Invariance of convex functions associated with a cardioid domain

Abstract

Sharp bounds are given for some functional including the second and third Hankel determinants, and the Zalcman functional a₂a₃-a₄ for convex functions related to a cardioid domain. We show that the sharp bounds for the third Hankel determinant for convex functions related to the cardioid domain and its inverse function are the same, but the corresponding sharp bounds for the second Hankel determinant and the Zalcman functional differ. The results presented therefore provide a new significant example of convex invariance, together with examples of non-invariance.