{"title":"A Conjecture on <i>H</i> <sub>3</sub>(1) for Certain Starlike Functions","authors":"Neha Verma, S. Sivaprasad Kumar","doi":"10.1515/ms-2023-0088","DOIUrl":null,"url":null,"abstract":"ABSTRACT We prove a conjecture concerning the third Hankel determinant, proposed by Kumar and Kamaljeet in [A cardioid domain and starlike functions , Anal. Math. Phys. 11 (2021), Art. 54], which states that | H 3 (1)| ≤ 1/9 is sharp for the class <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"> <m:msubsup> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>℘</m:mi> <m:mo>*</m:mo> </m:msubsup> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mi>z</m:mi> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mtext>/</m:mtext> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>≺</m:mo> <m:mi>φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>z</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>z</m:mi> <m:msup> <m:mi>e</m:mi> <m:mi>z</m:mi> </m:msup> </m:mrow> <m:mo>}</m:mo> </m:mrow> </m:math> . In addition, we also establish bounds for sixth and seventh coefficient, and | H 4 (1)| for functions in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"> <m:msubsup> <m:mi mathvariant=\"script\">S</m:mi> <m:mi>℘</m:mi> <m:mo>*</m:mo> </m:msubsup> </m:math> . The general bounds for two and three folds symmteric functions related with the Ma-Minda classes <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"> <m:msup> <m:mi mathvariant=\"script\">S</m:mi> <m:mo>*</m:mo> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mi>φ</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> of starlike functions are also obtained.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"59 1","pages":"0"},"PeriodicalIF":0.9000,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/ms-2023-0088","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
ABSTRACT We prove a conjecture concerning the third Hankel determinant, proposed by Kumar and Kamaljeet in [A cardioid domain and starlike functions , Anal. Math. Phys. 11 (2021), Art. 54], which states that | H 3 (1)| ≤ 1/9 is sharp for the class S℘*={zf′(z)/f(z)≺φ(z):=1+zez} . In addition, we also establish bounds for sixth and seventh coefficient, and | H 4 (1)| for functions in S℘* . The general bounds for two and three folds symmteric functions related with the Ma-Minda classes S*(φ) of starlike functions are also obtained.
期刊介绍：
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca. It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc. The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
