Mean Inequalities for Derivatives of the Generalised Exponential Integral Function
DOI:
https://doi.org/10.26713/jims.v17i3.3256Abstract
In this paper, among other things, we establish arithmetic, geometric and harmonic mean inequalities for derivatives of the generalised exponential integral function. The methods of proof rely heavily on monotonicity properties of certain functions associated with the generalised exponential integral function. The results obtained generalise some existing results in the literature.
Downloads
References
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, Dover Publications, Inc., New York, (1965).
D. A. Barry, J.-Y. Parlange and L. Li, Approximation for the exponential integral (Theis well function), Journal of Hydrology 227(1-4) (2000), 287 – 291, DOI: 10.1016/S0022-1694(99)00184-5.
P. K. Bhandari and S. K. Bissu, On some inequalities involving Turan-type inequalities, Cogent Mathematics 3(1) (2016), Article: 1130678, DOI: 10.1080/23311835.2015.1130678.
M. Bouali, A harmonic mean inequality for the q-gamma and q-digamma functions, Filomat 35(12) (2021), 4105 – 4119, DOI: 10.2298/FIL2112105B.
C. Chiccoli, S. Lorenzutta and G. Maino, Recent results for generalized exponential integrals, Computers & Mathematics with Applications 19(5) (1990), 21 – 29, DOI: 10.1016/0898-1221(90)90098-5.
C. Chiccoli, S. Lorenzutta and G. Maino, Concerning some integrals of the generalized exponential-integral function, Computers & Mathematics with Applications 23(11) (1992), 13 – 21, DOI: 10.1016/0898-1221(92)90065-P.
G. J. O. Jameson, The incomplete gamma functions, The Mathematical Gazette 100(548) (2016), 298 – 306, DOI: 10.1017/mag.2016.67.
W.-H. Li and F. Qi, Harmonic mean inequalities for generalized hyperbolic functions, Montes Taurus Journal of Pure and Applied Mathematics 6(3) (2024), 199 – 207, URL: https://mtjpamjournal.com/wp-content/uploads/2024/06/MTJPAM-D-23-00053.pdf.
S.-D. Lin, Y.-S. Chao and H. M. Srivastava, Some expansions of the exponential integral in series of the incomplete Gamma function, Applied Mathematics Letters 18(5) (2005), 513 – 520, DOI: 10.1016/j.aml.2004.03.016.
L. Matejícka, Proof of a conjecture on Nielsen’s ˇ β-function, Problemy Analiza-Issues of Analysis 8(26)(3) (2019), 105 – 111, DOI: 10.15393/j3.art.2019.6810.
K. Nantomah, A harmonic mean inequality for the exponential integral function, International Journal of Applied Mathematics 34(4) (2021), 647 – 652, DOI: 10.12732/ijam.v34i4.4.
K. Nantomah, A harmonic mean inequality concerning the generalized exponential integral function, Advances in Mathematics: Scientific Journal 10(9) (2021), 3227 – 3231, DOI: 10.37418/amsj.10.9.11.
F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark (editors), NIST Handbook of Mathematical Functions, Cambridge University Press, London, (2010).
I. Pinelis, L’hospital type rules for monotonicity, with applications, Journal of Inequalities in Pure and Applied Mathematics 3(1) (2002), Article 5, URL: https://www.emis.de/journals/JIPAM/images/010_01_JIPAM/010_01.pdf.
F. E. Prym, Zur Theorie der Gamma funktion, Journal für die reine und angewandte Mathematik 82(1877), 165 – 172.
A. Salem, A q-analogue of the exponential integral, Afrika Matematika 24 (2013), 117 – 125, DOI: 10.1007/s13370-011-0046-6.
B. Sroysang, On the n-th derivative of the exponential integral functions, Communications in Mathematics and Applications 4(2) (2013), 141 – 144, DOI: 10.26713/cma.v4i2.170.
W. T. Sulaiman, Turan inequalities for the exponential integral functions, Communications in Optimization Theory 1 (2012), 35 – 41, https://cot.mathres.org/issues/COT201203.pdf.
E. Yildirim, Monotonicity properties on k-digamma function and its related inequalities, Journal of Mathematical Inequalities 14(1) (2020), 161 – 173, DOI: 10.7153/jmi-2020-14-12.
L. Yin, L.-G. Huang, X.-L. Lin and Y.-L. Wang, Monotonicity, concavity, and inequalities related to the generalized digamma function, Advances in Difference Equations 2018 (2018), Article number: 246, DOI: 10.1186/s13662-018-1695-7.
T. Zenku, B. Jolevska-Tuneska and N. Tuneski, Results on the exponential integral, Sarajevo Journal of Mathematics 13(25)(1) (2017), 71 – 80, DOI: 10.5644/sjm.13.1.05.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
