Second Hankel determinants for the class of typically real functions (Q1669228)

Summary: 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 \(\operatorname{Im} z \operatorname{Im} f(z) \geq 0\) in the unit disk \(\Delta\). Main results are concerned with \(H_2(2)\) and \(H_2(3)\). The sharp upper and lower bounds are given. In general case, for \(n \geq 4\), the results are not sharp. Moreover, we present some remarks connected with typically real odd functions.