A refined cosπρ theorem

For an entire function f (z), let M(r) and m(r) be the maximum and minimum modulus, and let n(r) be the number of nonzero zeros of f (z) in |z| < r. Suppose that α and ρ are positive numbers, with 0 < ρ < 1, and that φ(r) is an increasing, unbounded function satisfying φ(r) = o(rρ) as r→∞. It is shown that if f (z) has order ρ, and n(r) − αrρ →−∞as r→∞, and |n(r) − αrρ| φ(r) for all large r, then lim r→∞ log m(r) −cosπρ logM(r) φ(r) log r −(1−cosπρ). An example shows that the constant on the right-hand side cannot be replaced by a number larger than −(1− cosπρ)/2.  2005 Elsevier Inc. All rights reserved