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...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2011/896483 |
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| author | Tie-Hong Zhao Yu-Ming Chu Hua Wang |
| author_facet | Tie-Hong Zhao Yu-Ming Chu Hua Wang |
| author_sort | Tie-Hong Zhao |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-97fae93df1dc4fd9a7b46a54747b4b36 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-97fae93df1dc4fd9a7b46a54747b4b362025-02-03T01:11:58ZengWileyAbstract and Applied Analysis1085-33751687-04092011-01-01201110.1155/2011/896483896483Logarithmically Complete Monotonicity Properties Relating to the Gamma FunctionTie-Hong Zhao0Yu-Ming Chu1Hua Wang2Department of Mathematics, Huzhou Teachers College, Huzhou 313000, ChinaDepartment of Mathematics, Huzhou Teachers College, Huzhou 313000, ChinaDepartment of Mathematics, Changsha University of Science and Technology, Changsha 410076, ChinaWe 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.http://dx.doi.org/10.1155/2011/896483 |
| spellingShingle | Tie-Hong Zhao Yu-Ming Chu Hua Wang Logarithmically Complete Monotonicity Properties Relating to the Gamma Function Abstract and Applied Analysis |
| title | Logarithmically Complete Monotonicity Properties Relating to the Gamma Function |
| title_full | Logarithmically Complete Monotonicity Properties Relating to the Gamma Function |
| title_fullStr | Logarithmically Complete Monotonicity Properties Relating to the Gamma Function |
| title_full_unstemmed | Logarithmically Complete Monotonicity Properties Relating to the Gamma Function |
| title_short | Logarithmically Complete Monotonicity Properties Relating to the Gamma Function |
| title_sort | logarithmically complete monotonicity properties relating to the gamma function |
| url | http://dx.doi.org/10.1155/2011/896483 |
| work_keys_str_mv | AT tiehongzhao logarithmicallycompletemonotonicitypropertiesrelatingtothegammafunction AT yumingchu logarithmicallycompletemonotonicitypropertiesrelatingtothegammafunction AT huawang logarithmicallycompletemonotonicitypropertiesrelatingtothegammafunction |