Q-analogue of a Linear Transformation Preserving Log-concavity

Log-concave and Log-convex sequences arise often in combinatorics, algebra, probability and statistics. There has been a considerable amount of research devoted to this topic in recent years. Let {xi}i≥0 be a sequence of non-negative real numbers. We say that {xi} is Log-concave ( Log-convex resp.) if and only if xi−1xi+1 ≤ xi (xi−1xi+1 ≥ xi resp.) for all i ≥ 1 (relevant results can see [2] and [4]). For our purpose, when a sequence is said to be Log-concave or Log-convex we always assume that it has no internal zeros, i.e., there are no three indices i < j < k such that xi, xk = 0 and xj = 0. This is a natural assumption for sequences since most of the Log-concave and Log-convex sequences of interest to us actually meet the condition. Thus a sequences is Log-concave (or Logconvex) if and only if xi−1xj+1 ≤ xixj ( or xi−1xj+1 ≥ xixj ) for any j ≥ i ≥ 1 ( see [1] ). Let f(x), g(x) ∈ R[x], we say f(x) ≥x g(x) if and only if f(x)−g(x) has nonnegative coefficients. We say that a linear transformation