  We find sufficient conditions for log-convexity and log-concavity for the
functions of the forms $a\mapsto\sum{f_k}(a)_kx^k$,
$a\mapsto\sum{f_k}\Gamma(a+k)x^k$ and $a\mapsto\sum{f_k}x^k/(a)_k$. The most
useful examples of such functions are generalized hypergeometric functions. In
particular, we generalize the Tur\'{a}n inequality for the confluent
hypergeometric function recently proved by Barnard, Gordy and Richards and
log-convexity results for the same function recently proved by Baricz. Besides,
we establish a reverse inequality which complements naturally the inequality of
Barnard, Gordy and Richards. Similar results are established for the Gauss and
the generalized hypergeometric functions. A conjecture about monotonicity of a
quotient of products of confluent hypergeometric functions is made.
