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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the zeros of $ L'+L\sp 2$ for certain rational functions $ L$


Author: T. Sheil-Small
Journal: Proc. Amer. Math. Soc. 107 (1989), 1013-1016
MSC: Primary 30D15; Secondary 30C15, 30D20
MathSciNet review: 984814
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Abstract: Let $ L$ be a nonconstant rational function whose poles are real, simple with each one having a positive residue. Then, if $ L' + {L^2}$ has no nonreal zeros, $ L$ has the form

$\displaystyle L(z) = \sum\limits_{k = 1}^n {\frac{{{\alpha _k}}}{{z - {x_k}}} - az + b,} $

$ {x_k}$ are real, $ {\alpha _k} > 0$ for $ 1 \leq k \leq n,a \geq 0$ and $ b$ is real. In particular, if $ P$ is a polynomial of degree $ \geq 2$, then $ P' + {P^2}$ has nonreal zeros. The result is applied to entire functions in connection with zeros of the derivatives.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0984814-0
PII: S 0002-9939(1989)0984814-0
Article copyright: © Copyright 1989 American Mathematical Society