What question did this study set out to answer?

This work aims to explore the properties of the Lambert series function relating to convexity and functional inequalities.

April 12, 2026Open Access

Inequalities for the Lambert series

Key Points

This work aims to explore the properties of the Lambert series function relating to convexity and functional inequalities.
Defined the classical Lambert series and transformed it using an exponential function.
Proved the strict convexity of the transformed Lambert series function.
Established the strict log-concavity of the same function on the interval [0, ∞).
Derived functional inequalities from these properties.
Confirmed that the function is strictly convex and log-concave on the specified interval.
Identified several functional inequalities related to the Lambert series.

Abstract

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.

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