Converging bounds of the entropy of multinomial distributions

Bound expressions of the entropy of multinomial distributions are studied. Practical random processes are often characterized by multinomial distributions, but it seems difficult to derive a closed-form formula of the entropy of multinomial distributions. An upper-bound and a lower-bound of the entropy have been studied so far, but those known bounds are separated by a noticeable gap that is proportional to one of two major parameters of the distribution. This study proposes improved bounds that are valid even for non-asymptotic parameters, and asymptotically converge to the true value of the entropy.