Short Remarks on Complete Monotonicity of Some Functions

• Main Author: Ladislav Matejíčka
• Format: Article
• Language: English
• Published: MDPI AG 2020-04-01
• Series: Mathematics
• Subjects: completely monotonic functions; laplace transform; inequality; Nielsen’s β-function; polygamma functions
• Online Access: https://www.mdpi.com/2227-7390/8/4/537

In this paper, we show that the functions <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>x</mi> <mi>m</mi> </msup> <mrow> <mo>|</mo> <msup> <mi>β</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are not completely monotonic on <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for all <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>β</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is the Nielsen’s <inline-formula> <math display="inline"> <semantics> <mi>β</mi> </semantics> </math> </inline-formula>-function and we prove the functions <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>x</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>|</mo> <msup> <mi>β</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>x</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>|</mo> <msup> <mi>ψ</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are completely monotonic on <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> for all <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> <mo>,</mo> </mrow> </semantics> </math> </inline-formula><inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>></mo> <mn>2</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> denotes the logarithmic derivative of Euler’s gamma function .