Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic
oleh: Lv Yu-Pei, Sun Tian-Chuan, Chu Yu-Ming
| Format: | Article |
|---|---|
| Diterbitkan: | SpringerOpen 2011-01-01 |
Deskripsi
<p>Abstract</p> <p>We prove that the function <it>F</it> <sub> <it>α,β</it> </sub>(<it>x</it>) = <it>x</it> <sup> <it>α</it> </sup>Γ<sup> <it>β</it> </sup>(<it>x</it>)/Γ(<it>βx</it>) is strictly logarithmically completely monotonic on (0, ∞) if and only if (<it>α</it>, <it>β</it>) ∈ {(<it>α</it>, <it>β</it>) : <it>β > </it>0, <it>β </it>≥ 2<it>α </it>+ 1, <it>β </it>≥ <it>α </it>+ 1}{(<it>α</it>, <it>β</it>) : <it>α </it>= 0, <it>β </it>= 1} and that [<it>F</it> <sub> <it>α,β</it> </sub>(<it>x</it>)]<sup>-1 </sup>is strictly logarithmically completely monotonic on (0, ∞) if and only if (<it>α</it>, <it>β</it>) ∈ {(<it>α</it>, <it>β </it>) : <it>β </it>> 0, <it>β </it>≤ 2<it>α </it>+ 1, <it>β </it>≤ <it>α </it>+ 1}{(<it>α</it>, <it>β </it>) : <it>α </it>= 0, <it>β </it>= 1}.</p> <p> <b>2010 Mathematics Subject Classification: </b>33B15; 26A48.</p>