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COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE DIVIDED DIFFERENCE OF PSI FUNCTIONS

Part of: Real functions Elementary classical functions Inequalities

Published online by Cambridge University Press:  18 January 2013

FENG QI ,
PIETRO CERONE  and
SEVER S. DRAGOMIR
FENG QI*
Affiliation:
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City 028043, Inner Mongolia Autonomous Region, PR China email qifeng618@hotmail.com, qifeng618@qq.com
PIETRO CERONE
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia email pcerone53@gmail.com, p.cerone@latrobe.edu.au
SEVER S. DRAGOMIR
Affiliation:
School of Engineering and Science, Victoria University, PO Box 14428, Melbourne, Victoria 8001, Australia School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa email sever.dragomir@vu.edu.au
*
✉qifeng618@gmail.com
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Abstract

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Necessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

Keywords

necessary and sufficient conditioncomplete monotonicitylogarithmically complete monotonicitydivided differenceratio of two gamma functionsinequalitypsi functionpolygamma function

MSC classification

Secondary: 26A48: Monotonic functions, generalizations 33B15: Gamma, beta and polygamma functions 26A51: Convexity, generalizations 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 309 - 319
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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