Examples of holomorphic, complex differentiable, always positive functions

Question

I am looking for classes of functions which are:

    1) holomorphic 
    2) |f(z)|>0 for all z
    3) complex differentiable (i.e. f(z)=mod(z) is not valid)

Particularly I am looking for functions whose complex derivatives are less complicated.

While conditions $(1)$ and $(3)$ may be too general, in particular I am interested in condition $(2)$, namely that for all complex numbers the function value is greater than zero.

For example, $\exp(z)$ would meet the conditions, because $|\exp(z)|$=$1>0$

If this question is too general, please let me know and I will think of further conditions to impose.

Answer 1

All such functions can be written in the form

$$f(z) = \exp(g(z))$$

where $g$ is holomorphic. Such functions certainly satisfy your condition, and the fact that all such functions can be written in this form follows from the existence of the logarithm of non-zero functions.