{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T10:26:30Z","timestamp":1760955990361,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2019,12,10]],"date-time":"2019-12-10T00:00:00Z","timestamp":1575936000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004569","name":"Ministerstwo Nauki i Szkolnictwa Wy\u017cszego","doi-asserted-by":"publisher","award":["030\/RID\/2018\/19"],"award-info":[{"award-number":["030\/RID\/2018\/19"]}],"id":[{"id":"10.13039\/501100004569","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we consider two functionals of the Fekete\u2013Szeg\u00f6 type      \u0398 f   ( \u03bc )  =  a 4  \u2212 \u03bc  a 2   a 3      and      \u03a6 f   ( \u03bc )  =  a 2   a 4  \u2212 \u03bc    a 3   2      for a real number    \u03bc    and for an analytic function     f  ( z )  = z +  a 2   z 2  +  a 3   z 3  + \u2026    ,     | z | &lt; 1    . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re      f ( z ) \/ z  &gt; \u03b1     or Re       f \u2032   ( z )   &gt; \u03b1    ,     \u03b1 \u2208 [ 0 , 1 )    . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss      \u0398 f   ( \u03bc )      and      \u03a6 f   ( \u03bc )      for close-to-convex functions such that      f \u2032   ( z )  = h  ( z )  \/   ( 1 \u2212 z )  2     , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of      \u0398 f   ( \u03bc )     ,      \u03a6 f   ( \u03bc )      or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions.<\/jats:p>","DOI":"10.3390\/sym11121497","type":"journal-article","created":{"date-parts":[[2019,12,10]],"date-time":"2019-12-10T10:52:41Z","timestamp":1575975161000},"page":"1497","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On the Fekete\u2013Szeg\u00f6 Type Functionals for Close-to-Convex Functions"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1693-5367","authenticated-orcid":false,"given":"Katarzyna","family":"Tra\u0327bka-Wi\u0229c\u0142aw","sequence":"first","affiliation":[{"name":"Mechanical Engineering Faculty, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7279-9582","authenticated-orcid":false,"given":"Pawe\u0142","family":"Zaprawa","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Faculty, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4612-1740","authenticated-orcid":false,"given":"Magdalena","family":"Gregorczyk","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Faculty, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9631-9785","authenticated-orcid":false,"given":"Andrzej","family":"Rysak","sequence":"additional","affiliation":[{"name":"Mechanical Engineering Faculty, Lublin University of Technology, ul. Nadbystrzycka 36, 20-618 Lublin, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,10]]},"reference":[{"key":"ref_1","unstructured":"Duren, P.L. (1983). Univalent Functions, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1155\/S0161171278000150","article-title":"On the definition of close-to-convex function","volume":"1","author":"Goodman","year":"1978","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1307\/mmj\/1028988895","article-title":"Close to convex schlicht functions","volume":"1","author":"Kaplan","year":"1952","journal-title":"Mich. Math. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1006\/jmaa.1999.6378","article-title":"Generalized Zalcman conjecture for starlike and typically real functions","volume":"234","author":"Ma","year":"1999","journal-title":"J. Math. Anal. 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