On the number of real critical points of logarithmic derivatives and the Hawaii conjecture
A real entire function \(f\) belongs to the class \(V_{2n}\) if \(f(z) = \exp(-az^{2n+2})g(z)\), where \(a \geq 0\) and \(g\) is a real entire function of genus \(\leq 2n + 1\) with only real zeros. The class \(U_{2n}\) can be defined recursively as \(U_0 = V_0\) and \(U_{2n} = V_{2n} - V_{2n-2}\) when \(n\geq 1\). A function \(\varphi\) is in the class \(U_{2n}^*\) whenever there exist \(f\in U_{2n}\) and a real polynomial \(p\) with no real zeros such that \(\varphi=p f\). The class \(U_{0}\) is called the Laguerre-Pólya class and is usually denoted by \(\mathcal{L}\text{-}\mathcal{P}\). The class \(U_{0}^*\) is frequently denoted by \(\mathcal{L}\text{-}\mathcal{P}^*\). For a given entire function \(\varphi \in U_{2n}^*\), the author establishes a connection between the number of real zeros of the functions \(Q[\varphi]:=(\varphi^{\prime}/\varphi)^{\prime}\) and \(Q_1[\varphi]:=(\varphi^{\prime\prime}/\varphi^{\prime})^{\prime}\). This investigation enables him to prove the Hawaii conjecture, first formulated by \textit{T. Craven}, the reviewer and \textit{W. Smith} [``The zeros of derivatives of entire functions and the Pólya-Wiman conjecture'', Ann. Math. (2) 125, 405--431 (1987; Zbl 0625.30036)]. The Hawaii conjecture states that the number of non-real zeros of a real polynomial \(p\) is not less than the number of real critical points of \(p^{\prime}/p\). In addition, the author shows that if the real entire function \(\varphi \in \mathcal{L}\text{-}\mathcal{P}^*\) has \(2m\) non-real zeros (counting multiplicities) and \(\varphi^{\prime}\) has \(2m_1\) non-real zeros, then \(Z_R(Q)\), the number of real zeros of \(Q[\varphi]\), satisfies \(2m-2m_1\leq Z_R(Q)\leq 2m\). This proves the generalized Hawaii conjecture. The author also derives several interesting results pertaining to the number of real critical points of the logarithmic derivatives of functions in \(U_{2n}^*\) (cf. Section 5).
scientific article; zbMATH DE number 6031949