We classify all functions F meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that F' has no zeros and F" only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation w" + H(z)w = 0, H entire. Introductionand statement of the main results. 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

The purpose of this note is to announce a partial solution to an old question of Pólya.To state concisely his question and our results we introduce the following notation: For each integer p > 0, denote by V 2 the class of entire functions of the form f{z) = exp (-az 2p + 2 )g(z) where a > 0 and g(z) is a constant multiple of a real entire function of genus < 2p + 1 with only real zeros. Recall that a real entire function is one which assumes real values on the real axis. Now set U 0 = V 0 , and for p > 1, set In

No 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.

ABSTRACT. We classify all functions F meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that F' has no zeros and F" only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation w" + H(z)w = 0, H entire. Introductionand statement of the main results. 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