Minimum codeword length and redundancy of Huffman codes

A tight upper bound on the redundancy r of Huffman codes, in terms of the minimum codeword length l, l≥1, is provided. The bound is a strictly decreasing function of l. For large l it yields r≤l−log(2l+1−1)+1+β+O(2−2l), where β≈0.0860.