{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,22]],"date-time":"2026-06-22T19:35:14Z","timestamp":1782156914385,"version":"3.54.5"},"reference-count":38,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,9]],"date-time":"2022-11-09T00:00:00Z","timestamp":1667952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004686","name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","doi-asserted-by":"publisher","award":["1621"],"award-info":[{"award-number":["1621"]}],"id":[{"id":"10.13039\/501100004686","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The primary objective of this study is to establish necessary conditions so that the normalized Fox\u2013Wright functions possess certain geometric properties, such as convexity and pre-starlikeness. In addition, we present a linear operator associated with the Fox\u2013Wright functions and discuss its k-uniform convexity and k-uniform starlikeness. Furthermore, some sufficient conditions were obtained so that this function belongs to the Hardy spaces. The results of this work are presumably new and illustrated by several consequences, remarks, and examples.<\/jats:p>","DOI":"10.3390\/axioms11110629","type":"journal-article","created":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T02:07:48Z","timestamp":1668046068000},"page":"629","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Certain Geometric Properties of the Fox\u2013Wright Functions"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1544-343X","authenticated-orcid":false,"given":"Anish","family":"Kumar","sequence":"first","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Jamshedpur, Jamshedpur 831014, Jharkhand, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4540-1601","authenticated-orcid":false,"given":"Saiful R.","family":"Mondal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, King Faisal University, Al Hasa 31982, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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, Jharkhand, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,9]]},"reference":[{"key":"ref_1","first-page":"167","article-title":"On certain geometric properties of normalized Mittag-Leffler functions","volume":"81","author":"Noreen","year":"2019","journal-title":"UPB Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Noreen, S., Raza, M., Liu, J.L., and Arif, M. (2019). Geometric properties of normalized Mittag-Leffler functions. Symmetry, 11.","DOI":"10.3390\/sym11010045"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13660-019-2044-4","article-title":"Certain geometric properties of Mittag-Leffler functions","volume":"2019","author":"Noreen","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6475","DOI":"10.2298\/FIL1818475P","article-title":"Radius of starlikeness and Hardy space of Mittag-Leffler functions","volume":"32","author":"Prajapat","year":"2018","journal-title":"Filomat"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Mondal, S.R. (2022). Subordination Involving Regular Coulomb Wave Functions. Symmetry, 14.","DOI":"10.3390\/sym14051007"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Mondal, S.R. (2022). On Lemniscate Starlikeness of the Solution of General Differential Equations. Mathematics, 10.","DOI":"10.3390\/math10183254"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1007\/s43037-020-00059-w","article-title":"Some geometric properties of a class of functions related to the Fox-Wright functions","volume":"14","author":"Mehrez","year":"2020","journal-title":"Banach J. Math. Anal."},{"key":"ref_8","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_9","doi-asserted-by":"crossref","unstructured":"Alarifi, N.M., and Mondal, S.R. (2022). On Geometric Properties of Bessel\u2013Struve Kernel Functions in Unit Disc. Mathematics, 10.","DOI":"10.3390\/math10142516"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1080\/17476933.2015.1079628","article-title":"Certain geometric properties of the Mittag-Leffler functions","volume":"61","author":"Bansal","year":"2016","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1112\/plms\/s2-27.1.389","article-title":"The asymptotic expansion of generalized hypergeometric functions","volume":"27","author":"Fox","year":"1928","journal-title":"Proc. Lond. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s13324-017-0198-0","article-title":"Inequalities for q-gamma function ratios","volume":"9","author":"Das","year":"2019","journal-title":"Anal. Math. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Andrews, G.E., Askey, R., and Roy, R. (1999). Special Functions, Encyclopedia of Mathematics and its Applications; Cambridge University Press.","DOI":"10.1017\/CBO9781107325937"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF02403202","article-title":"\u00dcber den Fundamentalsatz in der Teorie der Funktionen Ea(x)","volume":"29","author":"Wiman","year":"1905","journal-title":"Acta Math."},{"key":"ref_15","unstructured":"Duren, P.L. (1983). Univalent Functions, Springer. Grundlehren der Mathematischen Wissenschaften."},{"key":"ref_16","unstructured":"Pohlen, T. (2009). The Hadamard Product and Universal Power Series. [Ph.D. Thesis, Universit\u00e4t Trier]."},{"key":"ref_17","unstructured":"Ruscheweyh, S. (1982). Convolutions in Geometric Function Theory, Presses de l\u2019Universit\u00e9 de Montr\u00e9al. S\u00e9minaire de Math\u00e9matiques Sup\u00e9rieures."},{"key":"ref_18","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. Analyse Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"374","DOI":"10.2307\/1968451","article-title":"On the theory of univalent functions","volume":"37","author":"Robertson","year":"1936","journal-title":"Ann. Math. Ser. 2"},{"key":"ref_20","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_21","first-page":"647","article-title":"Conic regions and k-starlike functions","volume":"45","author":"Kanas","year":"2000","journal-title":"Rev. Roumaine Math. Pures Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1080\/10652460008819249","article-title":"Linear operators associated with k-uniformly convex functions","volume":"9","author":"Kanas","year":"2000","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_23","unstructured":"Duren, P.L. (1970). Theory of Hp Space, Academic Press."},{"key":"ref_24","first-page":"1","article-title":"On a one-parameter additive family of operators defined on analytic functions regular in the unit disk","volume":"22","author":"Komatu","year":"1979","journal-title":"Bull. Fac. Sci. Engrg. Chuo Univ. Ser. I Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"20","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_26","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1090\/S0002-9939-1963-0148892-5","article-title":"The radius of univalence of certain analytic functions. II","volume":"14","author":"MacGregor","year":"1963","journal-title":"Proc Am. Math. Soc."},{"key":"ref_27","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_28","first-page":"117","article-title":"Some starlike conditions for analytic functions","volume":"33","author":"Mocanu","year":"1988","journal-title":"Rev. Roumaine Math. Pures. Appl."},{"key":"ref_29","first-page":"117","article-title":"On uniformly convex functions","volume":"53","author":"Ravichandran","year":"2002","journal-title":"Ganita"},{"key":"ref_30","first-page":"127","article-title":"Bessel transforms and Hardy space of generalized Bessel functions","volume":"48","author":"Baricz","year":"2006","journal-title":"Mathematica"},{"key":"ref_31","first-page":"251","article-title":"Some applications of the Hadamard convolution in the theory of functions","volume":"40","author":"Stankiewicz","year":"1986","journal-title":"Ann. Univ. Mariae Curie-Sk\u0142odowska Sect. A"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1216\/rmjm\/1181071795","article-title":"Inclusion theorems for convolution product of second order polylogarithms and functions with the derivative in a halfplane","volume":"28","author":"Ponnusamy","year":"1998","journal-title":"Rocky Mountain J. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1307\/mmj\/1029000519","article-title":"The Hardy class of some univalent functions and their derivatives","volume":"17","author":"Eenigenburg","year":"1970","journal-title":"Michigan Math. J."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Kumar, A., Das, S., and Mehrez, K. (2022). Geometric Properties of a Certain Class of Mittag\u2013Leffler-Type Functions. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6020054"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Owa, S. (1992). The quasi-Hadamard products of certain analytic functions. Current Topics in Analytic Function Theory, World Scientific Publishing.","DOI":"10.1142\/9789814355896_0019"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1007\/s13370-020-00852-8","article-title":"Applications of fractional derivatives for Alexander integral operator","volume":"32","author":"Acu","year":"2021","journal-title":"Afr. Mat."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1016\/j.matcom.2022.07.011","article-title":"Integral transforms and probability distributions for a certain class of Fox-Wright type functions and its applications","volume":"203","author":"Kumar","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_38","first-page":"423","article-title":"The asymptotic expansion of integral functions defined by Taylor series. I","volume":"238","author":"Wright","year":"1940","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/629\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:13:30Z","timestamp":1760145210000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/11\/629"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,9]]},"references-count":38,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["axioms11110629"],"URL":"https:\/\/doi.org\/10.3390\/axioms11110629","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,9]]}}}