Does convolution preserve strict log-concavity?

Question

Suppose $f, g$ are strictly log-concave functions. Then the convolution $f * g$ will also be log-concave. However, will it also be strictly log-concave?

Thanks!

Answer 1

Follows by combining:

[1] Total Positivity of Mean Values and Hypergeometric Functions. B.C. Carlson and J.L. Gustafson. SIAM J. MATH. ANAL. Vol. 14, No. 2, March 1983.

[2] Convolutions of Logarithmically Concave Functions. Milan Merkle. Univ. Beograd, Publ. Elektrontehn. Fak. Ser. Mat. Vol. 9, 1998.