On the Redundancy Achieved by Huffman Codes

It has been recently proved that the redundancy r of any discrete memoryless source satisses r 1?H(p N), where p N is the least likely source letter probability. We prove that this bound is achieved only by sources consisting of two letters and that a sharper bound holds 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 improves on a previous result.