  In this paper, we study power series with coefficients equal to a product of
a generic sequence and an explicitly given function of a positive parameter
expressible in terms of the Pochhammer symbols. Four types of such series are
treated. We show that logarithmic concavity (convexity) of the generic sequence
leads to logarithmic concavity (convexity) of the sum of the series with
respect to the argument of the explicitly given function. The logarithmic
concavity (convexity) is derived from a stronger property, namely, positivity
(negativity) of the power series coefficients of the so-called generalized
Tur\'{a}nian. Applications to special functions such as the generalized
hypergeometric function and the Fox-Wright function are also discussed.
