An explicit bound below 2/π for the maximum term of an entire function

For a transcendental entire function f let μ(r) be its maximum term and M(r) its maximum modulus on |z| = r. Erdős asked for the largest possible value B of lim inf_r→∞ μ(r)/M(r). An argument of Clunie gives B ≤ 2/π = 0.636619…, and Clunie and Hayman proved B ≤ 2/π − c for an absolute constant c > 0 that they did not compute. We prove B ≤ 0.63005. We make the stability argument of Clunie and Hayman quantitative by testing f against trigonometric polynomials at one or both of two consecutive radii where the central index jumps. The proof is computer-assisted, and the theorem is also proved in the Lean proof assistant.

The certificates, both checkers, the programs that produced them and the Lean files are archived at doi:10.5281/zenodo.23716683 and available at github.com/tejasvi/maximum-term-bound (tag v4, commit 2c3f8bb7f3e1).