On the redundancy achieved by Huffman codes

It has been recently proved that the redundancy r of any discrete memoryless source satisfies r 1 \Gamma H(p N ), where p N is the least likely source letter probability. This bound is achieved only by sources consisting of two letters. We prove a sharper bound if the number of source letters is greater than two. Also provided is a new upper bound on r, as function of the two least likely source letter probabilities which improve on previous results