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