For a given real polynomial p we study the possible number of real roots of a differential polynomial \(H_{\varkappa }[p](x) = \varkappa \left( p'(x)\right) ^2-p(x)p''(x), \varkappa \in \mathbb {R}\) . In the special case when all real zeros of the polynomial p are simple, and all roots of its derivative \(p'\) are real and simple, the distribution of zeros of \(H_{\varkappa }[p]\) is completely described for each real \(\varkappa \) . We also provide counterexamples to two Boris Shapiro’s conjectures about the number of zeros of the function \(H_{\frac{n-1}{n}}[p]\) .
