1984
DOI: 10.2307/1999462
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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 f… Show more

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Cited by 9 publications
(13 citation statements)
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“…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.…”
Section: Continuation Of the Proof Of Theorem 16mentioning
confidence: 99%
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“…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.…”
Section: Continuation Of the Proof Of Theorem 16mentioning
confidence: 99%
“…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…”
Section: Proof Of Proposition 45mentioning
confidence: 99%
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“…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].…”
Section: Introductionmentioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%