On the hyper-order of meromorphic solutions to linear differential equations
This paper contains some order considerations for solutions to the linear differential equation \[ f^{(k)} + A_{k-1}(z) f^{(k-1)} + ... + A_0(z) f = 0 \] with entire coefficients in the complex plane, related to a recent paper by \textit{K.-H. Kwon} [Bull. Korean Math. Soc. 33, No. 3, 487-496 (1996; Zbl 0863.34007)]. Assume \(A_s\) is a dominating in the sense that \(\max\{\rho (A_j):j \neq s\} < \rho (A_s)\). Then the following results are proved: (1) If \(\rho (A_s)<\frac 12\), then for all transcendental solutions \(f\), the hyperorder \[ \rho_2(f):=\limsup_{r \to \infty} (\log^+\log^+ T(r,f)/\log r) \] satisfies \(\rho_2(f) \geq \rho(A_s)\) with equality provided the convergence exponent \(\lambda(f)\) for the zeros of \(f\) satisfies \(\lambda(f) < \rho(A_s)\). (2) Every solution \(f\) of finite order is either a polynomial of degree \(\leq s-1\), or \(\rho(f) \geq \rho(A_s)\). For a closely related paper, dealing with more general iterated order notions, see \textit{L. Kinnunen} [Southeast Asian Bull. Math. 22, No. 4, 385-405 (1998; Zbl 0934.34076)].
scientific article; zbMATH DE number 1286283