Iterating the Minimum Modulus: Functions of Order Half, Minimal Type

For a transcendental entire function f, the property that there exists $$r>0$$ r > 0 such that $$m^n(r)\rightarrow \infty $$ m n ( r ) → ∞ as $$n\rightarrow \infty $$ n → ∞ , where $$m(r)=\min \{|f(z)|:|z|=r\}$$ 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).

Generalized Bernoulli Wick differential equation

Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.