MATHEMATICS

V. I. POPOV

ON THE STARLIKENESS OF ARCS OF LEVEL LINES UNDER UNIVALENT CONFORMAL MAPPINGS

(Presented by Academician M. A. Lavrent'ev, 7 XII 1963)

For the class \(S\) of holomorphic functions univalent in the disk \(|z|<1\),
\[ w=f(z), \quad f(0)=0,\quad f'(0)=1, \]
the inequalities are well known
\[ \underline R=\frac{r}{(1+r)^2}\leq |f(re^{i\varphi})|\leq \frac{r}{(1-r)^2}=\overline R, \]
which indicate the minimal annulus in which all level lines
\[ L(f,r)=\{w\mid w=f(re^{i\varphi}),\ -\pi<\varphi\leq \pi\} \]
of functions \(f(z)\in S\) are situated for fixed \(r\), \(0<r<1\). I. E. Bazilevich and G. V. Koričkiĭ \((^1)\) singled out in this annulus the ring of starlikeness, proving the existence of an absolute constant \(d_s\) \((0.1005\leq d_s<0.134)\) such that every arc of the line \(L(f,r)\) lying in the annulus
\[ d_s\overline R\leq |w|\leq \overline R \]
is starlike* for every function \(f(z)\) of the class \(S\), and such that in the enlarged annulus
\[ (d_s-\varepsilon)\overline R\leq |w|\leq \overline R,\quad \varepsilon>0, \]
the line \(L(f,r)\), for some function \(f(z)\) from \(S\), will have a non-starlike arc.

Along with the problem of determining the exact value of the constant \(d_s\), I. E. Bazilevich and G. V. Koričkiĭ posed (see \((^2)\)) the following problem: for each \(r\), \(r_s=\operatorname{th}\pi/4<r<1\), find
\[ a_s(r)=\inf a(r) \]
over all \(a(r)<1\) having the property that any arc of the line \(L(f,r)\) belonging to the annulus
\[ a(r)\overline R\leq |w|\leq \overline R \]
is starlike, whatever the function \(f(z)\in S\).

In this article we give a formula determining \(a_s(r)\) for
\[ 0.709\ldots<r<1, \]
and establish the exact value of the constant \(a_s\).

Theorem. Every arc of the level line \(L(f,r)\), \(f\in S\), \(r_s<r<1\), lying in the annulus
\[ \lambda \underline R\overline R\leq |w|\leq \overline R, \]
where
\[ \lambda=\frac{5\sqrt5}{4}\,e^{-4\operatorname{arcctg}2}, \]
is starlike.

In the case where \(r_0<r<1\), where \(r_0\) is the unique root of the equation
\[ \ln \frac{\sqrt5\,y}{2a(1-y^2)^2} +2\ln\left(\sqrt{1-a^2y^2}+\sqrt{y^2-a^2}\right) +\arc\sin \frac{1+y^2}{2\sqrt2\,y} -3\arc\tg 3=0,\qquad a=\sqrt2-1, \tag{1} \]
there exists a function \(f(z)\in S\) whose level line \(L(f,r)\), at some point
\[ w_1=f(z_1),\quad |z_1|=r, \]
of the wider annulus
\[ (\lambda-\varepsilon)\underline R\overline R\leq |w|\leq \overline R,\quad \varepsilon>0, \]
is not starlike.

Proof. \(1^\circ\). Denote
\[ zf'(z)/f(z)=F(z) \]
and establish an upper estimate for the real functional prescribed on the class \(S\)
\[ J_1(f)=\ln |f(z_0)|-2\arg F(z_0) \tag{2} \]

* In all cases, when speaking of the starlikeness of an analytic curve, we mean its starlikeness with respect to the origin.

for fixed \(z_0\) on the circle \(|z|=r,\ a<r<1\). To this end, let us consider the functional (2) on the subclass \(S(k)\), dense in \(S\), of functions determined by integrals of the Löwner equation

\[ \frac{\partial f(z,t)}{\partial t} = -f(z,t)\frac{1+k(t)f(z,t)}{1-k(t)f(z,t)}, \qquad 0\leq t\leq \infty, \]

with a piecewise-continuous characteristic function \(k(t)\), \(|k(t)|=1\). Since on \(S(k)\) *

\[ \ln |f(z_0)| = \ln \frac{r}{1-r^2} - 2\int_0^r \cos\psi\,\frac{ds}{1-s^2}, \qquad \arg \frac{z_0 f'(z_0)}{f(z_0)} = -\int_0^r \Phi(\psi,s)\,\frac{ds}{1-s^2}, \]

where

\[ \Phi(\psi,s)=\frac{2(1-s^2)\sin\psi}{1-2s\cos\psi+s^2} \]

and \(\psi=\psi(s)\) is some real piecewise-continuous function on the segment \(0\leq s\leq r\), it follows that on the subclass \(S(k)\)

\[ J_1(f)=\ln \frac{r}{1-r^2} + 2\int_0^r G(\psi,s)\frac{ds}{1-s^2}, \tag{3} \]

where here \(G(\psi,s)=\Phi(\psi,s)-\cos\psi\). The greatest value of \(G(\psi,s)\), if the second argument is regarded as fixed, is attained at the point \(\psi=\varphi\), \(0\leq \varphi\leq \pi\), which is among the solutions of the equation

\[ \frac{2(1-s^4)\cos\varphi+4s^3-4s}{(1-2s\cos\varphi+s^2)^2} +\sin\varphi=0. \]

It is easy to verify that this equation decomposes into the following two equations:

\[ s^2+2s\sin\varphi-1=0, \]

\[ 2s\cos^2\varphi-2(1+s^2)\cos\varphi-(1-s^2)\sin\varphi+2s=0. \tag{4} \]

The solutions of the first of them,

\[ \varphi_1=\arcsin\frac{1-s^2}{2s},\qquad \varphi_2=\pi-\varphi_1, \]

which occur only for \(a\leq s\leq r\), correspond to one and the same value of the function \(G(\psi,s)\), equal to \(\dfrac{1+s^2}{2s}\).

We shall seek the solution of the second equation in the form

\[ \varphi_3=2\arctg\left(x\frac{1-s}{1+s}\right), \qquad 0\leq s\leq r, \tag{5} \]

regarding here \(x\) as a new nonnegative parameter. As a result of substituting (5) into (4), we find the formula

\[ \frac{1+s}{1-s} = \sqrt{\frac{x^4-x^3}{x+1}}, \tag{6} \]

from which it follows that, for \(s>0\),

\[ x>\frac{1+\sqrt5}{2}. \]

Since \(\dfrac{ds}{dx}>0\), the relation between the parameters \(x\) and \(s\) established by formula (6) is one-to-one, and therefore (5), with (6) taken into account, gives a solution of equation (4). To this solution there corresponds the value of \(G(\psi,s)\), equal to

\[ -\frac{x^2-6x-1}{x^2+1}. \]

\[ \text{* The derivation of the two formulas given below is contained in (2), p. 256; } |f(z_0,t)|=s,\quad \arg [k(t)f(z_0,t)]=\chi(t)=\psi(s). \]

On the interval \(a < s \leqslant r\) the difference

\[ G(\varphi_1,s)-G(\varphi_3,s)=\frac{2(x^2-2x-1)^2}{(x^2+1)(x^2-x-1)} \]

is positive and, thus,

\[ G(\varphi,s)= \begin{cases} G(\varphi_3,s), & \text{if } 0\leqslant s\leqslant a,\\ G(\varphi_1,s)=G(\varphi_2,s), & \text{if } a<s\leqslant r. \end{cases} \]

After replacing \(G(\psi,s)\) in (3) by \(G(\varphi,s)\) and carrying out the integration, we obtain in \(S(k)\), and hence also in \(S\), the following estimate for the functional (2):

\[ J_1(f)\leqslant \pi+\ln(\lambda R\overline{R}). \]

\(2^\circ\). By means of inessential changes in the preceding arguments one can establish that the functional

\[ J_2(f)=\ln |f(z_0)|+2\arg F(z_0), \qquad f\in S, \]

is bounded above by the constant \(\pi+\ln(\lambda R\overline{R})\). Thus, also for the functional

\[ J(f)=\ln |f(z_0)|+2\arg F(z_0), \qquad f\in S, \]

the estimate

\[ I(f)\leqslant \pi+\ln(\lambda R\overline{R}), \tag{7} \]

holds, independent of \(\arg z_0\). This estimate is sharp, since equality in it is attained for the function

\[ f(z)=\lim_{t\to\infty} e^t f(z,t), \]

where \(f(z,t)\) is the solution of the Löwner equation with characteristic function \(k(t)\), constructed with account of the initial condition \(f(z_0,0)=z_0\), from the function

\[ \psi(s,y)= \begin{cases} \varphi_3(s), & \text{if } 0\leqslant s\leqslant a,\\ \varphi_1(s), & \text{if } a<s\leqslant y \quad (a\leqslant y\leqslant r),\\ \varphi_2(s), & \text{if } y<s\leqslant r. \end{cases} \]

\(3^\circ\). To finish the proof of the first part of the theorem, let us note that \(L(f,r)\) is star-shaped with respect to the point \(f(z_0)\) if and only if \(2|\arg F(z_0)|\leqslant \pi\). Suppose now that \(\lambda R\overline{R}\leqslant |f(z_0)|\). Hence, on the basis of inequality (7), we find that at the point \(f(z_0)\), \(2|\arg F(z_0)|\leqslant \pi\).

\(4^\circ\). We proceed to the proof of the second part of the theorem. For the family of extremal functions giving equality in estimate (7), by direct computation we establish that

\[ \arg F(z_0)=-\int_0^r \Phi(\psi(s,y),s)\frac{ds}{1-s^2} =-\frac{1}{2}Q(r,y), \]

where

\[ Q(r,y)=\ln\frac{\sqrt{10}\,r}{1-r^2}-h(a)-h(r)+2h(y)+3\operatorname{arc\,ctg}3, \]

\[ h(s)=2\int \cos\varphi_1\,\frac{ds}{1-s^2}= \]

\[ =\arcsin\frac{1+s^2}{2\sqrt{2}\,s} +2\ln\bigl(\sqrt{1-a^2s^2}+\sqrt{s^2-a^2}\bigr) -\ln(1-s^2)+\mathrm{const}. \]

From the continuity and monotone increase of \(Q(y,y)\) on \([a,1]\) from \(Q(a,a)<\pi\) to \(Q(1,1)=+\infty\), there follows the existence on \((a,1)\) of a unique solution \(r_0\) of the equation \(Q(y,y)=\pi\), i.e., of equation (1). Fix in the interval \((r_0,1)\) some point \(\rho\). The function \(Q(\rho,y)\) is continuous and increases monotonically for \(a\leqslant y\leqslant \rho\), from \(Q(\rho,a)<Q(1,a)<\pi\) to \(Q(\rho,\rho)>\pi\). Therefore, for any number \(\delta>0\), \(\delta\leqslant Q(\rho,\rho)-\pi\), there will be found such a value

value \(y_0 = y_0(\delta)\) such that \(Q(\rho,y_0)=\pi+\delta\), i.e. \(-2\arg F(z_0)=\pi+\delta\). Taking this equality into account, from (7) we obtain that

\[ |f(z_0)|=\lambda e^{-\delta}R\overline{R}. \]

Thus, in the class \(S\) there exists a function \(f(z)\), depending on the choice of \(z_0\) and \(\delta\), with a level line \(L(f,r)\) having a non-star-shaped arc outside the annulus
\[ \lambda R\overline{R}\leq |w|\leq R. \]

The proof of the theorem is complete.

Remark. Computations show that \(\lambda=0.437\ldots<7/16\) \((\lambda-7/16=O(10^{-5}))\), and \(r_0=0.709\ldots\)

Corollary 1. If \(r\in(r_s,r_0)\), then
\[ a_s(r)\leq \lambda \frac{r}{(1+r)^2}; \]
if \(r\in(r_0,1)\), then
\[ a_s(r)=\lambda \frac{r}{(1+r)^2}. \]

Corollary 2. \(a_s=a_s(1)=\lambda/4=0.109\ldots\)

Corollary 3. If \(f(z)\in S\), then an arc of the level line \(L(f,r)\), \(r_s<r<1\), lying in the annulus

\[ \frac{7r^2}{16(1-r^2)^2}\leq |w|\leq \frac{r}{(1-r)^2}, \]

is star-shaped with respect to the origin.

Corollary 4. For the subclass \(S_p\subset S\) \((p=1,2,\ldots,\ S_1\equiv S)\) of \(p\)-symmetric functions \(f(z)\), the arcs of the level line \(L(f,r)\), \(r_{ps}=\sqrt[p]{r_s}<r<1\), are star-shaped in the annulus

\[ \frac{\sqrt[p]{\lambda}\,r^2}{(1-r^{2p})^{2/p}}\leq |w|\leq \frac{r}{(1-r^p)^{2/p}} . \tag{8} \]

Corollary 5. For the class \(\Sigma_{0p}\) of functions
\[ F(z)=\frac{1}{f(1/z)},\qquad f(z)\in S_p, \]
the arcs of the level line \(L(F,r)\), \(1<r<R_{ps}=r_{ps}^{-1}\), are star-shaped in the annulus

\[ \frac{(r^p-1)^{2/p}}{r}\leq |w|\leq \frac{(r^{2p}-1)^{2/p}}{\sqrt[p]{\lambda}\,r^2}. \tag{9} \]

Remark. In Corollaries 4 and 5, for \(\sqrt[p]{r_0}<r<1\) (respectively \(1<r<\sqrt[p]{r_0^{-1}}\)) one cannot replace \(\lambda\) by a smaller number without violating the indicated properties of the annuli (8) and (9).

Tomsk State University
named after V. V. Kuibyshev

Received
1 XII 1963

REFERENCES

1. I. E. Bazilevich, G. V. Koritskii, DAN, 140, No. 2, 279 (1961).
2. I. E. Bazilevich, G. V. Koritskii, Matem. sborn., 58 (100), 3, 249 (1962).