Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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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)
  • Received : 2019.05.23
  • Accepted : 2020.06.04
  • Published : 2020.07.31
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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.$$

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References

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