Analysis Problem of the Day 99

Today’s problem appeared as Problem 6 on the Texas A&M Winter 2016 Complex Analysis Qual:

Problem 6. Give an example of a non-polynomial entire surjective function $f: \mathbb{C} \to \mathbb{C}$ such that $f'$ is not surjective.

Solution: Recall that by Little Picard’s theorem, the image of an entire function can miss at most one point in the complex plane, and the prototypical example of such as function is $f(z)=e^z-c,$ missing the value $c.$ We thus claim that $g(z) = e^z+z$ with derivative $g'(z)=e^z+1,$ which misses the value $-1,$ is a surjective function. Suppose not. Then, for some $w \in \mathbb{C},$ $h(z)=e^z+z-w$ is an entire function of order $1$ with no zeros, and thus by the Hadamard factorization theorem takes the form $h(z) = e^{az+b}$ for some constants $a,b \in \mathbb{C}.$ Taking the derivative twice and three times on both sides yields $a^2 e^b = a^3 e^b,$ and since clearly $a \not =0,$ $a=1.$ Rearranging this and taking two derivatives again, we get $e^b = 1,$ so $z-w \equiv 0$ which is a contradiction. Thus, $e^z+z$ is an entire surjective function but its derivative is not, as desired.

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