Abstract
Abstract We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.
Highlights
This paper concerns transcendental entire functions of small order
Such functions have been studied extensively in classical complex analysis, ever since Wiman [27] observed that such functions have properties which, in some ways, resemble those of polynomials. Powerful results such as the version of the cos πρ theorem due to Barry [2] showed that, for such functions, the minimum modulus of the function on many circles centred at the origin is comparable in size to the maximum modulus of the function. These properties have led to functions of small order playing a key role in two major conjectures in complex dynamics: Baker’s conjecture explicitly concerns such functions and they were shown to have an unexpected link with Eremenko’s conjecture [11], one of the main drivers of research in transcendental dynamics, arising from the fact that for functions of small order the escaping set of the function is often connected; see [21] and [25]
To achieve the necessary control, we introduce a new technique for estimating the growth of certain positive harmonic functions from above; this technique may well have applications beyond our current purpose
Summary
This paper concerns transcendental entire functions of small order. Such functions have been studied extensively in classical complex analysis, ever since Wiman [27] observed that such functions have properties which, in some ways, resemble those of polynomials. The two estimates in Theorem 1.6 enable us to use any subharmonic function u ∈ K to obtain an entire function f with the same order, lower order and type class as u, and with the property that log |f | is uniformly bounded by u, provided (1.4) holds.
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