Inequalities for real-root polynomials and entire functions
The authors study the positivity of the operator \[ \bigcup_{2m} (f)={1\over 2} \sum^{2m}_{j=0} (-1)^{m+j}{2m\choose j} f^{(j)} f^{(2m-j)} \] that generalizes the classical Laguerre operator, and establish some new inequalities where \(\bigcup_{2m}(f)\) is positive for real polynomials with only real roots and for the Laguerre-Pólya class of entire functions. They indicate how their results can be applied to classical orthogonal polynomials. They also make some interesting conjectures about positivity of related operators (on polynomials with only real zeros). There is a comprehensive bibliography.
scientific article; zbMATH DE number 1888303