  We establish discrete and continuous log-concavity results for a biparametric
extension of the $q$-numbers and of the $q$-binomial coefficients. By using
classical results for the Jacobi theta function we are able to lift some of our
log-concavity results to the elliptic setting. One of our main ingredients is a
putatively new lemma involving a multiplicative analogue of Tur\'an's
inequality.
