{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:48:23Z","timestamp":1760230103707,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,11]],"date-time":"2022-07-11T00:00:00Z","timestamp":1657497600000},"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 the paper, we prove the completely monotonic (CM) property of some functions involving the function \u0394(l)=\u03c8\u2033(l)+\u03c8\u2032(l)2 and hence we deduce a new double inequality for it. Additionally, we study the CM degree of some functions involving the function \u03c8\u2032(l). Our new bounds takes priority over some of the recently published results.<\/jats:p>","DOI":"10.3390\/sym14071420","type":"journal-article","created":{"date-parts":[[2022,7,12]],"date-time":"2022-07-12T03:50:36Z","timestamp":1657597836000},"page":"1420","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Bounds and Completely Monotonicity of Some Functions Involving the Functions \u03c8\u2032(l) and \u03c8\u2033(l)"],"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"}]},{"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"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,11]]},"reference":[{"key":"ref_1","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_2","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":"2005","journal-title":"Math. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3011","DOI":"10.1088\/0305-4470\/39\/12\/010","article-title":"Summations for certain series containing the digamma function","volume":"39","author":"Miller","year":"2006","journal-title":"J. Phys. A Math. 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