Variance upper bounds and convolutions of [alpha]-unimodal distributions

The concept of [alpha]-uniform for generating an [alpha]-unimodal discrete distribution is introduced. Upper bounds for the variance of discrete [alpha]-unimodal distributions are established. Sufficient conditions for the convolution of [alpha]-unimodal distributions about a to be [alpha]-unimodal about the same mode are derived. These results are reduced to the usual discrete uniform and discrete unimodal distributions for [alpha] = 1. The paper includes other related results.