{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:07:49Z","timestamp":1760144869240,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T00:00:00Z","timestamp":1716854400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation of the Peoples Republic of China","award":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"],"award-info":[{"award-number":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"]}]},{"name":"Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region","award":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"],"award-info":[{"award-number":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"]}]},{"name":"Natural Science Foundation of Inner Mongolia of the Peoples Republic of China","award":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"],"award-info":[{"award-number":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"]}]},{"name":"Higher School Foundation of Inner Mongolia of the Peoples Republic of China","award":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"],"award-info":[{"award-number":["11561001","NJYT-18-A14","2018MS01026","NJZY2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this research article, we introduce new family RG of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acquire the first five coefficients for the functions belonging to this newly defined family, demonstrating their sharpness. Furthermore, we find the third Hankel determinant for functions in the class RG. Moreover, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the under-considered class RG are estimated.<\/jats:p>","DOI":"10.3390\/axioms13060359","type":"journal-article","created":{"date-parts":[[2024,5,28]],"date-time":"2024-05-28T07:38:39Z","timestamp":1716881919000},"page":"359","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Generalized Bounded Turning Functions Connected with Gregory Coefficients"],"prefix":"10.3390","volume":"13","author":[{"given":"Huo","family":"Tang","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Sciences, Chifeng University, Chifeng 024000, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-6051-4363","authenticated-orcid":false,"given":"Zeeshan","family":"Mujahid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1123-8578","authenticated-orcid":false,"given":"Nazar","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22500, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Mathematics Department, College of Science, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2239-6416","authenticated-orcid":false,"given":"Muhammad","family":"Ghaffar Khan","sequence":"additional","affiliation":[{"name":"Institute of Numerical Sciences, Kohat University of Sciences and Technology, Kohat 26000, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,28]]},"reference":[{"key":"ref_1","first-page":"940","article-title":"\u00dcber di\u00e9 Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. Sitzungsberichte Preuss","volume":"138","author":"Bieberbach","year":"1916","journal-title":"Akad. Der Wiss."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF02392821","article-title":"A proof of the Bieberbach conjecture","volume":"154","year":"1985","journal-title":"Acta Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1090\/S0002-9939-1977-0425097-1","article-title":"New subclasses of the class of close-to-convex functions","volume":"62","author":"Chichra","year":"1977","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_4","first-page":"157","article-title":"A unified treatment of some special classes of univalent functions","volume":"Volume I","author":"Li","year":"1994","journal-title":"Proceeding of the Conference on Complex Analysis (Tianjin, 1992)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2605","DOI":"10.1016\/j.camwa.2011.03.006","article-title":"Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers","volume":"61","author":"Dziok","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1203","DOI":"10.1016\/j.mcm.2012.10.023","article-title":"On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers","volume":"57","author":"Dziok","year":"2013","journal-title":"Math. Comput. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1515\/ms-2017-0289","article-title":"Sharp Coefficient bounds for starlike functions associated with the Bell numbers","volume":"69","author":"Kumar","year":"2019","journal-title":"Math. Slovaca"},{"key":"ref_8","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. Sci. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1007\/s40840-019-00784-y","article-title":"Certain class of starlike functions associated with modified sigmoid function","volume":"43","author":"Goel","year":"2020","journal-title":"Bull. Malays. Sci. Soc."},{"key":"ref_10","first-page":"531","article-title":"On generalized telephone number, their interpretations and matrix generators","volume":"10","year":"2017","journal-title":"Util. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1525","DOI":"10.1007\/s40840-020-01016-4","article-title":"Sharp coefficients bounds for starlike functions associated with generalized telephone numbers","volume":"44","author":"Deniz","year":"2021","journal-title":"Bull. Malays. Sci. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alahmade, A., Mujahid, Z., Tawfiq, F.M.O., Khan, B., Khan, N., and Tchier, F. (2023). Third Hankel Determinant for Subclasses of Analytic and m-Fold Symmetric Functions Involving Cardioid Domain and Sine Function. Symmetry, 15.","DOI":"10.3390\/sym15112039"},{"key":"ref_13","unstructured":"Berezin, I.S., and Zhidkov, N.P. (1965). Computing Methods, Addison-Wesley Publishing Company. Inc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1080\/00029890.1972.11993028","article-title":"Gregory\u2019s method for numerical integration","volume":"79","author":"Phillips","year":"1972","journal-title":"Amer. Math. Mon."},{"key":"ref_15","first-page":"337","article-title":"On the second Hankel determinant of areally mean p-valent functions","volume":"223","author":"Noonan","year":"1976","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1090\/S0002-9939-1969-0232926-9","article-title":"A coefficient inequality for certain classes of analytic functions","volume":"20","author":"Keogh","year":"1969","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_17","first-page":"619","article-title":"Hankel determinant for starlike and convex functions","volume":"1","author":"Janteng","year":"2007","journal-title":"Int. J. Math. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1186\/1029-242X-2013-281","article-title":"Bounds for the second Hankel determinant of certain univalent functions","volume":"2013","author":"Lee","year":"2013","journal-title":"J. Inequal. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/s00009-016-0829-y","article-title":"Third Hankel determinants for subclasses of univalent functions","volume":"14","author":"Zaprawa","year":"2017","journal-title":"Mediterr. J. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"767","DOI":"10.1007\/s40840-018-0683-0","article-title":"The bound of the Hankel determinant of the third kind for starlike functions","volume":"42","author":"Kwon","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Shi, L., Srivastava, H.M., Rafiq, A., Arif, M., and Ihsan, M. (2022). Results on Hankel determinants for the inverse of certain analytic functions subordinated to the exponential function. Mathematics, 10.","DOI":"10.3390\/math10193429"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1007\/s13398-022-01286-6","article-title":"The second Hankel determinant for subclasses of bi-univalent functions associated with a nephroid domain","volume":"116","author":"Srivastava","year":"2022","journal-title":"Real Acad. Cienc. Exactas F\u0131s. Natur. Ser. A Mat. (RACSAM)"},{"key":"ref_23","first-page":"2815","article-title":"Hermitian-Toeplitz and Hankel determinants for starlike functions associated with a rational function","volume":"23","author":"Srivastava","year":"2022","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"340","DOI":"10.3934\/math.2023016","article-title":"The Fekete-Szego functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator","volume":"8","author":"Srivastava","year":"2023","journal-title":"AIMS Math."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Avkhadiev, F.G., and Wirths, K.J. (2009). Schwarz-Pick Type Inequalities, Springer Science & Business Media.","DOI":"10.1007\/978-3-0346-0000-2"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1090\/S0002-9947-1985-0792819-9","article-title":"The De Branges theorem on univalent functions","volume":"290","author":"FitzGerald","year":"1985","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_27","first-page":"21","article-title":"A theorem of De Branges on univalent functions","volume":"13","author":"FitzGerald","year":"1987","journal-title":"Serdica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1007\/s11006-005-0149-1","article-title":"On Brennan\u2019s conjecture for a special class of functions","volume":"78","author":"Kayumov","year":"2005","journal-title":"Math. Notes"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1512\/iumj.1988.37.37035","article-title":"Inequalities for logarithmic coefficients of univalent functions and their derivatives","volume":"37","author":"Andreev","year":"1988","journal-title":"Indiana Univ. Math. J."},{"key":"ref_30","first-page":"6690027","article-title":"Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions","volume":"2021","author":"Alimohammadi","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_31","first-page":"5889","article-title":"On the logarithmic coefficients of Bazilevic functions","volume":"217","author":"Deng","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1090\/S0002-9939-07-08660-1","article-title":"A sharp inequality for the logarithmic coefficients of univalent functions","volume":"135","author":"Roth","year":"2017","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_33","first-page":"445","article-title":"The logarithmic coefficients of close-to-convex functions","volume":"3","author":"Ye","year":"2008","journal-title":"Bull. Inst. Math. Acad. Sin. (New Ser.)"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s00605-017-1024-3","article-title":"Logarithmic coefficients and a coefficient conjecture for univalent functions","volume":"185","author":"Ponnusamy","year":"2018","journal-title":"Monatshefte Fur Math."},{"key":"ref_35","first-page":"337","article-title":"Logarithmic coefficients of univalent functions","volume":"25","author":"Girela","year":"2000","journal-title":"Ann. Acad. Sci. Fenn. Ser. Math."},{"key":"ref_36","unstructured":"Duren, P.L. (2001). Grundlehren der Mathematischen Wissenschaften, Band 259, Springer."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF01448091","article-title":"Untersuchungen \u00dcber schlichte konforme Abbildungen des Einheitskreises I","volume":"89","author":"Lowner","year":"1923","journal-title":"Math. Ann."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF02772989","article-title":"Estimates for coefficients of univalent functions from integral means and Grunsky inequalities","volume":"87","author":"Yang","year":"1994","journal-title":"Isr. J. Math."},{"key":"ref_39","first-page":"251","article-title":"Coefficient bounds for the inverse of a function with derivatives in P","volume":"87","author":"Libera","year":"1983","journal-title":"Proc. Am. Soc."},{"key":"ref_40","unstructured":"Pommerenke, C. (1975). Studia Mathematica Mathematische Lehrbucher, Vandenhoeck and Ruprecht."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF03014795","article-title":"Uber den Variabilitatsbereich der Fourier\u2019schen Konstantenvon positiven harmonischen Funktionen","volume":"32","author":"Caratheodory","year":"1911","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1215\/21562261-2017-0015","article-title":"Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions","volume":"58","author":"Ohno","year":"2018","journal-title":"Kyoto J. Math."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.crma.2015.03.003","article-title":"Bound for the fifth coefficient of certain starlike functions","volume":"353","author":"Ravichandran","year":"2015","journal-title":"Comptes Rendus Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/359\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:49:34Z","timestamp":1760107774000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/359"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,28]]},"references-count":43,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060359"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060359","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,5,28]]}}}