Improvement of upper bound to the optimal average cost of the variable length code

We consider the optimal average cost of variable length source code averaged over a given probability distribution over source symbols. The problem was argued in Csiszar and Korner´s (1981) book. We find a upper bound to the optimal cost minus an ideal cost, where the ideal cost is the entropy of the source divided by a unique scalar that makes negative costs logarithmic probabilities. Our bound is better than the one given in the book