Third Hankel determinant of starlike and convex functions

Abstract

For an analytic function f of the form $f(z)= z+ a_2 z^2 +a_3 z^3 + \cdots$ satisfying either ${\text {Re}} \big ((f'(z))^{\alpha } (zf'(z)/f(z))^{(1-\alpha )}\big )> 0$ or ${\text {Re}} \big ((f'(z))^{\alpha } (1+ zf''(z)/f'(z))^{(1-\alpha )}\big )> 0$, the bounds for the third Hankel determinant $H_3(1)=a_3(a_2 a_4-a_3^2)- a_4 (a_4 -a_2 a_3) + a_5 (a_3 - a_2^2)$ are obtained. Our results include some previously known results.