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Mat. Zametki, 1971, Volume 10, Issue 1, Pages 41–52 (Mi mz7065)  

Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions

L. Kh. Burshtein

Far Eastern National University

Abstract: Let $T_r$ be the class of functions $f(z)=z+c_2z^2+…$, regular in the disk $|z|<1$, real on the diameter $-1<z<1$, and satisfying the condition $\operatorname{Im}f(z)\cdot\operatorname{Im}z>0$ in the remainder of the disk $|z|<1$. Let $z_f$ be the solution of $f(z)=\alpha f(a)$ on $T_r$, where $\alpha$ is any fixed complex number, $\alpha\ne0$, $\alpha\ne1$, $\alpha$ is any fixed real number, $|\alpha|<1$. We determine the region $D_{T_r}$ of values of the functional $z_f$ on the class $T_r$. Variation formulas for Stieltjes integrals due to G. M. Goluzin are used.

Full text: PDF file (817 kB)

English version:
Mathematical Notes, 1971, 10:1, 449–455

Bibliographic databases:

UDC: 517.54
Received: 28.10.1969

Citation: L. Kh. Burshtein, “Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions”, Mat. Zametki, 10:1 (1971), 41–52; Math. Notes, 10:1 (1971), 449–455

Citation in format AMSBIB
\Bibitem{Bur71}
\by L.~Kh.~Burshtein
\paper Roots of the equation $f(z)=\alpha f(a)$ for the class of typically-real functions
\jour Mat. Zametki
\yr 1971
\vol 10
\issue 1
\pages 41--52
\mathnet{http://mi.mathnet.ru/mz7065}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=286991}
\zmath{https://zbmath.org/?q=an:0221.30013}
\transl
\jour Math. Notes
\yr 1971
\vol 10
\issue 1
\pages 449--455
\crossref{https://doi.org/10.1007/BF01747068}


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