  In this paper we introduce the new class of generalized Volterra functions.
We prove some integral representations for them via Fox-Wright H-functions and
Meijer G-functions. From positivity conditions on the weight in these
representations, we found sufficient conditions on parameters of the
generalized Volterra function to prove its complete monotonicity. As
applications we prove some Tur\'an type inequalities for generalized Volterra
functions and derive closed-form integral representations for a family of
convergent Mathieu-type series defined in terms of generalized Volterra
functions.
