{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T18:14:56Z","timestamp":1769192096943,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T00:00:00Z","timestamp":1723075200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we scrutinize a collection of meromorphic functions known as normal families, prove the theorem that normal families share a holomorphic function, and present several illustrative counterexamples.<\/jats:p>","DOI":"10.3390\/axioms13080538","type":"journal-article","created":{"date-parts":[[2024,8,8]],"date-time":"2024-08-08T12:13:14Z","timestamp":1723119194000},"page":"538","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Normality Criterion for Sharing a Holomorphic Function"],"prefix":"10.3390","volume":"13","author":[{"given":"Sheng","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing University, Chongqing 400031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaojun","family":"Huang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Chongqing University, Chongqing 400031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,8]]},"reference":[{"key":"ref_1","unstructured":"Yang, L. (1993). Value Distribution Theory, Springer."},{"key":"ref_2","unstructured":"Hayman, W.K. (1964). Meromorphic Functions, Clarendon Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"487","DOI":"10.24033\/asens.652","article-title":"Sur les families de fonctions analytiques qui admettent des valeurs exceptionnelles dansun domaine","volume":"29","author":"Montel","year":"1912","journal-title":"Ann. \u00c9cole. Norm. Sup."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Schiff, J.L. (1993). Normal Families, Springer.","DOI":"10.1007\/978-1-4612-0907-2"},{"key":"ref_5","unstructured":"Carath\u00e9odory, C. (1960). Theory of Functions of a Complex Variable, Chelsea Publ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1007\/s40315-014-0054-2","article-title":"Normal families: A geometric perspective","volume":"14","author":"Beardon","year":"2014","journal-title":"Comput. Methods Funct. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s40315-021-00363-y","article-title":"An extension to Montel\u2019s three omitted value theorem","volume":"21","author":"Beardon","year":"2021","journal-title":"Comput. Methods Funct. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/BF02392392","article-title":"Boundary behaviour and normal meromorphic functions","volume":"97","author":"Lehto","year":"1957","journal-title":"Acta Mathematica"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1007\/BF01199014","article-title":"Sharing values and normality","volume":"59","author":"Schwick","year":"1992","journal-title":"Arch. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/BF02384496","article-title":"Normality and shared values","volume":"38","author":"Pang","year":"2000","journal-title":"Ark. Mat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/BF03320991","article-title":"Normal families and shared values of meromorphic functions II","volume":"1","author":"Fang","year":"2001","journal-title":"Comput. Methods Funct. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1007\/BF03321856","article-title":"Normal families and shared values of meromorphic functions III","volume":"2","author":"Fang","year":"2002","journal-title":"Comput. Methods Funct. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1080\/17476930802124666","article-title":"Some normality criteria of meromorphic functions","volume":"8","author":"Zhang","year":"2008","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1007\/s40840-014-0089-6","article-title":"Normality criteria of meromorphic functions sharing a holomorphic function","volume":"38","author":"Meng","year":"2015","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1007\/s40840-017-0515-7","article-title":"Normal families and shared functions concerning hayman\u2019s question","volume":"42","author":"Deng","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_16","unstructured":"Singh, C.K., and Bharti, N. (2023). Value distribution of certain differential polynomials leading to some normality criteria. arXiv."},{"key":"ref_17","unstructured":"Zhang, X., and Xu, J. (2014). Further results on normal families of meromorphic functions concerning shared values. arXiv."},{"key":"ref_18","first-page":"1507","article-title":"Normal families of meromorphic functions concerning shared fixed-points","volume":"6","author":"Liu","year":"2011","journal-title":"Int. Math. Forum"},{"key":"ref_19","first-page":"409","article-title":"Shared values and normal families","volume":"50","author":"Liu","year":"2007","journal-title":"Acta Math. Sin. Chin. Ser."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"287","DOI":"10.2298\/FIL1602287X","article-title":"Normal functions and shared sets","volume":"30","author":"Xu","year":"2016","journal-title":"Filomat"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"227","DOI":"10.2306\/scienceasia1513-1874.2020.027","article-title":"Normal families of meromorphic functions which share a set","volume":"46","author":"Cai","year":"2020","journal-title":"Sci. Asia"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s10114-012-0600-7","article-title":"A normal criterion about two families of meromorphic functions concerning shared values","volume":"29","author":"Liu","year":"2013","journal-title":"Acta Math. Sin. (Engl. Ser.)"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00605-021-01559-z","article-title":"Normality relationships between two families concerning shared values","volume":"196","author":"Sun","year":"2021","journal-title":"Mon. Heft. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1007\/s40315-021-00376-7","article-title":"Normality concerning shared values between two families","volume":"21","author":"Chang","year":"2021","journal-title":"Comput. Methods Funct. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"355","DOI":"10.4171\/rmi\/176","article-title":"On the singularities of the inrerse to a meromorphic function of finite order","volume":"11","author":"Bergweiler","year":"1995","journal-title":"Rev. Mate. Iberoamericana"},{"key":"ref_26","first-page":"8","article-title":"On the kth derivatives of meromorphic functions and rational functions","volume":"4","author":"Yang","year":"2014","journal-title":"J. East China Norm. Univ. (Nat. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/BF02563879","article-title":"Picard values and normal families of meromorphic func- tions with multiple zeros","volume":"14","author":"Wang","year":"1998","journal-title":"Acta Math. Sin. Chin. Ser."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1112\/S002460939900644X","article-title":"Normality families and shared values","volume":"32","author":"Pang","year":"2000","journal-title":"Bull. London Math. Soc."},{"key":"ref_29","first-page":"1","article-title":"Uniqueness of L functions with special class of meromorphic functions via sharing of pre-image sets","volume":"2","author":"Kundu","year":"2023","journal-title":"Rend. Del Circ. Mat. Palermo Ser."},{"key":"ref_30","first-page":"1","article-title":"Investigations on uniqueness of meromorphic functions in the k-punctured complex plane","volume":"2","author":"Maity","year":"2024","journal-title":"Rend. Del Circ. Mat. Palermo Ser."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"20230103","DOI":"10.1515\/dema-2023-0103","article-title":"On the equation fn + (f\u2033)m\u22611","volume":"56","author":"Dang","year":"2023","journal-title":"Demonstr. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20230142","DOI":"10.1515\/dema-2023-0142","article-title":"Some properties of a class of holomorphic functions associated with tangent function","volume":"57","author":"Khan","year":"2024","journal-title":"Demonstr. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/538\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:32:19Z","timestamp":1760110339000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/538"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,8]]},"references-count":32,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080538"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080538","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,8]]}}}