Non-real zeroes of real entire derivatives

A real entire function belongs to the Laguerre-Pólya class LP if it is the limit of a sequence of real polynomials with real zeroes. By building upon results that resolved a long-standing conjecture of Wiman, a number of conditions are established under which a real entire function f must belong to the class LP, or to one of the related classes U _2p ^*. These conditions typically involve the non-real zeroes of f and its derivatives, or those of the differential polynomial f f″−a(f′)².