Abstract
Full Text
MATHEMATICS
M. I. REDKOV
RANGE OF VALUES OF THE FUNCTIONAL
\[ I=\ln \frac{w^\lambda \varphi'(w)^{1-\lambda}\varphi'(0)^\nu} {\varphi(w)^\lambda|\varphi(w)|^\kappa} \]
ON CERTAIN CLASSES OF BOUNDED UNIVALENT FUNCTIONS
(Presented by Academician M. A. Lavrent'ev on February 26, 1960)
We consider the following classes of functions: \(S_1\) is the class of functions
\(\varphi(\omega)=\beta\omega+\beta_2\omega^2+\ldots+\beta_n\omega^n+\ldots,\ \beta>0\), regular, univalent, and satisfying the condition \(|\varphi(\omega)|<1\) in the disk \(|\omega|<1\); \(S_1[|\varphi(w)|]\) is the class of functions \(\varphi(\omega)\) from \(S_1\) with prescribed \(|\varphi(w)|\) at a fixed point \(w\) of the disk \(|\omega|<1\); \(S_1(\beta)\) is the class of functions \(\varphi(\omega)\) from \(S_1\) with prescribed coefficient \(\beta=\varphi'(0)\). The following theorems are proved by the method proposed by P. P. Kufarev \((^1)\).*
Theorem 1. The boundary of the range of values of the functional
\[ I=\ln \frac{w^\lambda\varphi'(w)^{1-\lambda}\varphi'(0)^\nu} {\varphi(w)^\lambda} \]
in the class \(S_1[|\varphi(w)|]\) is determined by the equation
\[ I_0=\int_{\rho_0}^{r} \left[ \lambda\frac{1+q}{1-q} -(1-\lambda)\frac{1-q^2+2q}{(1-q)^2} -\nu \right] \frac{|1-q|^2}{1-\rho^2}\,\frac{d\rho}{\rho}, \tag{1} \]
where \(I_0\) is a boundary point of the domain; \(r=|w|\); \(\rho_0=|\varphi(w)|\); \(q,\ |q|=\rho\), is a root of the equation
\[ \overline{B}_0 q-B_0\overline{q} =(1-\lambda)(1-\rho^2) \left[ \frac{qx}{(1-q)^2} - \frac{\overline{q}\,\overline{x}}{(1-\overline{q})^2} \right], \tag{2} \]
in which
\[ B_0=\nu\,\frac{x+\overline{x}}{2} +(1-2\lambda)\frac{x-\overline{x}}{2}, \qquad x=e^{-i\alpha}, \]
\(\alpha\) is a parameter, \(-\pi<\alpha\leq\pi\).
Corollary 1. The boundary of the range of values of the functional
\[ I=\ln\frac{w\varphi'(w)}{\varphi(w)} \]
in the class \(S_1[|\varphi(w)|]\) is determined by the equation
\[ I_0=e^{i\alpha}\ln\frac{(1-r)(1+\rho_0)}{(1+r)(1-\rho_0)}, \qquad -\pi<\alpha\leq\pi . \]
Corollary 2. The boundary of the range of values of the functional
\[ I=\ln \varphi'(w)\varphi'(0) \]
* By the logarithm is meant that single-valued branch for which \(\operatorname{Im} I\to 0\) as \(w\to 0\); \(\lambda,\nu,\kappa\) are arbitrary real numbers, and \(w\) is any fixed point of the disk \(|w|<1\).
in the class \(S_1[|\varphi(w)|]\) consists of the curve
\[ I_0= \begin{cases} - e^{i\alpha}\ln \dfrac{r^2(1-\rho_0^2)}{\rho_0^2(1-r^2)}, & |\alpha|\leqslant 2\arcsin \rho_0;\\[1.2em] \ln \dfrac{4\rho_0^2(1-\rho_0^2)}{\sin^2\alpha} \pm 4i\left(\dfrac{|\alpha|}{2}-\arcsin \rho_0\right) - e^{i\alpha}\ln \dfrac{r^2\operatorname{ctg}^2\dfrac{\alpha}{2}}{1-r^2}, & 2\arcsin \rho_0\leqslant |\alpha|\leqslant 2\arcsin r;\\[1.2em] \ln \dfrac{\rho_0^2(1-\rho_0^2)}{r^2(1-r^2)} \pm 4i(\arcsin r-\arcsin \rho_0), & 2\arcsin r\leqslant |\alpha|\leqslant \pi, \end{cases} \]
and of the straight-line segment joining the endpoints of this curve.
Corollary 3. The boundary of the range of values of the functional
\[ I=\ln \frac{w^2\varphi'(w)\varphi'(0)}{\varphi(w)^2} \]
in the class \(S_1[|\varphi(w)|]\) is determined by the equation
\[ I_0=e^{i\alpha}\ln \frac{1-r^2}{1-\rho_0^2},\qquad -\pi<\alpha\leqslant \pi. \]
The results formulated in Corollaries 1, 2, and 3 were obtained earlier by N. A. Lebedev \((^2)\) by the method of parametric representations.
Corollary 4. The boundary of the range of values of the functional
\[ I=\ln \frac{w\varphi'(0)^\nu}{\varphi(w)} \]
in the class \(S_1[|\varphi(w)|]\) is determined by the equation
\[ I_0=\ln \left(\frac{1-r^2}{1-\rho_0^2}\right)^\nu \left(\frac{r_0}{\rho_0}\right)^{1-\nu} -\frac{\nu^2\cos\alpha+i\sin\alpha}{\sqrt{\nu^2\cos^2\alpha+\sin^2\alpha}} \ln \frac{(1+r)(1-\rho_0)}{(1-r)(1+\rho_0)}, \qquad -\pi<\alpha\leqslant \pi. \]
Theorem 2. The boundary of the range of values of the functional
\[ I=\ln \frac{w^\lambda\varphi'(w)^{1-\lambda}\varphi'(0)^\nu}{\varphi(w)|\varphi(w)|^\chi} \]
in the class \(S_1\) is determined by the equation
\[ I_0=\int_{\rho_0}^{r} \left[ \lambda\frac{1+q}{1-q} -(1-\lambda)\frac{1-q^2+2q}{(1-q)^2} -\nu \right] \frac{|1-q|^2\,d\rho}{1-\rho^2}\,\frac{1}{\rho} -\ln \rho_0^\chi, \tag{3} \]
where \(q\), \(|q|=\rho\), is a root of equation (2); \(\rho_0\) is either a root of equation (4) satisfying the condition \(0<\rho_0\leqslant r\), or zero, if equation (4) has no such roots. In the latter case the boundary point does not belong to the set of values of the functional \(I\).
The equation for determining \(\rho_0\) has the form
\[ \left| \begin{array}{ccccccc} b_0 & b_1 & b_2 & b_3 & b_4 & 0 & 0\\ 0 & b_0 & b_1 & b_2 & b_3 & b_4 & 0\\ 0 & 0 & b_0 & b_1 & b_2 & b_3 & b_4\\ 4b_0 & 3b_1 & 2b_2 & b_3 & 0 & 0 & 0\\ 0 & 4b_0 & 3b_1 & 2b_2 & b_3 & 0 & 0\\ 0 & 0 & 4b_0 & 3b_1 & 2b_2 & b_3 & 0\\ 0 & 0 & 0 & 4b_0 & 3b_2 & 2b_2 & b_3 \end{array} \right|=0, \tag{4} \]
where
\[ b_0=(1-2\lambda)\frac{\chi-\bar{\chi}}{2}+\nu\frac{\chi+\bar{\chi}}{2},\qquad b_1=\left[\chi\left(1-\lambda-\nu-\frac{\chi}{2}\right)+ \right. \]
\[ \left. +\bar{\chi}\left(1-2\lambda-\nu-\frac{\chi}{2}\right) \right]\rho_0^2 +\chi\left(-1+2\lambda-\nu+\frac{\chi}{2}\right) +\bar{\chi}\left(1-\lambda-\nu+\frac{\chi}{2}\right), \]
\[ b_2=(-1+2\lambda+\nu+\kappa)\frac{x+\bar{x}}{2}\rho_0^4 +2(\nu+\lambda-1)(x+\bar{x})\rho_0^2 +(1-2\lambda+\nu-\kappa)\frac{x+\bar{x}}{2},\quad b_3=\rho_0^2 b_1,\quad b_4=\rho_0^4 b_0. \]
Corollary 1. The boundary of the range of values of the functional
\[ I=\ln \frac{w\varphi'(0)^\nu}{\varphi(w)|\varphi(w)|^\kappa}, \qquad \nu\ne \kappa+1, \]
in the class \(S_1\) is determined by the equation
\[ I_0=\ln \left(\frac{1-r^2}{1-\rho_0^2}\right)^\nu \frac{r^{1-\nu}}{\rho_0^{\,1-\nu+\kappa}} -\frac{\nu^2\cos\alpha+i\sin\alpha}{\sqrt{\nu^2\cos^2\alpha+\sin^2\alpha}}\, \ln\frac{(1+r)(1-\rho_0)}{(1-r)(1+\rho_0)}, \]
where
\[ \rho_0= \frac{(1+\kappa-\nu)\cos\alpha} {\sqrt{\nu^2\cos^2\alpha+\sin^2\alpha} +\sqrt{(1+\kappa)^2\cos^2\alpha+\sin^2\alpha}}, \]
and the parameter \(\alpha\) varies so that \(0<\rho_0\le r\).
Corollary 2. The boundary of the range of values of the functional
\[ I=\ln\frac{w\varphi'(0)^\nu}{\varphi(w)|\varphi(w)|^{\nu-1}} \]
in the class \(S_1\) is determined by the equation
\[ I_0=\ln(1-r^2)^\nu r^{1-\nu} -\frac{\nu^2\cos\alpha+i\sin^2\alpha} {\sqrt{\nu^2\cos^2\alpha+\sin^2\alpha}}\, \ln\frac{(1+r)}{(1-r)},\qquad -\pi<\alpha\le\pi, \]
and the boundary itself does not belong to the set of values of the functional \(I\) in the class \(S_1\).
Corollary 3. The boundary of the range of values of the functional
\[ I=\ln\frac{w\varphi'(w)}{\varphi(w)|\varphi(w)|^{2\kappa}},\qquad \kappa\ne 0, \]
in the class \(S_1\) is determined by the equation
\[ I_0=-e^{i\alpha}\ln\frac{(1+r)(1-\rho_0)} {(1-r)(1+\rho_0)} -\ln\rho_0^{2\kappa}, \]
where
\[ \rho_0=\frac{2\kappa\cos\alpha}{1+\sqrt{1+4\kappa^2\cos^2\alpha}}, \]
and the parameter \(\alpha\) varies so that \(0<\rho_0\le r\).
Corollary 4. The boundary of the range of values of the functional
\[ I=\ln\frac{w\varphi'(w)}{\varphi(w)} \]
in the class \(S_1\) is determined by the equation
\[ I_0=e^{i\alpha}\ln\frac{1+r}{1-r},\qquad -\pi<\alpha\le\pi, \]
and the boundary itself does not belong to the set of values of the functional \(I\).
Corollary 5. The boundary of the range of values of the functional
\[ I=\ln\frac{w^2\varphi'(w)\varphi'(0)} {\varphi(w)^2|\varphi(w)|^{3\kappa}},\qquad \kappa\ne 0, \]
in the class \(S_1\) is determined by the equation
\[ I_0=e^{i\alpha}\ln\frac{1-r^2}{1-\rho_0^2} -\ln\rho_0^{3\kappa}, \]
where
\[ \rho_0=\sqrt{\frac{-\frac{3}{2}\kappa\cos\alpha} {1+\frac{3}{2}\kappa\cos\alpha}}, \]
and the parameter \(\alpha\) varies so that \(0<\rho_0\le r\).
Corollary 6. The boundary of the range of values of the functional
\[ I=\ln \frac{w^2\varphi'(w)\varphi'(0)}{\varphi(w)^2} \]
in the class \(S_1\) is determined by the equation
\[ I_0=e^{i\alpha}\ln(1-r^2),\qquad -\pi<\alpha\leqslant \pi, \]
and the boundary itself does not belong to the set of values of the functional \(I\).
Corollary 7. If \(r>1/\sqrt2\), then the boundary of the range of values of the functional
\[ I=\ln\varphi'(w)\varphi'(0) \]
in the class \(S_1\) consists of the curve
\[ I_0= \begin{cases} \displaystyle \ln\operatorname{ctg}^2\alpha \pm \pi i -e^{i\alpha}\ln\frac{r^2\operatorname{ctg}^2\frac{\alpha}{2}}{1-r^2}, & \displaystyle \frac{\pi}{2}<|\alpha|\leqslant 2\arcsin r, \\[1.2em] \displaystyle \ln\frac{\cos^2\alpha}{4r^2(1-r^2)} \pm 4i\left(\arcsin r-\frac{|\alpha|}{2}+\frac{\pi}{4}\right), & \displaystyle 2\arcsin r\leqslant |\alpha|\leqslant \pi, \end{cases} \]
and the segment of the straight line connecting the points
\[ \ln\frac{1}{4r^2(1-r^2)} \pm 4i\left(\arcsin r-\frac{\pi}{4}\right). \]
If \(r\leqslant 1/\sqrt2\), then the boundary is determined by the equation
\[ I_0=\ln\frac{\cos^2\alpha}{4r^2(1-r^2)} \pm 4i\left(\arcsin r-\frac{|\alpha|}{2}+\frac{\pi}{4}\right), \qquad \frac{\pi}{2}<|\alpha|\leqslant 2\arcsin r+\frac{\pi}{2}. \]
Theorem 3. The boundary of the range of values of the functional
\[ I=\ln\frac{w^\lambda\varphi'(w)^{1-\lambda}} {\varphi(w)^\lambda|\varphi(w)|^\varkappa} \]
in the class \(S_1(\beta)\) is determined by the equation
\[ I_0=\int_{\rho_0}^{r} \left[ \lambda\frac{1+q}{1-q} -(1-\lambda)\frac{1-q^2+2q}{(1-q)^2} \right] \frac{|1-q|^2}{1-\rho^2}\,\frac{d\rho}{\rho} -\ln\rho_0^\varkappa, \]
where \(q\), \(|q|=\rho\), is determined from equation (2) for
\[ B_0=(1-2\lambda)\frac{x-\bar{x}}{2}+k, \]
and the modulus \(\rho\) of the boundary function and the real constant \(k\) are found from the system of two equations:
\[ 1)\quad \ln\frac{1}{\beta} = \int_{\rho_0}^{r} \frac{|1-q|^2}{1-\rho^2}\,\frac{d\rho}{\rho}; \]
\[ 2)\quad \text{equation (4), in which instead of } v\frac{x+\bar{x}}{2} \text{ there stands } k. \]
Corollary. The boundary of the range of values of the functional
\[ I=\ln\frac{w|\varphi(w)|^{-\varkappa}}{\varphi(w)} \]
in the class \(S_1(\beta)\) is determined by the equation
\[ I_0= \ln\frac{r}{\rho_0^{1+\varkappa}} - \frac{i\sin\alpha(1-\rho_0^2)} {(1+\rho_0^2)\sqrt{(1+\varkappa)^2\cos^2\alpha+\sin^2\alpha} -2\rho_0(1+\varkappa)\cos\alpha} \ln\frac{(1+r)(1-\rho_0)}{(1-r)(1+\rho_0)}, \]
where \(\rho_0\) is found from the relation
\[ \ln\frac{\rho_0(1-r^2)}{\beta r(1-\rho_0^2)} = \frac{ (1+\varkappa)(1+\rho_0^2)\cos\alpha -2\rho_0\sqrt{(1+\varkappa)^2\cos^2\alpha+\sin^2\alpha} }{ (1+\rho_0^2)\sqrt{(1+\varkappa)^2\cos^2\alpha+\sin^2\alpha} -2\rho_0(1+\varkappa)\cos\alpha } \ln\frac{(1+r)(1-\rho_0)}{(1-r)(1+\rho_0)}. \]
Tomsk State University
named after V. V. Kuibyshev
Received
23 II 1960
CITED LITERATURE
- P. P. Kufarev, DAN, 107, No. 5, 633 (1956).
- N. A. Lebedev. Vestn. LGU, No. 11, issue 4, 3 (1955).