Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682212Special Issue20211213New results of modern concept on the fourth-Hankel determinant of a certain subclass of analytic functions22432255611610.22075/ijnaa.2021.6116ENSarab DakhilTheyabDepartment of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, IraqWaggas GalibAtshanDepartment of Mathematics, College of Science, University of Al-Qadisiyah, Diwaniyah, IraqHabeeb KareemAbdullahDepartment of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, IraqJournal Article20211015A form for the fourth Hankel determinant is given in this paper as<br />$$H_4 (1)=\begin{vmatrix}<br /> 1 & \mathfrak{a}_2 & \mathfrak{a}_3 & \mathfrak{a}_4 \\<br /> \mathfrak{a}_2 & \mathfrak{a}_3 & \mathfrak{a}_4 & \mathfrak{a}_5 \\<br /> \mathfrak{a}_3 & \mathfrak{a}_4 & \mathfrak{a}_5 & \mathfrak{a}_6 \\<br /> \mathfrak{a}_4 & \mathfrak{a}_5 & \mathfrak{a}_6 & \mathfrak{a}_7 \\<br /> \end{vmatrix}$$<br />The modern concept of the fourth Hankel determinant is studied for the subclass of analytic functions $\mu \left(\beta ,\lambda ,t\right)$ defined here using the concept of subordination. Bounds on the coefficients $\left|a_n\right|$ with n = 2,3,4, 5,6,7 for the functions in this newly introduced class are given and the upper bound of the fourth Hankel determinant for this class is obtained. Lemmas used by the authors of this paper improve the results from a previously published paper. Interesting particular cases are given in the corollaries of the main theorems.https://ijnaa.semnan.ac.ir/article_6116_4c1d29373dad4c1adaafa7641ed8ca86.pdf