Let the t ' th iteration of a real entire function f(x) with f(0) = 0 and f ' (0) = 1 be given by

f^[t](x) = 1/n ( f( t^a_1 * x)/(t^a_1) + f( t_a^2 * x)/(t^a_2) + ... + f( t^a_n * x)/(t^a_n) )

for some integer n > 0 and where the a_n are positive reals.

for some interval t element of [a,b].

Clearly it automaticly holds for (lim) t = 0 or t = 1.

Many related ideas can be made.

For instance solve for

\( f(f(x)) = ( f(2x)/2 + f(4x)/4 )/2. \)

***

I had the idea

lim

f^[-n]( ( f( t f^[n](x) )/t + f( t^2 f^[n](x) )/t^2 )/2 )

as a kind of koenings type function giving

f^[t](x) = lim f^[-n]( ( f( t f^[n](x) )/t + f( t^2 f^[n](x) )/t^2 )/2 )

for 0 < t =< 1.

regards

tommy1729

f^[t](x) = 1/n ( f( t^a_1 * x)/(t^a_1) + f( t_a^2 * x)/(t^a_2) + ... + f( t^a_n * x)/(t^a_n) )

for some integer n > 0 and where the a_n are positive reals.

for some interval t element of [a,b].

Clearly it automaticly holds for (lim) t = 0 or t = 1.

Many related ideas can be made.

For instance solve for

\( f(f(x)) = ( f(2x)/2 + f(4x)/4 )/2. \)

***

I had the idea

lim

f^[-n]( ( f( t f^[n](x) )/t + f( t^2 f^[n](x) )/t^2 )/2 )

as a kind of koenings type function giving

f^[t](x) = lim f^[-n]( ( f( t f^[n](x) )/t + f( t^2 f^[n](x) )/t^2 )/2 )

for 0 < t =< 1.

regards

tommy1729