In the note a new class of functions called $\alpha$-quasi-concave is introduced and it is proved that the convolution of two unimodal functions each having some additional concavity properties belongs to this class. Other results concerning the convolution of unimodal functions are also studied and the extensions of some of them are also given in the paper.
"Some Remarks About the Convolution of Unimodal Functions." Ann. Probab. 12 (2) 640 - 645, May, 1984. https://doi.org/10.1214/aop/1176993312