Characterizations of arcsin and related distributions based on a new generalized unimodality

A general characterization for -unimodal distributions was provided in [Alamatsaz, M.H. 1985: A Note on an Article by Artikis, Acta Mathematica Hungarica 45, 159-162] and its extension to a multivariate case in [Alamatsaz, M.H. 1993: On Characterizations of Exponential and Gamma Distributions, Statistics and Probability Letters 17, 315-319]. Here, by solving the related equations, another generalization for unimodality is presented. As a result of this generalization, a simpler proof of a conjecture, as well as a characterization for generalized arcsin distributions and some generalizations of the author's earlier works, is obtained. Last, but not the least, it is shown that some elementary methods can be more powerful than some more advanced techniques.