Uniqueness of meromorphic functions concerning fixed points

Abstract

<abstract><p>In this paper, we study a uniqueness question of meromorphic functions concerning fixed points and mainly prove the following theorem: Let $ f $ and $ g $ be two nonconstant meromorphic functions, let $ n, k $ be two positive integers with $ n &gt; 3k+10.5-\Theta_{\min}(k+6.5) $, if $ \Theta_{\min}\geq \frac{2.5}{k+6.5} $, otherwise $ n &gt; 3k+8 $, and let $ (f^{n})^{(k)} $ and $ (g^{n})^{(k)} $ share $ z $ CM, $ f $ and $ g $ share $ \infty $ IM, then one of the following two cases holds: If $ k = 1 $, then either $ f(z) = c_{1}e^{cz^{2}} $, $ g(z) = c_{2}e^{-cz^{2}} $, where $ c_{1}, c_{2} $ and $ c $ are three constants satisfying $ 4n^{2}(c_{1}c_{2})^{n}c^{2} = -1 $, or $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $; if $ k\geq2 $, then $ f = tg $ for a constant $ t $ such that $ t^{n} = 1 $. Our results extend and improve some results due to ^{[<xref ref-type="bibr" rid="b8">8</xref>,<xref ref-type="bibr" rid="b9">9</xref>,<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b24">24</xref>]}.</p></abstract>