{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:12:14Z","timestamp":1760227934060,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,2]],"date-time":"2022-05-02T00:00:00Z","timestamp":1651449600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In this paper, we present the following rational approximations for Bateman\u2019s G-function G(w)=1w+2w2+\u2211j=1n4\u03b1jw2\u22122j\u22121+O1w2n+2, where \u03b11=14, and \u03b1j=(1\u221222j+2)B2j+2j+1+\u2211\u03bd=1j\u22121(1\u221222j\u22122\u03bd+2)B2j\u22122\u03bd+2\u03b1\u03bdj\u2212\u03bd+1,j&gt;1. As a consequence, we introduced some new bounds of G(w) and a completely monotonic function involving it.<\/jats:p>","DOI":"10.3390\/sym14050929","type":"journal-article","created":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T08:21:25Z","timestamp":1651652485000},"page":"929","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Rational Approximations and Bounds for Bateman\u2019s G-Function"],"prefix":"10.3390","volume":"14","author":[{"given":"Omelsaad","family":"Ahfaf","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5918-1913","authenticated-orcid":false,"given":"Mansour","family":"Mahmoud","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed","family":"Talat","sequence":"additional","affiliation":[{"name":"Mathematics and Computer Sciences Department, Faculty of Science, Port Said University, Port Said 42526, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,2]]},"reference":[{"key":"ref_1","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1981). Higher Transcendental Functions, McGraw-Hill Inc.. California Institute of Technology-Bateman Manuscript Project, 1953\u20131955."},{"key":"ref_2","unstructured":"Andrews, G.E., Askey, R.A., and Roy, R. (1999). Special Functions, Cambridge University Press. Encyclopedia of Mathematics and Its Applications 71."},{"key":"ref_3","first-page":"118","article-title":"Some best approximation formulas and inequalities for the Bateman\u2019s G-function","volume":"27","author":"Hegazi","year":"2019","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kiryakova, V. (2021). A guide to special functions in fractional calculus. Mathematics, 9.","DOI":"10.3390\/math9010106"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1515\/gmj-2016-0037","article-title":"Bounds for Bateman\u2019s G-function and its applications","volume":"23","author":"Mahmoud","year":"2016","journal-title":"Georgian Math. J."},{"key":"ref_6","first-page":"672","article-title":"On some inequalities of the Bateman\u2019s G-function","volume":"22","author":"Mahmoud","year":"2017","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_7","first-page":"1165","article-title":"Some approximations of the Bateman\u2019s G-function","volume":"23","author":"Mahmoud","year":"2017","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_8","first-page":"23","article-title":"Generalized Bateman\u2019s G-function and its bounds","volume":"24","author":"Mahmoud","year":"2018","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_9","first-page":"970","article-title":"Completely monotonic functions involving Bateman\u2019s G-function","volume":"29","author":"Mahmoud","year":"2021","journal-title":"J. Comput. Anal. Appl."},{"key":"ref_10","first-page":"41","article-title":"A sharp inequality involving the psi function","volume":"22","author":"Mortici","year":"2010","journal-title":"Acta Univ. Apulensis"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1090\/S0025-5718-04-01675-8","article-title":"Some properties of the gamma and psi functions with applications","volume":"74","author":"Qiu","year":"2004","journal-title":"Math. Comp."},{"key":"ref_12","first-page":"445","article-title":"Some classes of completely monotonic functions","volume":"27","author":"Alzer","year":"2002","journal-title":"Ann. Acad. Sci. Fenn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1006\/jmaa.1996.0443","article-title":"Completely monotonic and related functions","volume":"204","author":"Haeringen","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1186\/s13662-020-03051-8","article-title":"A necessary and sufficient condition for sequences to be minimal completely monotonic","volume":"2020","author":"Wang","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_15","unstructured":"Widder, D.V. (1946). The Laplace Transform, Princeton University Press."},{"key":"ref_16","unstructured":"Baker, G.A., and Graves\u2013Morris, P.R. (1996). Pad\u00e9 Approximants, Cambridge University Press. [2nd ed.]."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Brezinski, C. (2002). Computational Aspects of Linear Control, Kluwer.","DOI":"10.1007\/978-1-4613-0261-2"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.cam.2014.07.007","article-title":"New representations of Pad\u00e9, Pad\u00e9-type, and partial Pad\u00e9 approximants","volume":"284","author":"Brezinski","year":"2015","journal-title":"J. Comput. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2149","DOI":"10.1016\/j.cam.2009.09.044","article-title":"Completely monotonicity of some functions involving polygamma functions","volume":"233","author":"Qi","year":"2010","journal-title":"J. Comput. Appl. Math."},{"key":"ref_20","unstructured":"Apostol, T.M. (1967). Calculus, Volume I, One-Variable Calculus, with an Introduction to Linear Algebra, John Wiley & Sons. [2nd ed.]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/929\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:05:25Z","timestamp":1760137525000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/929"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,2]]},"references-count":20,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050929"],"URL":"https:\/\/doi.org\/10.3390\/sym14050929","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,5,2]]}}}