{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:39:54Z","timestamp":1760146794448,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,15]],"date-time":"2024-12-15T00:00:00Z","timestamp":1734220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100018693","name":"Horizon Europe\u2019s project VIBraTE","doi-asserted-by":"publisher","award":["101086815"],"award-info":[{"award-number":["101086815"]}],"id":[{"id":"10.13039\/100018693","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work explores the geometric properties of the Turanian of the modified Bessel function of the first kind (TMBF). Using the properties of the digamma function, we establish conditions under which the normalized TMBF satisfies starlikeness, convexity, k-starlikeness, k-uniform convexity, pre-starlikeness, lemniscate starlikeness, and convexity, and under which exponential starlikeness and convexity are obtained. By combining methods from complex analysis, inequalities, and functional analysis, the article advances the theory of Bessel functions and hypergeometric functions. The established results could be useful in approximation theory and bounding the behavior of functions.<\/jats:p>","DOI":"10.3390\/axioms13120874","type":"journal-article","created":{"date-parts":[[2024,12,17]],"date-time":"2024-12-17T06:26:12Z","timestamp":1734416772000},"page":"874","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Geometric Nature of the Tur\u00e1nian of Modified Bessel Function of the First Kind"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2345-4705","authenticated-orcid":false,"given":"Samanway","family":"Sarkar","sequence":"first","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur 831014, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8694-0535","authenticated-orcid":false,"given":"Dimiter","family":"Prodanov","sequence":"additional","affiliation":[{"name":"Laboratory of Neurotechnology (PAML-LN), Institute for Information and Communication Technologies (IICT), Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1544-343X","authenticated-orcid":false,"given":"Anish","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Bhilai, Bhilai 492015, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2270-8109","authenticated-orcid":false,"given":"Sourav","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur 831014, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1080\/10652460903516736","article-title":"Starlikeness and convexity of generalized Bessel functions","volume":"21","author":"Baricz","year":"2010","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2133","DOI":"10.1016\/j.aml.2011.06.014","article-title":"Certain geometric properties of normalized Bessel functions","volume":"24","author":"Prajapat","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_3","first-page":"179","article-title":"Geometric Properties of Generalized Bessel Functions","volume":"35","author":"Mondal","year":"2012","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s40840-015-0180-7","article-title":"Close-to-convexity of some special functions and their derivatives","volume":"39","author":"Baricz","year":"2016","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"211","DOI":"10.3906\/mat-1610-41","article-title":"Bounds for radii of starlikeness and convexity of some special functions","volume":"42","author":"Baricz","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sarkar, S., Das, S., and Mondal, S.R. (2024). Geometric Properties of Normalized Galu\u00e9 Type Struve Function. Symmetry, 16.","DOI":"10.3390\/sym16020211"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s11139-018-0105-9","article-title":"Geometric and monotonic properties of hyper-Bessel functions","volume":"51","author":"Baricz","year":"2020","journal-title":"Ramanujan J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1007\/s00025-017-0668-6","article-title":"Bounds for radii of starlikeness of some q-Bessel functions","volume":"72","author":"Baricz","year":"2017","journal-title":"Results Math."},{"key":"ref_9","first-page":"355","article-title":"Bounds for radii of convexity of some q-Bessel functions","volume":"57","author":"Aktas","year":"2020","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1186\/s13660-022-02891-0","article-title":"Normalized generalized Bessel function and its geometric properties","volume":"2022","author":"Zayed","year":"2022","journal-title":"J. Inequalities Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2715","DOI":"10.1007\/s40840-021-01082-2","article-title":"Geometric properties of the products of modified Bessel functions of the first kind","volume":"44","author":"Mehrez","year":"2021","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_12","unstructured":"Arfken, G., and Weber, H.J. (1985). Mathematical Methods for Physicists, Academic Press."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1090\/S0002-9904-1948-09017-6","article-title":"On an inequality of P. Tur\u00e1n concerning Legendre polynomials","volume":"54","year":"1948","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"991","DOI":"10.7153\/mia-2017-20-63","article-title":"Properties of the Tur\u00e1nian of modified Bessel functions","volume":"20","author":"Mezo","year":"2017","journal-title":"Math. Inequalities Appl."},{"key":"ref_15","unstructured":"Duren, P.L. (1983). Univalent functions. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/S0377-0427(99)00018-7","article-title":"Conic regions and k-uniform convexity","volume":"105","author":"Kanas","year":"1999","journal-title":"J. Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"87","DOI":"10.4064\/ap-56-1-87-92","article-title":"On uniformly convex functions","volume":"56","author":"Goodman","year":"1991","journal-title":"Ann. Polon. Math."},{"key":"ref_18","first-page":"647","article-title":"Conic domains and starlike functions","volume":"45","author":"Kanas","year":"2000","journal-title":"Rev. Roum. Math. Pures Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1090\/S0002-9939-1993-1128729-7","article-title":"Uniformly convex functions and a corresponding class of starlike functions","volume":"118","author":"Ronning","year":"1993","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_20","first-page":"101","article-title":"Radius of convexity of some subclasses of strongly starlike functions","volume":"19","author":"Stankiewicz","year":"1996","journal-title":"Zesz. Nauk. Politech. Rzesz. Mat"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/s40840-014-0026-8","article-title":"On a subclass of strongly starlike functions associated with exponential function","volume":"38","author":"Mendiratta","year":"2015","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1090\/S0002-9939-1963-0148891-3","article-title":"The radius of univalence of certain analytic functions","volume":"14","author":"MacGregor","year":"1963","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1090\/S0002-9939-1964-0158985-5","article-title":"A class of univalent functions","volume":"15","author":"MacGregor","year":"1964","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s13324-022-00678-6","article-title":"Logarithmically completely monotonic functions related to the q-gamma function and its applications","volume":"12","author":"Mehrez","year":"2022","journal-title":"Anal. Math. Phys."},{"key":"ref_25","first-page":"168","article-title":"Convolutions in geometric function theory","volume":"Volume 83","author":"Ruscheweyh","year":"1982","journal-title":"S\u00e9minaire de Math\u00e9matiques Sup\u00e9rieures [Seminar on Higher Mathematics]"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/BF02803399","article-title":"Convolution multipliers and starlike functions","volume":"41","author":"Silverman","year":"1982","journal-title":"J. D\u2019Anal. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/874\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:52:33Z","timestamp":1760115153000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/12\/874"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,15]]},"references-count":26,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["axioms13120874"],"URL":"https:\/\/doi.org\/10.3390\/axioms13120874","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,15]]}}}