  In this paper our aim is to present some monotonicity and convexity
properties for the one dimensional regularization of the Coulomb potential,
which has applications in the study of atoms in magnetic fields and which is in
fact a particular case of the Tricomi confluent hypergeometric function.
Moreover, we present some Tur\'an type inequalities for the function in the
question and we deduce from these inequalities some new tight upper bounds for
the Mills ratio of the standard normal distribution.
