Fixed points of the \(l\)th power of differential polynomials generated by solutions of differential equations (Q2574317)

Let \(f\) be a nontrivial solution of the second-order differential equation \[ f''+A(z)f =0, \] where \(A(z)\) is an entire function. A classic research problem is to study the exponent of convergence of zeros of \(f\). \textit{Z. Chen} [Acta Math. Sci. (Chin. Ed.) 20, 425--432 (2000; Zbl 0980.30022)] studied the exponent of convergence of fixed-points of \(f\). In this paper, the authors studies the (hyper)-exponent of convergence of the distinct fixed-points of \(L^{l}(f)\), where \(L(f) = a_kf^{(k)} + a_{k-1}f^{k-1} +\dots+a_0f,\) \(a_n\), \(n=1,\dots,k\), are constants. By mainly using Wiman-Valiron theory, they prove their main result (Theorem 1): (1) If \(A(z)\) is a polynomial of degree \(n\), then the exponent of convergence of the distinct fixed-points of \(L^l(f)\) is \(\frac{n+2}{2}\). (2) If \(A(z)\) is a transcendental entire function with order \(\sigma<+\infty\), then the exponent of convergence of the distinct fixed-points of \(L^l(f)\) is infinite and the hyper-exponent of convergence of the distinct fixed-points of \(L^l(f)\) is \(\sigma\). Concerning the case \(l\geq 2\), which is a special case of composite functions, see \textit{C.-C. Yang} and \textit{J.-H. Zheng}'s paper [J. Anal. Math. 68, 59--93 (1996); correction ibid. 72, 311--312 (1997; Zbl 0863.30035)].