Entire functions sharing simple $a$-points with their first derivative

Andreas Schweizer

Entire functions sharing simple $a$-points with their first derivative

Complex Variables 2013-12-04 v3

Authors: Andreas Schweizer

Abstract

We show that if a complex entire function $f$ and its derivative $f'$ share their simple zeroes and their simple $a$ -points for some nonzero constant $a$ , then $f\equiv f'$ . We also discuss how far these conditions can be relaxed or generalized. Finally, we determine all entire functions $f$ such that for 3 distinct complex numbers $a_1,a_2,a_3$ every simple $a_j$ -point of $f$ is an $a_j$ -point of $f'$ .

Keywords

univalent functions function theory

Cite

@article{arxiv.1104.2382,
  title  = {Entire functions sharing simple $a$-points with their first derivative},
  author = {Andreas Schweizer},
  journal= {arXiv preprint arXiv:1104.2382},
  year   = {2013}
}

Comments

v3: 11 pages, corrected a typo in Theorem 2', updated address; refereed version, but note that the journal version carries my old address and has a finer division into sections and a different numbering of the theorems

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