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