Abstract :
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
