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This analytical study proves the convexity and log-concavity of the Lambert series function, indicating functional inequalities.
The classical Lambert series is given by L(x)= ∑ ν =1^∞ xν1-xν, |x|<1. 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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Alzer et al. (2026) studied this question.
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