For a transcendental entire function f , the property that there exists r > 0 such that m n (r) → ∞ as n → ∞, where m(r) = min{|f (z)| : |z| = r}, is related to conjectures of Eremenko and of Baker, for both of which order 1/2 minimal type is a significant rate of growth. We show that this property holds for functions of order 1/2 minimal type if the maximum modulus of f has sufficiently regular growth and we give examples to show the sharpness of our results by using a recent generalisation of Kjellberg's method of constructing entire functions of small growth, which allows rather precise control of m(r).