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Fixed points of the \(l\)th power of differential polynomials generated by solutions of differential equations. (English) Zbl 1090.34071

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 \(f\). 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 C.-C. Yang and J.-H. Zheng’s paper [J. Anal. Math. 68, 59–93 (1996); correction ibid. 72, 311–312 (1997; Zbl 0863.30035)].

MSC:

34M10 Oscillation, growth of solutions to ordinary differential equations in the complex domain
30D15 Special classes of entire functions of one complex variable and growth estimates
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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