Second Hankel Determinants for the Class of Typically Real Functions

Abstract

We discuss the Hankel determinants $H_2(n)=a_n{a}_{n+2}-{\left(a_{n+1}\right)}^2$ for typically real functions, that is, analytic functions which satisfy the condition $\mathrm{Im} z \mathrm{Im}f(z)\ge 0$ in the unit disk Δ. Main results are concerned with $H_2(2)$ and $H_2(3)$. The sharp upper and lower bounds are given. In general case, for $n\ge 4$, the results are not sharp. Moreover, we present some remarks connected with typically real odd functions.