What non-zero complex analytic functions have all derivatives non-zero, everywhere?

Update: Here's a proof in progress (I hope)...

Since C is open (in itself), every zero-free analytic function f(z) has an analytic logarithm.

Thus, f(z) = e^(g(z)) for some analytic g(z) on C.

Note that f' = g' e^g.

Since we want f'(z) ≠ 0 for all z in C, this forces

g'(z) ≠ 0 for all z in C.

Thus, g'(z) = e^(h(z)) for some analytic h(z) on C.

Next, we look at the second derivative.

f'' = (g'' + (g')^2) e^g

= (h' e^h + (e^h)^2) e^g

= (h' + e^h) e^(g+h).

Since we also want f''(z) ≠ 0 for all z in C, we require that

h' + e^h ≠ 0.

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Major case:

This is certainly satisfied if h(z) is a nonzero constant

Then, g'(z) = k for some nonzero k in C.

==> g(z) = kz + r for some r in C.

==> f(z) = e^(kz + r) = Ce^(kz) for some nonzero constants C and k.

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Note that f^(n)(z) = Ck^n e^(kz) is never 0 for all z in C.

Hence, we've found one such family of functions.

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Still to be addressed:

Are there other functions which satisfy h' + e^h ≠ 0 for all z in C which will fit the criteria for this posted question?

(This is certainly false if h(z) = az + b with a ≠ 0.)