Log-convexity and strong q-log-convexity for some triangular arrays

We give a criterion for the log-convexity (resp. the strong q-log-convexity) of the first column of certain infinite triangular array (An,k)0⩽k⩽n of nonnegative numbers (resp. of polynomials in q with nonnegative coefficients), for which the recurrence relation is of the formAn,k=fkAn−1,k−1+gkAn−1,k+hkAn−1,k+1. This allows a unified treatment of the log-convexity of the Catalan-like numbers, as well as that of the q-log-convexity of some classical polynomials. In particular, we obtain simple proofs of the q-log-convexity of Narayana polynomials.