Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures

Abstract

A function f is in the class $ V_2p $ iff $ f(z) = e^{-az^{2p+2}}g(z) $ where a S 0 and g is a constant multiple of a real entire function of genus h 2 p + 1 with only real zeros. The class $ U_2p $ is defined as follows: $ U_0 = V_0 $ , $ U_{2p} = V_{2p}-V_{2p-2} $ . Functions in the class $ U_{2p}^{*} $ are represented as $ g(z) = c(z)f(z) $ where $ f\\in U_{2p} $ and c is a real polynomial with no real zeros. Every real entire function g , of finite order with at most finitely many non-real zeros satisfies $ g\\in U_{2p}^{*} $ for a unique p . We show the exact number of non-real zeros of f" , for $ f\\in U_{2p} $ , in terms of the number of non-real zeros of f' and a geometrical condition on the components of Im Q ( z ) > 0, where $ \\displaystyle Q(z) = z-({f(z)}/{f'(z)}) $ . Further, for a subclass of $ f\\in U_{2p} $ , we show necessary and sufficient conditions for f" to have exactly 2 p non-real zeros. For a subclass of $ U_{2p}^{*} $ we show that if f' has only real zeros, then f" has exactly 2 p non-real zeros. For $ f\\in U_{2p}^{*} $ we show that 2 p is a lower bound for the number of non-real zeros of $ f^{(k)} $ for k S 2.