Disproving of a Shash hypothesis concerning univalent functions (in Ukrainian)

M.M. Sheremeta

It is disproved the following Shah conjecture: if for an increasing sequence $(n_p)$ such that $\Sigma_{p=1}^\infty\frac1{n^p}=\infty$ a function $f$ and all its derivatives $f^{(n_p)}$ are univalent in $\Bbb D=\{z\,:\,|z| < 1\}$, then $f$ is an entire function. A necessary and sufficient condition is given that for every univalent in $\Bbb D$ function $f$ the univalence of all the derivatives $f^{(n_p)}$ implies that $f$ is entire.

1. Shah S.M. Analytic functions with univalent derivatives and entire functions of exponential type // Bull. Amer. Math. Soc. 1972. V.78, N.2. P.110–118.

2. Miller S.S. Problems in complex function theory // Complex Anal. Proc. S.U.N.Y. Brockport conf. New York-Basel. 1978. P.167–177.

3. Шеремета М.М. О целых функциях с однолистыми в круге производными // Укр. мат. журн. 1991. Т.43, N.3. С.400–406. Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290602, Ukraine

	Disproving of a Shash hypothesis concerning univalent functions (in Ukrainian)
Author	M.M. Sheremeta Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290602, Ukraine
Abstract	It is disproved the following Shah conjecture: if for an increasing sequence $(n_p)$ such that $\Sigma_{p=1}^\infty\frac1{n^p}=\infty$ a function $f$ and all its derivatives $f^{(n_p)}$ are univalent in $\Bbb D=\{z\,:\,\|z\| < 1\}$, then $f$ is an entire function. A necessary and sufficient condition is given that for every univalent in $\Bbb D$ function $f$ the univalence of all the derivatives $f^{(n_p)}$ implies that $f$ is entire.
Keywords	Shah conjecture, increasing sequence, derivatives, univalent function, unit disk, entire function
DOI	doi:10.30970/ms.2.1.46-48
Reference	1. Shah S.M. Analytic functions with univalent derivatives and entire functions of exponential type // Bull. Amer. Math. Soc. 1972. V.78, N.2. P.110–118. 2. Miller S.S. Problems in complex function theory // Complex Anal. Proc. S.U.N.Y. Brockport conf. New York-Basel. 1978. P.167–177. 3. Шеремета М.М. О целых функциях с однолистыми в круге производными // Укр. мат. журн. 1991. Т.43, N.3. С.400–406. Department of Mechanics and Mathematics, Lviv University, Universytetska 1, Lviv, 290602, Ukraine
Pages	46-48
Volume	2
Issue	1
Year	1993
Journal	Matematychni Studii
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