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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic | Lv et al. Journal of Inequalities and Applications 2011 2011 36 http www.journalofinequalitiesandapplications.eom content 2011 1 36 Journal of Inequalities and Applications a SpringerOpen Journal RESEARCH Open Access Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic Yu-Pei Lv Tian-Chuan Sun and Yu-Ming Chu Correspondence chuyuming2005@yahoo.com.cn Department of Mathematics Huzhou Teachers College Huzhou 313000 PR China Abstract We prove that the function Fap x x rb x r bx is strictly logarithmically completely monotonic on 0 if and only if a b e a b b 0 b 2a 1 b a 1 a b a 0 b 1 and that Fa b x -1 is strictly logarithmically completely monotonic on 0 if and only if a b e a b b 0 b 2a 1 b a 1 a b a 0 b 1 . 2010 Mathematics Subject Classification 33B15 26A48. Keywords completely monotonic logarithmically completely monotonic gamma function 1 Introduction For real and positive values of x the Euler gamma function r and its logarithmic derivative the so-called digamma functions are defined by r x 1.1 0 r x r x X -Y 0 e-t e-xt 1 - e- dt 1.2 where g 0.5772 is the Euler s constant. For extension of these functions to complex variable and for basic properties see 1 . Over the last half century many authors have established inequalities and monotonicity for these functions 2-22 . We know that a real-valued function f I R is said to be completely monotonic on I if f has derivatives of all orders on I and -1 nf n x 0 1.3 for all x e I and n 0. Moreover f is said to be strictly completely monotonic if inequalities 1.3 are strict. We also know that a positive real-valued function f I 0 is said to be logarithmically completely monotonic on I iff has derivatives of all orders on I and its logarithm log f satisfies Springer 2011 Lv et al licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http creativecommons.org licenses by 2.0 which .
