We extend the improved Schwarz inequality of Dragomir [1, Theorem 2] to any power p ≥ 2, |(x, e)| (lxlp − |(x, e)|p)1/p  p lxlplylp − |(x, y)|p ≥ det   |(y, e)| (lylp − |(y, e)|p)1/p for any vectors x, y, e ∈ Cn with lel = 1. Applications to n-tuples of complex numbers are also included.