Logarithmically Complete Monotonicity Properties Relating to the Gamma Function
We prove that the function fα,β(x)=Γβ(x+α)/xαΓ(βx) is strictly logarithmically completely monotonic on (0,∞) if (α,β)∈{( α,β):1/α≤β≤1, α≠1}∪{(α,β):0<β≤1,φ1(α,β)≥0,φ2(α,β)≥0} and [fα,β(x)]-1 is strictly logarithmically completely monotonic on (0,∞) if (α,β)∈{(α,β):0<α≤1/2,0<β≤1}∪{(α,β):1≤β≤1...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/896483 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | We prove that the function fα,β(x)=Γβ(x+α)/xαΓ(βx) is strictly logarithmically completely monotonic on (0,∞) if (α,β)∈{( α,β):1/α≤β≤1, α≠1}∪{(α,β):0<β≤1,φ1(α,β)≥0,φ2(α,β)≥0} and [fα,β(x)]-1 is strictly logarithmically completely monotonic on (0,∞) if (α,β)∈{(α,β):0<α≤1/2,0<β≤1}∪{(α,β):1≤β≤1/α≤2,α≠1}∪{(α,β):1/2≤α<1,β≥1/(1-α)}, where φ1(α,β)=(α2+α-1)β2+(2α2-3α+1)β-α and φ2(α,β)=(α-1)β2+(2α2-5α+2)β-1. |
|---|---|
| ISSN: | 1085-3375 1687-0409 |