Entire functions of small order of growth (Q643209)

The authors study the growth of composite entire functions. They prove that if \(f\) is a transcendental entire function and \(F\) is an entire function satisfying \[ \log M(r,F)=K(\log r)^p(1+o(1)),\tag{\(*\)} \] then, \[ \log M(r,F(f))=K\left(\log M(r,f)\right)^p(1+o(1)). \] Then, the authors apply this to the functional equation \[ f(sz)=F(f(z)), s\in \mathbb{C}, |s|>1, \] where \(F\) satisfies the equation (\(*\)) and prove that if \(f\) is a solution of the above functional equation, then \[ \log\log M(r,f)=A(r)r^{\rho}, \rho=\frac{\log p}{\log|s|}, \] where \(K_1<A(r)<K_2\) for some constants \(K_1\) and \(K_2\).