Inequalities for gamma and q-gamma functions of complex arguments

We prove that the function Gamma(x + a)Gamma(x + b)/vertical bar Gamma(x + c + iy)vertical bar(2), a + b = 2c, and its q-analogue are of the form e-(h(x, y)) and h is completely monotonic in x. In particular both Gamma(x + a)Gamma(x + b)/vertical bar Gamma(x + c + iy)vertical bar(2) and Gamma(q)(x + a)Gamma(q) (x + b)/vertical bar Gamma(q)(x + c + iy)vertical bar(2) are Laplace transforms of infinitely divisible distributions. We also extend Lerch's inequality to the q-gamma function.