On the Second Hankel Determinant for the kth-Root Transform of Analytic Functions

Abstract

Let $f$ be a normalized analytic function in the open unit disk of the complex plane satisfying $zf'(z)/f(z)$ is subordinate to a given analytic function $\varphi$. A sharp bound is obtained for the second Hankel determinant of the $k$th-root transform $z\left[f\left(z^k\right)/z^k\right]^\frac{1}{k}$. Best bounds for the Hankel determinant are also derived for the $k$th-root transform of several other classes, which include the class of $\alpha$-convex functions and $\alpha$-logarithmically convex functions. These bounds are expressed in terms of the coefficients of the given function $\varphi$, and thus connect with earlier known results for particular choices of $\varphi$.