Meromorphic solutions of the differential equation $(f^{n})^{(k)}(g^{n})^{(k)}=1$ and its applications
Volume 131 / 2023
Abstract
In 1997, C. C. Yang and X. H. Hua [Ann. Acad. Sci. Fenn. Math. 22 (1997), 395–406] proved that if $f$ and $g$ are two non-constant meromorphic functions such that $f^nf’g^n g’= 1,$ where $n$ is a positive integer satisfying $n\geq 6,$ then $f$ and $g$ are transcendental entire functions such that $g(z) = c_1e^{cz}$ and $f(z) = c_2 e^{-cz},$ where $c_1$, $c_2$ and $c$ are three complex constants such that $(c_1 c_ 2 )^{ n+1}c^2 = -1.$ By using Zalcman’s Lemma, we prove that if $f$ and $g$ are two non-constant meromorphic functions such that $(f^n)^{(k)}(g^n)^{(k)}=1,$ where $n$ and $k$ are positive integers satisfying $n \gt 2k,$ then $f(z)=c_1e^{cz}$ and $g(z)=c_2e^{-cz},$ where $c_1,$ $c_2$ and $c$ are three complex constants satisfying $(-1)^k(c_1c_2)^n(nc)^{2k}=1.$ Applying this result, we completely resolve a uniqueness question of meromorphic functions involving certain non-linear differential polynomials. As applications, we also improve a result from Yang and Hua’s cited paper and study a periodicity question of non-constant meromorphic functions involving certain non-linear differential polynomials, where the periodicity question is related to a conjecture of Yang, reported by Q. Wang and P. C. Hu [Acta Math. Sci. 38 (2018), 209–214], and the differential-difference versions of Yang’s conjecture proposed by X. L. Liu and R. Korhonen [Bull. Austral. Math. Soc. 101 (2020), 453–465]. We also discuss a gap in the proof of a result of S. S. Bhoosnurmath and R. S. Dyavanal [Comput. Math. Appl. 53 (2007), 1191–1205].
