Uniqueness for meromorphic solutions of Schwarzian differential equation

Dan-Gui Yao; Zhi-Bo Huang; Ran-Ran Zhang; Dan-Gui Yao; Zhi-Bo Huang; Ran-Ran Zhang

doi:10.3934/math.2021727

AIMS Mathematics

2021, Volume 6, Issue 11: 12619-12631. doi: 10.3934/math.2021727

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Uniqueness for meromorphic solutions of Schwarzian differential equation

1.
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China
2.
Department of Mathematics, Guangdong University of Education, Guangzhou, 510303, China

Received: 09 April 2021 Accepted: 17 August 2021 Published: 02 September 2021
MSC : 30D35

Let $ f $ be a meromorphic function, $ R $ be a nonconstant rational function and $ k $ be a positive integer. In this paper, we consider the Schwarzian differential equation of the form

$ \begin{align*} \left[\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^{2}\right]^{k} = R(z). \end{align*} $

We investigate the uniqueness of meromorphic solutions of the above Schwarzian differential equation if the meromorphic solution $ f $ shares three values with any other meromorphic function.
- differential equation,
- meromorphic function,
- sharing value,
- uniqueness
Citation: Dan-Gui Yao, Zhi-Bo Huang, Ran-Ran Zhang. Uniqueness for meromorphic solutions of Schwarzian differential equation[J]. AIMS Mathematics, 2021, 6(11): 12619-12631. doi: 10.3934/math.2021727

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Abstract

Let $ f $ be a meromorphic function, $ R $ be a nonconstant rational function and $ k $ be a positive integer. In this paper, we consider the Schwarzian differential equation of the form

$ \begin{align*} \left[\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^{2}\right]^{k} = R(z). \end{align*} $

We investigate the uniqueness of meromorphic solutions of the above Schwarzian differential equation if the meromorphic solution $ f $ shares three values with any other meromorphic function.

References

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