The periodicity on a transcendental entire function with its differential polynomials
Let \(f(z), z\in \mathbb{C},\) be a periodic entire function. This obviously implies that \(f^{(k)}\), or more generally, polynomials of \(f\) and its derivatives are periodic entire functions with the same period. Simple examples show that the converse implication is not always true. There is a number of results describing different situations where the converse implication is true. Yang's Conjecture given in \textit{P. Li} et al. [Houston J. Math. 45, No. 2, 431--437 (2019; Zbl 1428.30030)] states that if \(f\) is a transcendental entire function and \(k\) a positive integer such that \(ff^{(k)}\) is periodic, then \(f\) is also a periodic function. The authors prove several results related to this conjecture. In particular, they prove that if \(f\) is a transcendental entire function, \(k\) is a non-negative integer, \(a_1,\dots,a_n, a_n\ne0,n\geq2,\) are constants such that the function \((a_nf^n + \dots+ a_1f)^{(k)}\) is periodic, then \(f\) is also a periodic function.
scientific article; zbMATH DE number 7393075