Sharp bounds for initial coefficients and the second Hankel determinant
Bull. Korean Math. Soc. 2020 Vol. 57, No. 4, 839-850
https://doi.org/10.4134/BKMS.b190520
Published online July 9, 2020
Printed July 31, 2020
Rosihan M. Ali, See Keong Lee, Milutin Obradovi\'{c}
Universiti Sains Malaysia; Universiti Sains Malaysia; Bulevar Kralja Aleksandra 73
Abstract : For functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ belonging to particular classes, this paper finds sharp bounds for the initial coefficients $a_{2}, \, a_{3}, \, a_{4},$ as well as the sharp estimate for the second order Hankel determinant $H_{2}(2)=a_{2}a_{4}-a_{3}^{2}.$ Two classes are treated: first is the class consisting of $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in the unit disk $\mathbb{D}$ satisfying $$\left|\left(\frac{z}{f(z)}\right)^{1+\alpha}f'(z)-1\right|<\lambda, \quad 0<\alpha <1, \, 0 < \lambda \leq 1.$$ The second class consists of Bazilevi\v{c} functions $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ in $\mathbb{D}$ satisfying $${\rm Re}\left\{\left(\frac{f(z)}{z}\right)^{\alpha-1}f'(z)\right\}>0, \quad \alpha >0.$$
Keywords : Coefficient estimates, Hankel determinants, univalent functions, Bazilevi\v{c} functions
MSC numbers : Primary 30C45, 30C50
Supported by : The first author gratefully acknowledged support from a USM research university grant 1001.PMATHS.8011101. The second author acknowledged support from a USM research university grant 1001.PMATHS.8011038. The work of the third author was supported by MNZZS Grant, No. ON174017, Serbia
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