Abstract The classical Lambert series is given by L (x) = =₁^ x^ 1-x^{ }, |x| L (x) = ∑ ν = 1 ∞ x ν 1 - x ν, | x | 1. We prove that the function t L (1-e^-t) t ↦ L (1 - e - t) is strictly convex and strictly log-concave on [0, ) [ 0, ∞). Moreover, we use these results to deduce some functional inequalities for the Lambert series.
Alzer et al. (Fri,) studied this question.