A conjecture for x >> 2 and functions with decreasing positive derivatives.
let some f(x) have dominant term a_n x^n
let f(f(x)) have dominant term b_m x^m.
conjecture A :
If |f(x)| < exp^[1/3](x)
then f(f(x)) ~ a_n (a_n x^n)^n = b_m x^m
thus b_m = a_n ^ (n+1) and m = n^2.
conjecture B :
(reverse of A)
If |f(f(x))| < exp^[2/3](x)
then b_m = a_n ^ (n+1) and n = sqrt(m)+o(1).
conjecture C :
If |f(f(x))| < exp^[2/3](x)
then b_m = a_n ^ (n+1) and n = sqrt(m)+o(1).
and if a_n = b_m^{1/(n+1)} does not converge fast enough , then f(x) is not entire and there is a complex z with |z|<1 such that its nearest singularity is of type a_0 + a_1 x + ...
I havent considered it alot , it might need modification or perhaps even very false.
But I wanted to share it now.
regards
tommy1729
let some f(x) have dominant term a_n x^n
let f(f(x)) have dominant term b_m x^m.
conjecture A :
If |f(x)| < exp^[1/3](x)
then f(f(x)) ~ a_n (a_n x^n)^n = b_m x^m
thus b_m = a_n ^ (n+1) and m = n^2.
conjecture B :
(reverse of A)
If |f(f(x))| < exp^[2/3](x)
then b_m = a_n ^ (n+1) and n = sqrt(m)+o(1).
conjecture C :
If |f(f(x))| < exp^[2/3](x)
then b_m = a_n ^ (n+1) and n = sqrt(m)+o(1).
and if a_n = b_m^{1/(n+1)} does not converge fast enough , then f(x) is not entire and there is a complex z with |z|<1 such that its nearest singularity is of type a_0 + a_1 x + ...
I havent considered it alot , it might need modification or perhaps even very false.
But I wanted to share it now.
regards
tommy1729
