On the existence of relative fix points (Q2708411)

The authors introduce the following classes of entire functions:NEWLINENEWLINENEWLINEClass I: \(f\) is an entire transcendental function.NEWLINENEWLINENEWLINEClass II: For \(a\neq b\) \(f\) is analytic in \(\widehat{\mathbb{C}}\setminus\{a, b\}\) and has an essential singularity at \(b\) and a singularity at \(a\) and \(f\) omits \(a\) and \(b\) in \(\widehat{\mathbb{C}}\setminus\{a, b\}\). (For example take \(a= 0\) and \(b=\infty\).)NEWLINENEWLINENEWLINELet \(f\) and \(\Phi\) be functions of class II and define NEWLINE\[NEWLINE\begin{alignedat}{2} & f_1:=f,\quad &&\Phi_1:= \Phi,\\ & f_2:= f\circ\Phi_1,\quad &&\Phi_2:= \Phi\circ f_1,\\ & f_3:= f\circ\Phi_2,\quad &&\Phi_3:= \Phi\circ f_2\end{alignedat}NEWLINE\]NEWLINE and so on. Then they prove: If \(f\) and \(\Phi\) belong to class II, then \(f\) has infinitely many relative fixed points of exact order \(n\) for every \(n\in\mathbb{N}\), provided \(T(r,\Phi_n)/T(r, f_n)\) is bounded.NEWLINENEWLINENEWLINEA point \(c\) is called a relative fixed point of \(f\) of exact order \(n\) with respect to \(\Phi:\Leftrightarrow f_n(c)= c\) but \(f_k(c)\neq c\) for \(k= 1,2,\dots, n-1\).