Refinements of lower bounds for polygamma functions
Authors:
BaiNi Guo and Feng Qi
Journal:
Proc. Amer. Math. Soc. 141 (2013), 10071015
MSC (2010):
Primary 33B15; Secondary 26D07
Published electronically:
August 9, 2012
MathSciNet review:
3003692
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In the paper, some lower bounds for polygamma functions are refined. Moreover, several open problems are posed.
 1.
Horst
Alzer, Sharp inequalities for the digamma and polygamma
functions, Forum Math. 16 (2004), no. 2,
181–221. MR 2039096
(2005d:33003), 10.1515/form.2004.009
 2.
Horst
Alzer, Sharp inequalities for the harmonic numbers, Expo.
Math. 24 (2006), no. 4, 385–388. MR 2313126
(2007m:11041), 10.1016/j.exmath.2006.02.001
 3.
Necdet
Batir, On some properties of digamma and polygamma functions,
J. Math. Anal. Appl. 328 (2007), no. 1,
452–465. MR 2285562
(2008c:33001), 10.1016/j.jmaa.2006.05.065
 4.
Necdet
Batir, Some new inequalities for gamma and polygamma
functions, JIPAM. J. Inequal. Pure Appl. Math. 6
(2005), no. 4, Article 103, 9. MR 2178284
(2006k:33001)
 5.
P.
S. Bullen, Handbook of means and their inequalities,
Mathematics and its Applications, vol. 560, Kluwer Academic Publishers
Group, Dordrecht, 2003. Revised from the 1988 original [P. S. Bullen, D. S.
Mitrinović and P. M. Vasić, Means and their inequalities,
Reidel, Dordrecht; MR0947142]. MR 2024343
(2005a:26001)
 6.
Neven
Elezović, Carla
Giordano, and Josip
Pečarić, The best bounds in Gautschi’s
inequality, Math. Inequal. Appl. 3 (2000),
no. 2, 239–252. MR 1749300
(2001g:33001), 10.7153/mia0326
 7.
BaiNi
Guo, RongJiang
Chen, and Feng
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 trigamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655667; Available online at http://dx.doi.org/10.4134/JKMS.2011.48.3.655.
 9.
BaiNi
Guo and Feng
Qi, A simple proof of logarithmic convexity of extended mean
values, Numer. Algorithms 52 (2009), no. 1,
89–92. MR
2533996 (2010h:26044), 10.1007/s1107500892597
 10.
B.N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), no. 9, 12081216; Available online at http://dx.doi.org/10.1016/
j.jat.2011.04.003.
 11.
BaiNi
Guo and Feng
Qi, Some properties of the psi and polygamma functions, Hacet.
J. Math. Stat. 39 (2010), no. 2, 219–231. MR 2681248
(2011g:33001)
 12.
BaiNi
Guo and Feng
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. MR 2604236
(2011c:33004), 10.4134/BKMS.2010.47.1.103
 13.
BaiNi
Guo, Feng
Qi, and H.
M. Srivastava, Some uniqueness results for the nontrivially
complete monotonicity of a class of functions involving the polygamma and
related functions, Integral Transforms Spec. Funct.
21 (2010), no. 11, 849–858. MR 2739394
(2012c:26013), 10.1080/10652461003748112
 14.
Ji
Chang Kuang, Changyong budengshi, 2nd ed., Hunan Jiaoyu
Chubanshe, Changsha, 1993 (Chinese, with English summary). With a preface
by Shan Zhen Lu. MR 1305610
(95j:26001)
 15.
Feng
Qi, Bounds for the ratio of two gamma functions, J. Inequal.
Appl. , posted on (2010), Art. ID 493058, 84. MR 2611044
(2011d:33004), 10.1155/2010/493058
 16.
Feng
Qi, Pietro
Cerone, Sever
S. Dragomir, and H.
M. Srivastava, Alternative proofs for monotonic and logarithmically
convex properties of oneparameter mean values, Appl. Math. Comput.
208 (2009), no. 1, 129–133. MR
2490776, 10.1016/j.amc.2008.11.023
 17.
F. Qi and B.N. Guo, A short proof of monotonicity of a function involving the psi and exponential functions, Available online at http://arxiv.org/abs/0902.2519.
 18.
Feng
Qi and BaiNi
Guo, Completely monotonic functions involving divided differences
of the di and trigamma functions and some applications, Commun. Pure
Appl. Anal. 8 (2009), no. 6, 1975–1989. MR 2552160
(2010j:33002), 10.3934/cpaa.2009.8.1975
 19.
Feng
Qi and BaiNi
Guo, Necessary and sufficient conditions for functions involving
the tri and tetragamma functions to be completely monotonic, Adv. in
Appl. Math. 44 (2010), no. 1, 71–83. MR 2552656
(2010i:33007), 10.1016/j.aam.2009.03.003
 20.
F. Qi and B.N. Guo, Refinements of lower bounds for polygamma functions, Available online at http://arxiv.org/abs/0903.1966.
 21.
Feng
Qi, Senlin
Guo, and BaiNi
Guo, Complete monotonicity of some functions involving polygamma
functions, J. Comput. Appl. Math. 233 (2010),
no. 9, 2149–2160. MR 2577754
(2010j:33003), 10.1016/j.cam.2009.09.044
 22.
Feng
Qi and QiuMing
Luo, A simple proof of monotonicity for extended mean values,
J. Math. Anal. Appl. 224 (1998), no. 2,
356–359. MR
1637478, 10.1006/jmaa.1998.6003
 23.
Feng
Qi, SenLin
Xu, and Lokenath
Debnath, A new proof of monotonicity for extended mean values,
Int. J. Math. Math. Sci. 22 (1999), no. 2,
417–421. MR 1695308
(2000c:26019), 10.1155/S0161171299224179
 24.
Edgar
M. E. Wermuth, Some elementary properties of infinite
products, Amer. Math. Monthly 99 (1992), no. 6,
530–537. MR 1166002
(93h:40001), 10.2307/2324060
 1.
 H. Alzer, Sharp inequalities for the digamma and polygamma functions, Forum Math. 16 (2004), 181221. MR 2039096 (2005d:33003)
 2.
 H. Alzer, Sharp inequalities for the harmonic numbers, Expo. Math. 24 (2006), no. 4, 385388. MR 2313126 (2007m:11041)
 3.
 N. Batır, On some properties of digamma and polygamma functions, J. Math. Anal. Appl. 328 (2007), no. 1, 452465. 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, DordrechtBostonLondon, 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), 239252. 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, 124134. MR 2331512 (2009e:33004)
 8.
 B.N. Guo and F. Qi, A class of completely monotonic functions involving divided differences of the psi and trigamma functions and some applications, J. Korean Math. Soc. 48 (2011), no. 3, 655667; 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), 8992; Available online at http://dx.doi.org/10.1007/s1107500892597. 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, 12081216; 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, 219231. 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, 103111; 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 nontrivially complete monotonicity of a class of functions involving the polygamma and related functions, Integral Transforms Spec. Funct. 21 (2010), no. 11, 103111; Available online at http://dx.doi.org/10.1080/10652461003748112. MR 2739394
 14.
 J.C. Kuang, Chángyòng Bùděngshì (Applied Inequalities), 3rd ed., Shāndōng Kēxué Jìshù Chūban Shè (Shandong Science and Technology Press), Ji'nan City, Shandong Province, China, 2004 (Chinese). MR 1305610 (95j:26001)
 15.
 F. Qi, Bounds for the ratio of two gamma functions, J. Inequal. Appl. 2010 (2010), Article ID 493058, 84 pages; Available online at http://dx.doi.org/10.1155/2010/493058. MR 2611044 (2011d:33004)
 16.
 F. Qi, P. Cerone, S. S. Dragomir, and H. M. Srivastava, Alternative proofs for monotonic and logarithmically convex properties of oneparameter mean values, Appl. Math. Comput. 208 (2009), no. 1, 129133; Available online at http://dx.doi.org/10.1016/j.amc.2008.11.023. MR 2490776
 17.
 F. Qi and B.N. Guo, A short proof of monotonicity of a function involving the psi and exponential functions, Available online at http://arxiv.org/abs/0902.2519.
 18.
 F. Qi and B.N. Guo, Completely monotonic functions involving divided differences of the di and trigamma functions and some applications, Commun. Pure Appl. Anal. 8 (2009), no. 6, 19751989; Available online at http://dx.doi.org/10.3934/cpaa.2009.8.1975. MR 2552160 (2010j:33002)
 19.
 F. Qi and B.N. Guo, Necessary and sufficient conditions for functions involving the tri and tetragamma functions to be completely monotonic, Adv. Appl. Math. 44 (2010), no. 1, 7183; Available online at http://dx.doi.org/10.1016/j.aam.2009.03.003. MR 2552656 (2010i:33007)
 20.
 F. Qi and B.N. Guo, Refinements of lower bounds for polygamma functions, Available online at http://arxiv.org/abs/0903.1966.
 21.
 F. Qi, S. Guo, and B.N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 21492160; Available online at http://dx.doi.org/10.1016/j.cam.2009.09.044. MR 2577754 (2010j:33003)
 22.
 F. Qi and Q.M. Luo, A simple proof of monotonicity for extended mean values, J. Math. Anal. Appl. 224 (1998), no. 2, 356359. MR 1637478
 23.
 F. Qi, S.L. Xu, and L. Debnath, A new proof of monotonicity for extended mean values, Int. J. Math. Math. Sci. 22 (1999), no. 2, 417421. MR 1695308 (2000c:26019)
 24.
 E. M. E. Wermuth, Some elementary properties of infinite products, Amer. Math. Monthly 99 (1992), no. 6, 530537. MR 1166002 (93h:40001)
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Additional Information
BaiNi 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/S000299392012113875
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
Published electronically:
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.
