<p>The classical Lambert series is given by <Equation ID="Equ40"> <EquationSource Format="TEX">$ L(x)= \sum _{\nu =1}^\infty \frac{x^{\nu }}{1-x^{\nu }}, \quad |x|<1. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>L</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <mfrac> <msup> <mi>x</mi> <mi>ν</mi> </msup> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mi>x</mi> <mi>ν</mi> </msup> </mrow> </mfrac> <mo>,</mo> <mspace width="1em" /> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We prove that the function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t\mapsto L(1-e^{-t})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>↦</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is strictly convex and strictly log-concave on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we use these results to deduce some functional inequalities for the Lambert series.</p>

Inequalities for the Lambert series

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2026年4月10日

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Horst Alzer,
Hans W. Volkmer

摘要

The classical Lambert series is given by $ L(x)= \sum _{\nu =1}^\infty \frac{x^{\nu }}{1-x^{\nu }}, \quad |x|<1. $ $L (x) = \sum_{ν = 1}^{\infty} \frac{x^{ν}}{1 - x^{ν}}, | x | < 1 .$ We prove that the function $t\mapsto L(1-e^{-t})$ $t \mapsto L (1 - e^{- t})$ is strictly convex and strictly log-concave on $[0,\infty )$ $[0, \infty)$ . Moreover, we use these results to deduce some functional inequalities for the Lambert series.