Bounded Entire Functions Are Constant

This question is about entire complex functions that are bounded on the whole complex plane, and asks the candidate to show that such a function cannot be anything but constant. It sits at the foundation of complex analysis, formalizing the idea that analyticity plus a global constraint on size completely rigidifies the function. The candidate must reason globally from local analytic structure, bridging the gap between behavior on large circles and the impossibility of nontrivial growth at infinity. This is a classic theorem-level question that often appears in advanced pure mathematics and PhD qualifying exams.

The solution leans heavily on power series expansions, Cauchy's integral formula, and the maximum modulus principle or related growth estimates. An interviewer is looking for fluency in moving between local series coefficients and global bounds, and for comfort with estimating derivatives via contour integrals. They also watch for structural understanding: recognizing how boundedness constrains all higher-order terms, organizing a clean argument without handwaving, and articulating the conceptual message that entire functions are extremely rigid under global size conditions.

What it tests

The core structure behind this problem class is the interplay between analyticity (holomorphicity) and boundedness for functions defined on the entire complex plane. Analytic functions are tightly controlled by their local behavior: the value of the function everywhere is determined by its power series expansion at any point, and the coefficients of this expansion are dictated by the function's values on circles of arbitrary radius. When a function is bounded everywhere, the coefficients of higher powers in its expansion must shrink rapidly enough to prevent unbounded growth as $|z|$ increases. This is not just a technicality: the global boundedness forces all non-constant terms to vanish, since any nonzero coefficient would eventually dominate and violate the bound. The principle is that entire functions cannot 'hide' growth in their tails if they are globally bounded—their entire structure is revealed by their maximum modulus on arbitrarily large circles.