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J. K. Langley 《Journal d'Analyse Mathématique》2005,97(1):357-396
Letf be a real meromorphic function of infinite order in the plane such thatf has finitely many poles. Then for eachk≥3, at least one off andf (k) has infinitely many non-real zeros. Together with a result of Edwards and Hellerstein, this establishes the analogue for higher derivatives of a conjecture going back to Wiman around 1911. 相似文献
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A number of results are proved concerning non-real zeros of derivatives of real and strictly non-real meromorphic functions in the plane. 相似文献
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We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their
numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or the Khovanskii
bounds for these systems. When the support is exactly a circuit whose affine span is ℤn, this bound is 2n+1, while the Khovanskii bound is exponential in n2. The bound 2n+1 can be attained only for non-degenerate circuits. Our methods involve a mixture of combinatorics, geometry, and arithmetic.
Part of work done at MSRI was supported by NSF grant DMS-9810361.
Work of Sottile is supported by the Clay Mathematical Institute.
Sottile and Bihan were supported in part by NSF CAREER grant DMS-0134860.
Bertrand is supported by the European research network IHP-RAAG contract HPRN-CT-2001-00271. 相似文献
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The behaviour of real zeroes of the Hurwitz zeta-function
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Steven R. Adams David A. Cardon 《Proceedings of the American Mathematical Society》2007,135(12):3857-3866
We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem, we construct sums of exponential functions having only real zeros, we construct polynomials having zeros only on the unit circle, and we obtain the three-term recurrence relation for an arbitrary family of real orthogonal polynomials. We discuss a similarity of this result with the Lee-Yang Circle Theorem from statistical mechanics. Also, we state several open problems.
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M. N. Sheremeta 《Mathematical Notes》1975,18(3):823-827
Suppose thatf(z) is an entire transcendental function with real Taylor coefficients, M(r)= max¦f(z)¦on¦z¦=r, and {n} is the sequence of sign changes of the coefficients. We will show that if (1/n)<, then
.Translated from Matematicheskie Zametki, Vol. 18, No. 3, pp. 395–402, September, 1975.The author thanks A. A. Gol'dberg for his useful comments. 相似文献
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J. K. Langley 《Journal d'Analyse Mathématique》2009,107(1):107-140
Let f be a real entire function with finitely many non-real zeros, not of the form f = Ph with P a polynomial and h in the Laguerre-Pólya class. Lower bounds are given for the number of non-real zeros of f″ + ω f, where ω is a positive real constant. 相似文献
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Luis Bernal-González 《Archiv der Mathematik》1997,68(2):145-150
We prove in this note a generalization of a theorem due to G. Herzog on zero-free universal entire functions. Specifically,
it is shown that, if a nonnegative integer q and a nonconstant entire function φ of subexponential type are given, then there is a residual set in the class of entire functions with zero-free derivatives
of orders q and q + 1, such that every member of that set is universal with respect to φ (D), where D is the differentiation operator.
This work is supported in part by DGICYT grant PB93-0926. 相似文献
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Let ? and f be functions in the Laguerre-Pólya class. Write ?(z)=e−αz2?1(z) and f(z)=e−βz2f1(z), where ?1 and f1 have genus 0 or 1 and α,β?0. If αβ<1/4 and ? has infinitely many zeros, then ?(D)f(z) has only simple real zeros, where D denotes differentiation. 相似文献
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In this paper, we prove some uniqueness theorems of entire functions and their derivatives that share an entire of function whose order is less than their order. The results in this paper improve those given by G. G. Gundersen and L. Z. Yang, J. P. Wang, J. M. Chang, and Y. Z. Zhu, X. M. Li and H. X. Yi, and others. Some examples are provided to show that the results in this paper are the best possible. 相似文献
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In this paper, we prove a theorem on the growth of a solution of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible. 相似文献
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Arie Harel Su Namn Jacob Sturm 《Proceedings of the American Mathematical Society》1999,127(2):563-567
It is shown that if is an entire function of order less than one, all of whose zeros are real, then the minimal root of is an increasing function of which accelerates as increases.
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In this paper, we study the growth of solutions of a first order linear differential equation. From this we verify that a
conjecture given by Brück is true under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems
of a nonconstant entire function and its first derivative sharing a nonzero constant CM. The results in this paper also improve
some known results. Some examples show that the results in this paper are best possible.
Project supported by the NSFC (NO. A0324617) and the RFDP (NO. 20060422049).
Received: 20 April 2006 Revised: 4 February 2007 相似文献