EXTENSION OF HOLOMORPHIC FUNCTIONS ONTO A SPECIAL DOMAIN

Abstract. We present a modified version of the Arakelyan’s result: a rela-tionship between holomorphic extension of a holomorphic function on the unit disc onto the domain C \ [1,∞) and its Taylor coefficients ’ interpolation. This paper aims at explaining some ideas and simplifying the proof from the paper of N. U. Arakelyan (see [2]) in the case of the domain C \ [1,∞). The problem is to characterize these holomorphic functions on the unit disc, which extend holomorphically on C \ [1,∞). There are known several such conditions, all connected with the existence of a holomorphic function on the half-plane or the plane, interpolating Taylor coefficients of the given function and having controlled growth at infinity. This growth is measured by the (inner) exponential type, the indicator function or the order. 1. Notation and results By D we denote the unit open disc on the complex plane and by T — the unit circle. Moreover, let H: = {z ∈ C: Re z ≥ 0}, ∆(θ1, θ2): = {z ∈ C: θ1 ≤ arg z ≤ θ2} ∪ {0}, −pi ≤ θ1 ≤ θ2 ≤ pi, where arg: C \ {0} − → (−pi, pi], continuous in C \ (−∞, 0], is the main argument. Sets ∆(θ1, θ2) we shall call closed sectors. Let log be the standard branch of the logarithm (continuous in C\(−∞, 0]) that is log z: = log |z|+ i arg z, z 6 = 0. Put log 0: = − ∞ and log+ t: = max{log t, 0} for t ≥ 0. The interior of a set S ⊂ C we denote by intS. Let G ⊂ C be a closed unbounded connected set and let a function ϕ be holomor-phic in an open neighborhood of G (which we denote ϕ ∈ O(G)). The exponential type of ϕ on G is defined as (1) ETG(ϕ): = lim sup z∈G, z→∞ log+ |ϕ(z)| |z | = max lim sup z∈G, z→∞ log |ϕ(z)