Reality of the zeros of an entire function and its derivatives

Abstract

Abstract. In 1914 Pólya raised the problem of classifying the entire functions which together with all their derivatives have only real zeros. In earlier work Hellerstein and Williamson settled this problem for entire functions which are real on the real axis. We complete the classification in all cases and show that it is sufficient to consider the function and its first two derivatives.

Real meromorphic functions and linear differential polynomials

“…Since a 0 = 0, while M is transcendental with finitely many poles, a standard application of Clunie's lemma [14, p. 68] to (17) shows that G is transcendental. …”

“…69-73] (see also [22]) to (17), with the Nevanlinna characteristic replaced by that of Tsuji, then yields (6), where the function a satisfies T(r, a) = S(r, M ). If a is constant then Proposition 3.1 gives (18), so assume that a is non-constant.…”

“…Functions with real poles, for which some derivatives have only real zeros, were treated in a number of papers including [17,18,21,38]. The following result [31,32] may be viewed as an improvement of a theorem from [18].…”

“…Then since the poles of f have bounded multiplicities we obtain a setF 0 ⊆ [1, ∞) of infinite measure such that r = O(N (r, f )) = O(T (r, M ))on F 0 . Applying the Tumura-Clunie method again to(17) gives(6), by Lemma 5.2, where T (r, a) = o(T (r, M )) on F 0 . If a is constant then we obtain (4) from Proposition 3.1 and Lemma 4.4.…”

Real zeros of derivatives of meromorphic functions and solutions of second order differential equations

“…In a series of papers [3,4,6] the authors recently settled an old conjecture of Pólya by characterizing those entire functions / for which /, /' and /" have only real zeros. These results may be summarized by Observe that the functions given by (0.1) are (constant multiples of) real entire functions, while those of the form (0.2)-(0.5) are strictly nonreal entire functions (/ is real if z real implies f(z) real, while / is strictly nonreal if it is not a constant multiple of a real function).…”