Extreme control for a functional on classes of analytical functions

G. D. Sadritdinova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 2 (2014), pp. 29-34

Résumé

Let $S$ be the class of holomorphic univalent functions $f(z)$ normalized by conditions $f(0)=0$, $f'(0)=1$ in a unit circle $E=\{z\colon|z|<1\}$ functions $f(z)$, rated conditions $f(0)=0$, $f'(0)=1$. Let $S_p$ ($p=1,2,\dots$) is a subclass of the class $S$ of functions possessing $p$-multiple symmetry of rotation with respect to zero, that is, such that $$ f\left(e^{i\frac{2\pi k}p}z\right)=e^{i\frac{2\pi k}p}f(z),\qquad k=1,2,\dots,p-1. $$ The subclass $S_p$ is distinguished as an independent class of functions, and $S_1=S$. We consider Loewner's equation \begin{gather*} \frac{d\zeta(z,\tau)}{d\tau}=-\zeta(z,\tau)\frac{\mu^p(\tau)+\zeta^p(z,\tau)}{\mu^p(\tau)-\zeta^p(z,\tau)},\qquad \zeta(z,0)=z\\ |z|<1,\qquad0\le\tau<\infty, \end{gather*} in which control function $\mu(\tau)$, $|\mu(\tau)|=1$, is continuous or piecewise-continuous on $[0,\infty)$. Functions $f(z)=\lim_{\tau\to\infty}e^\tau\zeta(z,\tau)$ which we call limiting for solutions of the Loewner equation form a dense subclass of the class $S_p$. In this article the problem of finding control functions leading to boundary functions of the functional $I=\ln\left|\frac{f(z)}z\right|$ in Loewner's equation on classes $S$ and $S_p$ is solved by the parametrical method. The set of values of this functional does not depend on $\operatorname{arg}z$ therefore, from now on we suppose $z=r$, $0<r<1$. Executing some transformations over Loewner's equation, introducing the designations $$ |\zeta(r,\tau)|=\rho(r,\tau),\qquad \zeta(r,\tau)\bar\mu(\tau)=\rho(r,\tau)y(r,\tau) $$ and substituting $\rho=\left(\frac{1-s}{1+s}\right)^\frac 1p$ and $y=\left(\frac{i+t}{i-t}\right)^\frac 1p$, we have $$ \ln\left|\frac{f(r)}r\right|=\frac 1p\int^1_\sigma g(s,t)\,ds-\frac 1p\ln(1-r^{2p}), $$ where $g(s,t)=\frac{t^2-1}{t^2+1}\cdot\frac 1s$, $\sigma=\frac{1-r^p}{1+r^p}$. The condition $g'_t(s,t)=0$ yields $t(s)=0$ and $t(s)=\infty$. The solution $t(s)=0$ leads to extreme control functions $\mu=1^{1/p}$, providing a minimum to the studied functional. Function $f(z)=\frac z{(1+z^p)^{2/p}}\in S_p$, as applied to the functional $I$, is a boundary function at which the functional reaches the minimum value. As $t(s)=\infty$, we find extreme control functions $\mu=(-1)^{1/p}$, leading to a maximum of the functional $I$. The boundary function $f(z)=\frac z{(1-z^p)^{2/p}}\in S_p$ provides a maximum to the functional $I$. Setting everywhere $p=1$, we find extreme control functions for the functional $I$ on the class $S$.

Détail

Keywords: boundary function, class of univalent holomorphic functions, the maximum value of the functional, minimal value of the functional, Loewner's equation, extreme control function.

Bibliographie

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