Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, A-∞ (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in A_∞ (B _X; X) is the supremum of the set |x*(h(x))| : x ε X, x x* ε X x*, || x x* parallel = x = x* (x) = 1. We prove that every complex Banach space X with the Radon-Nikodým property satisfies that the subset of numerical radius attaining functions in A_∞ (B _X; X) is dense in {A}_∞ (B _X; X). We also show the denseness of the numerical radius attaining elements of A_u (B_c0 } ; c ₀ ) in the whole space, where B_c0 } ; c₀ is the subset of functions in B_c0 } ; c₀ ) which are uniformly continuous on the unit ball. For C(K) we prove a denseness result for the subset of the functions in {A_∞ (B _C(K); C(K)) which are weakly uniformly continuous on the closed unit ball. For a certain sequence space X, there is a 2-homogenous polynomial P from X to X such that for every R > e, P cannot be approximated by bounded and numerical radius attaining holomorphic functions defined on RB _X . If Y satisfies some isometric conditions and X is such that the subset of norm attaining functions of A _∞ (B _X; ℂ) is dense in A}_∞ B_c0 } ; c₀ (B _X; ℂ), then the subset of norm attaining functions in {A}_∞ (B _X; Y) is dense in the whole space.