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http://dx.doi.org/10.4134/BKMS.b190520

SHARP BOUNDS FOR INITIAL COEFFICIENTS AND THE SECOND HANKEL DETERMINANT  

Ali, Rosihan M. (School of Mathematical Sciences Universiti Sains Malaysia)
Lee, See Keong (School of Mathematical Sciences Universiti Sains Malaysia)
Obradovic, Milutin (Department of Mathematics Faculty of Civil Engineering University of Belgrade)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.4, 2020 , pp. 839-850 More about this Journal
Abstract
For functions f(z) = z + a2z2 + a3z3 + ⋯ belonging to particular classes, this paper finds sharp bounds for the initial coefficients a2, a3, a4, as well as the sharp estimate for the second order Hankel determinant H2(2) = a2a4 - a23. Two classes are treated: first is the class consisting of f(z) = z + a2z2 + a3z3 + ⋯ in the unit disk 𝔻 satisfying $$\|\(\frac{z}{f(z)}\)^{1+{\alpha}}\;f^{\prime}(z)-1\|<{\lambda},\;0<{\alpha}<1,\;0<{\lambda}{\leq}1.$$ The second class consists of Bazilevič functions f(z) = z+a2z2+a3z3+⋯ in 𝔻 satisfying $$Re\{\(\frac{f(z)}{z}\)^{{\alpha}-1}\;f^{\prime}(z)\}>0,\;{\alpha}>0.$$
Keywords
Coefficient estimates; Hankel determinants; univalent functions; $Bazilevi{\check{c}}$ functions;
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