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Lithuanian Mathematical Journal

, Volume 58, Issue 2, pp 212–218 | Cite as

The second Hankel determinant for alpha-convex functions

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Abstract

Let S denote the class of analytic univalent functions in 𝔻:= {z ϵ ℂ: |z| < 1} normalized so that \( f(z)=z+{\sum}_{n=2}^{\infty }{a}_n{z}^n. \) Let C and S be the subclasses of S consisting of convex and starlike functions, respectively. For real α, the class M α of alpha-convex functions f ∈ S defined by

\( \operatorname{Re}\left\{\left(1-a\right)\frac{zf^{\hbox{'}}(z)}{f(z)}+a\left(\frac{zf^{\hbox{'}\hbox{'}}(z)}{f^{\hbox{'}}(z)}+1\right)\right\}>0, \) z ϵ 𝔻,

is well known, so thatM1 = C andM0 = S * . We give bounds for the second Hankel determinant \( {H}_2(2)=\left|{a}_2{a}_4-{a}_3^2\right| \) when f ∈ M α and α ≥ 0, thus extending the well-known results in the cases α = 0 and α = 1. We also give bounds for a wider class of functions.

Keywords

univalent functions starlike function convex function alpha-convex function Hankel determinant 

MSC

30C45 30C55 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural SciencesUniversity of RzeszówRzeszówPoland
  2. 2.Department of MathematicsSwansea UniversitySwanseaUnited Kingdom

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