True and false questions whether combinations of entire functions are still entire functions

Question

Let $f,g$ be entire functions. Decide which of the following are still entire functions.

(a) $f(z)^3$

(b) $(fg)(z)$

(c) $(f/g)(z)$

(d) $5f(z) + ig(z)$

(e) $f(1/z)$

(f) $g(z^2 + 2)$

(g) $f(g(z))$

My thoughts are

(a) Yes, since $(f(z)^3)' = 3f'(z)f(z)^2$ and so on and so on for any $n$.

(b) This follows from product rule I believe.

(c) Not if $f(z) = g(z) = x$. Then we would get $\frac{f(z)}{g(z)} = \frac{x}{x}$. So not true.

(d) Sums of differentiable functions.

(e) $f(z) = x$ is a counter example like above. All use real analysis.

(f) I think this is chain rule again, but I can't see why translations would make a function not entire.

(g) Chain rule again.

Answer 1

Entire functions are closed under composition, +, -, * and positive integer powers. This will get a few.