The second Hankel determinant for alpha-convex functions

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.