Dense-lineability of sets of Birkhoff-universal functions with rapid decay

Let A be an unbounded Arakelian set in the complex plane whose complement has infinite inscribed radius, and ψ be an increasing positive function on the positive real numbers. We prove the existence of a dense linear manifold M of entire functions all of whose non-zero members are Birkhoff-universal, such that each function in M has overall growth faster than ψ and, in addition, exp(α|z|)f(z)→0 (z→∞, z∈A) for all α<1/2 and f∈M. With slightly more restrictive conditions on A, we get that the last property also holds for the action Tf of certain holomorphic operators T. Our results unify, extend and complete recent work by several authors.