Variance upper bounds and convolutions of α-unimodal distributions

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