Im considering that f(f(z)) - exp(z) is small in a region without holes.

The idea of holes fascinates me.

Many conjectures come to my mind though the quality of them is not yet certain.

For instance

CONJECTURE HOLE A

If g(z) is a fake (entire) half-iterate of exp(z) on the positive real line and all the derivatives are positive then g(g(z)) - exp(z) is small in a region without holes.

Problem is also that things are only defined intuitive.

For instance the presence of a hole or not depends on your " definition of small ".

Only depending on your definition of small is if there is 1 hole or 2 holes close together. ( perhaps name that a pseudohole or fake hole ?)

Im not sure if the concept of holes is relevant to the location of zero's of f(z) , some (conjectured) uniqueness conditions , equivalence principles and other stuff discussed sofar , or if it is just a new subject.

Maybe I did to much topology lately.

regards

tommy1729

The idea of holes fascinates me.

Many conjectures come to my mind though the quality of them is not yet certain.

For instance

CONJECTURE HOLE A

If g(z) is a fake (entire) half-iterate of exp(z) on the positive real line and all the derivatives are positive then g(g(z)) - exp(z) is small in a region without holes.

Problem is also that things are only defined intuitive.

For instance the presence of a hole or not depends on your " definition of small ".

Only depending on your definition of small is if there is 1 hole or 2 holes close together. ( perhaps name that a pseudohole or fake hole ?)

Im not sure if the concept of holes is relevant to the location of zero's of f(z) , some (conjectured) uniqueness conditions , equivalence principles and other stuff discussed sofar , or if it is just a new subject.

Maybe I did to much topology lately.

regards

tommy1729