Bulletin of the
Korean Mathematical Society BKMS

Let $\mathcal U^+$ be the class of analytic functions $f$ such that $\frac{z}{f(z)}$ has real and positive coefficients and $f^{-1}$ be its inverse. In this paper we give sharp estimates of the initial coefficients and initial logarithmic coefficients for $f$, as well as, sharp estimates of the second and the third Hankel determinant for $f$ and $f^{-1}$. We also show that the Zalcman conjecture holds for functions $f$ from $\mathcal U^+$.

Keywords: Univalent, real coefficients, logarithmic coefficients, coefficient estimates, Hankel determinant, Zalcman conjecture

MSC numbers: 30C45, 30C50, 30C55