In 1950 P. Turán used the expansion of complex polynomials in terms of Hermite polynomials \(H_n\) to study the location of their zeros in certain strips.
Here, among other results, the authors prove the following Theorem. Assume that a complex polynomial \(f(x)\) has the Hermite expansion \(f(x)=\sum_0^n b_jH_j(x)\) and let \(\mu_1\), …, \(\mu_n\) be positive real numbers such that \(\mu_1+ \cdots+\mu_n\leq1\). Then all the roots of \(f\) lie in the strip \[ |\operatorname{Im} z|\leq {1\over2}\,\max_{1\leq j\leq n}\left( |b_{n_j}|\over \mu_j |b_n| \right)^{1/j}. \] The proofs of these new results combine some ideas of Turán and the classical results of Fujiwara, Bailleu, Cowling and Thron.