{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T04:10:13Z","timestamp":1774498213409,"version":"3.50.1"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,3]],"date-time":"2019-01-03T00:00:00Z","timestamp":1546473600000},"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>The aim of this paper is to investigate certain properties such as convexity of order    \u03bc   , close-to-convexity of order     1 + \u03bc    \/2 and starlikeness of normalized Mittag\u2013Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity of Mittag\u2013Leffler functions with respect to certain starlike functions. Furthermore, we find the conditions for the above-mentioned function to belong to the Hardy space     H p    . Some of our results improve the results in the literature.<\/jats:p>","DOI":"10.3390\/sym11010045","type":"journal-article","created":{"date-parts":[[2019,1,3]],"date-time":"2019-01-03T11:11:56Z","timestamp":1546513916000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Geometric Properties of Normalized Mittag\u2013Leffler Functions"],"prefix":"10.3390","volume":"11","author":[{"given":"Saddaf","family":"Noreen","sequence":"first","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}]},{"given":"Mohsan","family":"Raza","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Government College University, Faisalabad 38000, Pakistan"}]},{"given":"Jin-Lin","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Yangzhou University, Yangzhou 225002, China"}]},{"given":"Muhammad","family":"Arif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,3]]},"reference":[{"key":"ref_1","first-page":"554","article-title":"Sur la nouvelle fonction E\u03b1x","volume":"137","year":"1903","journal-title":"C. 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