Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

Zhao, Tie-Hong; Chu, Yu-Ming; Wang, Hua

Logarithmically Complete Monotonicity Properties Relating to the Gamma Function

Tie-Hong Zhao, Yu-Ming Chu and Hua Wang

Abstract and Applied Analysis, 2011, vol. 2011, 1-13

Abstract:

We prove that the function ð ‘“ ð ›¼ , ð ›½ ( ð ‘¥ ) = Î“ ð ›½ ( ð ‘¥ + ð ›¼ ) / ð ‘¥ ð ›¼ Î“ ( ð ›½ ð ‘¥ ) is strictly logarithmically completely monotonic on ( 0 , âˆž ) if âˆš ( ð ›¼ , ð ›½ ) âˆˆ { ( ð ›¼ , ð ›½ ) âˆ¶ 1 / ð ›¼ â‰¤ ð ›½ â‰¤ 1 , ð ›¼ â‰ 1 } âˆª { ( ð ›¼ , ð ›½ ) âˆ¶ 0 < ð ›½ â‰¤ 1 , ð œ‘ 1 ( ð ›¼ , ð ›½ ) â‰¥ 0 , ð œ‘ 2 ( ð ›¼ , ð ›½ ) â‰¥ 0 } and [ ð ‘“ ð ›¼ , ð ›½ ( ð ‘¥ ) ] âˆ’ 1 is strictly logarithmically completely monotonic on ( 0 , âˆž ) if âˆš ( ð ›¼ , ð ›½ ) âˆˆ { ( ð ›¼ , ð ›½ ) âˆ¶ 0 < ð ›¼ â‰¤ 1 / 2 , 0 < ð ›½ â‰¤ 1 } âˆª { ( ð ›¼ , ð ›½ ) âˆ¶ 1 â‰¤ ð ›½ â‰¤ 1 / âˆš ð ›¼ â‰¤ 2 , ð ›¼ â‰ 1 } âˆª { ( ð ›¼ , ð ›½ ) âˆ¶ 1 / 2 â‰¤ ð ›¼ < 1 , ð ›½ â‰¥ 1 / ( 1 âˆ’ ð ›¼ ) } , where ð œ‘ 1 ( ð ›¼ , ð ›½ ) = ( ð ›¼ 2 + ð ›¼ âˆ’ 1 ) ð ›½ 2 + ( 2 ð ›¼ 2 âˆ’ 3 ð ›¼ + 1 ) ð ›½ âˆ’ ð ›¼ and ð œ‘ 2 ( ð ›¼ , ð ›½ ) = ( ð ›¼ âˆ’ 1 ) ð ›½ 2 + ( 2 ð ›¼ 2 âˆ’ 5 ð ›¼ + 2 ) ð ›½ âˆ’ 1 .

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:896483

DOI: 10.1155/2011/896483

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