Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, A(infinity) (B-X; Y) will denote the space of bounded and continuous functions from BX to Y that are holomorphic on the open unit ball. The numerical radius of an element h in A(infinity) (B-X; X) is the spectrum of the set {vertical bar x*(h(x))vertical bar : x is an element of X, x* is an element of X*, parallel to x*parallel to = parallel to x parallel to =x*(x) = 1} we prove that every complex Banach space X with the Radon-Nikodym property satisfies that the subset of numerical radius attaining functions in A(infinity) (B-X; X). We also show that the denseness of the numerical radius attaining elements of A(u)(B-c0 ;c(0)) which are uniformly continuous on the unit ball. For C(K) we prove a denseness result for the subset of the functions in A(infinity) (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 every R > e, P cannot be approximated by bounded and numerical radius attaining holomorphic functions defined on RBX. If Y satifies some isometric conditions and X is such that the subset of norm attaining functions of A(infinity) (B-X; C) is dense in A(infinity) (B-X; C), then the subset of norm attaining functions in A infinity(B-X; Y) in dense in the whole space.