  We show that the duals of Banach algebras of scalar-valued bounded
holomorphic functions on the open unit ball $B_E$ of a Banach space $E$ lack
weak$^*$-strongly exposed points. Consequently, we obtain that some Banach
algebras of holomorphic functions on an arbitrary Banach space have the
Daugavet property which extends the observation of P. Wojtaszczyk. Moreover, we
present a new denseness result by proving that the set of norm-attaining
vector-valued holomorphic functions on the open unit ball of a dual Banach
space is dense provided that its predual space has the metric $\pi$-property.
Besides, we obtain several equivalent statements for the Banach space of
vector-valued homogeneous polynomials to be reflexive, which improves the
result of J. Mujica, J. A. Jaramillo and L. A. Moraes. As a byproduct, we
generalize some results on polynomial reflexivity due to J. Farmer.
