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Refinements of lower bounds for polygamma functions
Authors:
Bai-Ni Guo and Feng Qi
Journal:
Proc. Amer. Math. Soc. 141 (2013), 1007-1015
MSC (2010):
Primary 33B15; Secondary 26D07
Posted:
August 9, 2012
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Additional Information
Abstract: In the paper, some lower bounds for polygamma functions are refined. Moreover, several open problems are posed.
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- 1.
- H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), 181-221. MR 2039096 (2005d:33003)
- 2.
- H. Alzer, Sharp inequalities for the harmonic numbers, Expo. Math. 24 (2006), no. 4, 385-388. MR 2313126 (2007m:11041)
- 3.
- N. Batır, On some properties of digamma and polygamma functions, J. Math. Anal. Appl. 328 (2007), no. 1, 452-465. MR 2285562 (2008c:33001)
- 4.
- N. Batır, Some new inequalities for gamma and polygamma functions, J. Inequal. Pure Appl. Math. 6 (2005), no. 4, Art. 103; Available online at http://www.emis.de/journals/
JIPAM/article577.html. MR 2178284 (2006k:33001)
- 5.
- P. S. Bullen, Handbook of Means and Their Inequalities, Mathematics and its Applications, Volume 560, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003. MR 2024343 (2005a:26001)
- 6.
- N. Elezović, C. Giordano, and J. Pečarić, The best bounds in Gautschi's inequality, Math. Inequal. Appl. 3 (2000), 239-252. MR 1749300 (2001g:33001)
- 7.
- B.-N. Guo, R.-J. Chen, and F. Qi, A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 2, 124-134. MR 2331512 (2009e:33004)
- 8.
- B.-N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655-667; Available online at http://dx.doi.org/10.4134/JKMS.2011.48.3.655.
- 9.
- B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms 52 (2009), 89-92; Available online at http://dx.doi.org/10.1007/s11075-008-9259-7. MR 2533996 (2010h:26044)
- 10.
- B.-N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), no. 9, 1208-1216; Available online at http://dx.doi.org/10.1016/
j.jat.2011.04.003.
- 11.
- B.-N. Guo and F. Qi, Some properties of the psi and polygamma functions, Hacet. J. Math. Stat. 39 (2010), no. 2, 219-231. MR 2681248 (2011g:33001)
- 12.
- B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103-111; Available online at http://dx.doi.org/10.4134/bkms.2010.47.1.103. MR 2604236 (2011c:33004)
- 13.
- B.-N. Guo, F. Qi, and H. M. Srivastava, Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 (2010), no. 11, 103-111; Available online at http://dx.doi.org/10.1080/10652461003748112. MR 2739394
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- F. Qi and B.-N. Guo, Necessary and sufficient conditions for functions involving the tri- and tetra-gamma functions to be completely monotonic, Adv. Appl. Math. 44 (2010), no. 1, 71-83; Available online at http://dx.doi.org/10.1016/j.aam.2009.03.003. MR 2552656 (2010i:33007)
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- F. Qi and B.-N. Guo, Refinements of lower bounds for polygamma functions, Available online at http://arxiv.org/abs/0903.1966.
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- F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149-2160; Available online at http://dx.doi.org/10.1016/j.cam.2009.09.044. MR 2577754 (2010j:33003)
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- 23.
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Additional Information
Bai-Ni Guo
Affiliation:
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, People’s Republic of China
Email:
bai.ni.guo@gmail.com
Feng Qi
Affiliation:
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, People’s Republic of China
Email:
qifeng618@gmail.com
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11387-5
PII:
S 0002-9939(2012)11387-5
Keywords:
Refinement,
lower bound,
polygamma function,
inequality,
mean,
open problem
Received by editor(s):
December 13, 2009
Received by editor(s) in revised form:
March 2, 2011, and August 4, 2011
Posted:
August 9, 2012
Additional Notes:
The second author was partially supported by the China Scholarship Council and the Science Foundation of Tianjin Polytechnic University
Communicated by:
Walter Van Assche
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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