On maximum modulus and maximum term of composition of entire functions. (English) Zbl 0757.30036

It is well-known that \(\lim_{r\to\infty} {{\log\log M(r,f\circ g)} \over {\log\log M(r,f)}}=\infty\) whenever \(f\), \(g\) are transcendental entire functions. The authors ask whether the corresponding upper limit becomes finite if \(R\) is sufficiently large and the maximum modulus \(M(r,f)\) above will be replaced by \(M(R,f)\). They give a number of results related to this question. A typical example is the following theorem:
Let \(f\), \(g\) be entire functions of positive lower order and of finite order. Then \[ \lim_{r\to\infty} {{\log\log M(r,f\circ g)} \over {\log\log M(r^ A,f)}}=\infty, \qquad \lim_{r\to\infty} {{\log\log M(r,f\circ g)} \over {\log\log M(r^ A,g)}}=\infty \text{ for every } A>0. \] Moreover, the conclusion remains valid, if the maximum modulus will be replaced by maximum term from the Wiman-Valiron theory. On the other hand, if \(f\), \(g\) are entire functions of finite order such that \(\rho_ g<\rho_ f\), then \[ \liminf_{r\to\infty} {{\log\log M(r,f\circ g)} \over {\log\log M(\exp(r^{\rho_ f}),f)}}=0. \]
Reviewer: I.Laine (Joensuu)


30D20 Entire functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory