We denote N, R, C the sets of natural, real and complex numbers respectively. Let (
λn),
n ∈ N be an unbounded sequence of complex numbers. Costakis has proved the following result. There exists an entire function
f with the following property: for every
x, y ∈ R with
0 < x < y, every
θ ∈(0,1) and every
a ∈ C there is a subsequence of natural numbers
(mn),
n ∈ N such that, for every compact subset
L ⊆ C ,
In the present paper we show that the constant function
a cannot be replaced by any non-constant entire function
G. This is so even if one demands the convergence in (*) only for a single radius
r and a single positive number
θ. This result is related with the problem of existence of common universal vectors for an uncountable family of sequences of translation operators.