Real Zeros of Derivatives of Meromorphic Functions and Solutions of Second Order Differential Equations

Abstract

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

Real meromorphic functions and linear differential polynomials

“…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. If z 0 is a pole of f in the upper half-plane, of multiplicity p, then (5) and (6) imply that (10) holds at z 0 , with finitely many choices for the integer p, because the poles of f have bounded multiplicities.…”

“…Let the negative real zeros of G be −a 1 −a 2 · · · , repeated according to multiplicity. Using (18) we may therefore write G(z) = S(z)e bz Π 1 (z), Π 1 (z) = ∞ j=1…”

“…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].…”

Non-real zeros of real differential polynomials

“…All meromorphic functions f in the plane for which all derivatives f (k) (k 0) have only real zeros were determined by Hinkkanen [14][15][16], while functions with real poles, for which some of the derivatives have only real zeros, were considered in several papers including [11,12,31]. All meromorphic functions f in the plane for which all derivatives f (k) (k 0) have only real zeros were determined by Hinkkanen [14][15][16], while functions with real poles, for which some of the derivatives have only real zeros, were considered in several papers including [11,12,31].…”

Non-real zeros of derivatives of real meromorphic functions of infinite order

“…Meromorphic functions f in the plane for which all derivatives f (k) (k 0) have only real zeros were determined by Hinkkanen [13,14,15], while functions for which some derivatives have only real zeros have been considered in several papers including [11,12,27]. Meromorphic functions f in the plane for which all derivatives f (k) (k 0) have only real zeros were determined by Hinkkanen [13,14,15], while functions for which some derivatives have only real zeros have been considered in several papers including [11,12,27].…”