Composition of entire functions with finite iterated order
Let \(f\) and \(g\) be two entire functions. The authors investigate the growth of the composite function \(f(g)\) with iterated order. For sufficiently large \(r\), define \(\log_1r=\log r\) and \(\log_{n+1}r=\log(\log_n r)\), \(n\in\mathbb{N}\). The iterated order and the iterated lower order are defined by \(\rho_n(f)=\lim\sup_{r\to \infty}\log_{n+1}M(r,f)/\log r\) and \(\mu_n(f)=\lim\inf_{r\to\infty}\log_{n+1} M(r,f)/\log r\), respectively. The finiteness degree \(i(f)\) of the order is defined by \(i(f)=0\) if \(f\) is a polynomial, \(i(f)=\min\{n\in\mathbb{N}:\rho_n(f)< \infty\}\) if \(f\) is transcendental with \(\rho_n(f)< \infty\) for some \(n\in\mathbb{N}\), \(i(f)=\infty\) if \(\rho_n(f)=\infty\) for all \(\in \mathbb{N}\). See e.g., [\textit{L. Kinnunen}, Southeast Asian Bull. Math. 22, No. 4, 385--405 (1998; Zbl 0934.34076)]. One of the main results is the follwing: If \(f\) and \(g\) are of finite iterated order with \(i(f)=p\), \(i(g)=q\) and \(\mu_p(f)>0\), then \(i(f(g))=p+q\) and \(\rho_{p+g} (f(g))=\rho_q(g)\). The authors give an application to the factorization problem of the solutions of linear differential equations.
scientific article; zbMATH DE number 5549428