## 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)

### MSC:

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