Improving Gallager's upper bound on Huffman codes redundancy

We propose the first single bound that supersedes Gallager's (1978) upper bound on the redundancy of binary Huffman codes with the given largest source symbol probability ranging from 0 to 0.5. We define the linear logarithm and linear logarithm entropy. We find the maximal difference between the linear and the ordinary logarithms. We prove that the "redundancy" of a binary Huffmann code with respect to the linear logarithm entropy is no more than the largest source symbol probability. We therefore establish a better upper bound than Gallager's on the redundancy of binary Huffman codes.