Geodesic
A solution to Qi's eighth open problem on complete monotonicity
Problemy analiza, Tome 10 (2021) no. 3, pp. 108-112.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, the complete monotonicity of $\frac{1}{\arctan x}$ is proved. This problem was posted by F. Qi and R. P. Agarwal as the eighth open problem of collection of eight open problems.
Keywords: logarithmically completely monotonic functions, polygamma function, inequalities, Stieltjes function. 
@article{PA_2021_10_3_a7,
     author = {A. Venkata Lakshmi},
     title = {A solution to {Qi's} eighth open problem on complete monotonicity},
     journal = {Problemy analiza},
     pages = {108--112},
     publisher = {mathdoc},
     volume = {10},
     number = {3},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PA_2021_10_3_a7/}
}
TY  - JOUR
AU  - A. Venkata Lakshmi
TI  - A solution to Qi's eighth open problem on complete monotonicity
JO  - Problemy analiza
PY  - 2021
SP  - 108
EP  - 112
VL  - 10
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PA_2021_10_3_a7/
LA  - en
ID  - PA_2021_10_3_a7
ER  - 
%0 Journal Article
%A A. Venkata Lakshmi
%T A solution to Qi's eighth open problem on complete monotonicity
%J Problemy analiza
%D 2021
%P 108-112
%V 10
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PA_2021_10_3_a7/
%G en
%F PA_2021_10_3_a7
A. Venkata Lakshmi. A solution to Qi's eighth open problem on complete monotonicity. Problemy analiza, Tome 10 (2021) no. 3, pp. 108-112. http://geodesic.mathdoc.fr/item/PA_2021_10_3_a7/

[1] B. Ravi, A. Venkata Lakshmi, “Completely monotonicity of class functions involving the polygamma and related functions”, Asian-European Journal of Mathematics, 14:04 (2021), 2150064 | DOI | MR | Zbl

[2] D. V. Widder, “The Stieltjes transform”, Trans. Amer. Math. Soc., 43 (1939), 7–60 | DOI | MR

[3] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1946 | MR

[4] Feng Qi, Ravi P. Agarwal, “On complete monotonicity for several classes of functions related to ratios of gamma functions”, Journal of Inequalities and Applications, 1 (2019), 1–42 | DOI | MR

[5] Kunle Adegoke, Olawanle Layeni, “The Higher Derivatives Of The Inverse Tangent Function and Rapidly Convergent BBP-Type Formulas For Pi”, App. Math. E-Notes, 10 (2010), 70–75 | MR | Zbl

[6] K. Splindler, “A short proof of the formula of Faá di Bruno”, Elem. Math., 60 (2005), 33–35 | DOI | MR

[7] R. L. Schilling, R. Song, Z. Vondraček, Bernstein Functions. Theory and Applications, 2nd edition, De Gruyter, Berlin, 2012 | MR | Zbl

[8] W. P. Johnson, The curious history of Faá di Bruno's formula, The AMS. Math. Monthly, 2020 | DOI | MR

[9] H. Alzer, “On some inequalities for the gamma and psi functions”, Math. Comput., 66 (1997), 373–389 | DOI | MR | Zbl

[10] H. Alzer, G. Jameson, “A harmonic inequality for the digmma function and related results”, Rend. Sem. Mat. Univ. Padova, 137 (2017), 203–209 | DOI | MR | Zbl

[11] A. Laforgia, P. Natalini, “Exponential, gamma and polygamma functions: simple proofs of classical and new inequalities”, J. Math. Anal. Appl., 407 (2013), 495–504 | DOI | MR | Zbl

[12] W. E. Clark, Mourad. E. H. Ismail, “Inequalities involving gamma and psi functions”, Anal. Appl., 1:1 (2003), 129–140 | DOI | MR | Zbl