1^*(a)\) is determined by the equality

\[ R=\min\{r^*,\ r^{**}\}, \]

where \(r^*\) and \(r^{**}\) are, respectively, the least positive roots of the equations

\[ \varphi^*(r)=a^2-2ar+2ar^3-r^4=0 \]

and

\[ \varphi^{**}(r)=a^2-(4-a^2)r^2+3r^4-r^6=0. \]

If \(R=r^*\), then the extremal function is

\[ f^*(z)= \frac{e^{2i\alpha_0}}{2r^*(a-ar^{*2}+r^{*3})} \left[ \frac{ar^*-(a+r^*)e^{i\alpha}z+r^{*2}e^{2i\alpha}z^2} {1-r^*e^{i\alpha}z} \right]^2, \]

where \(\alpha_0\) and \(\alpha\) are arbitrary real numbers. If \(R=r^{**}\), then the extremal function is

\[ f^{**}(z)= \frac{1}{2r^{**2}(4-3r^{**2}+r^{**4})} \left[ a(1+r^{**2})e^{i\alpha_0}+ e^{i\alpha_1} \frac{ r^{**2}(4-3r^{**2}+r^{**4})z -2(1+r^{**2})r^{**}e^{i\alpha}z^2 +(1+r^{**2})r^{**2}e^{2i\alpha}z^3 } {(1-r^{**}e^{i\alpha}z)^2} \right]^2, \]

where \(\alpha_0\), \(\alpha_1\), and \(\alpha\) are arbitrary real numbers. The indicated functions have zeros of the derivatives on \(|z|=r^*\) and \(|z|=r^{**}\), respectively.

In conclusion I take this opportunity to thank N. A. Lebedev for his help in writing this paper.

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

Received
20 VI 1957

REFERENCES

1. G. M. Goluzin, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 18 (1946).