Abstract
The principal goal of this paper is to investigate and report results concerning the following problem. Determine the family of all real entire functions of positive order, φ, in the Laguerre–Pólya class, such that if p is an arbitrary, non-constant real polynomial which has no zeros in common with φ, then the entire function f=φ+p possesses some non-real zeros. Ramifications of the results obtained are also considered in relation to the Hermite–Poulain Theorem and the theory of multiplier sequences.
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References
I. N. Baker, Entire functions with linearly distributed values, Math. Z., 86 (1964), 263–267.
R. P. Boas, Jr., Entire Functions, Academic Press (New York, 1954).
T. Craven and G. Csordas, On a converse of Laguerre’s theorem, Electron. Trans. Numer. Anal., 5 (1997), 7–17.
T. Craven and G. Csordas, Differential operators of infinite order and the distribution of zeros of entire functions, J. Math. Anal. Appl., 186 (1994), 799–820.
T. Craven and G. Csordas, Composition theorems, multiplier sequences and complex zero decreasing sequences, in: G. Barsegian, I. Laine and C. C. Yang (Eds.) Value Distribution Theory and Its Related Topics, Kluwer Press (2004), pp. 131–166.
G. Csordas and C. C. Yang, Finite Fourier transforms and the zeros of the Riemann ξ-function, in: Advances in Analysis, World Sci. Publ. (Hackensack, NJ, 2005), pp. 109–119.
R. Duffin and A. C. Schaeffer, Some properties of functions of exponential type, Bull. Amer. Math. Soc., 44 (1938), 236–240.
A. Edrei, Meromorphic functions with three radially distributed values, Trans. Amer. Math. Soc., 78 (1955), 276–293.
G. H. Hardy, On the zeros of a class of integral functions, Messenger of Math., 34 (1904), 97–101.
L. Hörmander, Some inequalities for functions of exponential type, Math. Scand., 3 (1955), 21–27.
J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc., 25 (1923), 325–332.
S. Kimura, On the value distribution of entire functions of order less than one, Kōdai Math. Sem. Rep., 28 (1976/77), 28–32.
T. Kobayashi, Distribution of values of entire functions of lower order less than one, Kōdai Math. Sem. Rep., 28 (1976/77), 33–37.
T. Kobayashi, Distribution of values of entire functions of lower order less than one, Kodai Math. J., 2 (1979), 54–81.
T. Kobayashi, Factorization Theory of Meromorphic Functions and Related Topics, Lecture Notes in Pure and Appl. Math. 78, Dekker (New York, 1982), pp. 35–54.
B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Mono. 5, Amer. Math. Soc (Providence, RI, 1964; revised ed. 1980).
N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften (Berlin, 1963).
G. Pólya, Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins, J. Reine Angew. Math., 145 (1915), 224–249.
G. Pólya, Über trigonometrische Integrale mit nur reellen Nullstellen, J. Reine Angew. Math., 158 (1927), 6–18.
G. Pólya, Collected Papers, Vol. II Location of Zeros (R. P. Boas, ed.), MIT Press (Cambridge, MA, 1974).
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Csordas, G., Yang, CC. Polynomial perturbations of transcendental entire functions in the Laguerre–Pólya class. Acta Math Hung 136, 240–251 (2012). https://doi.org/10.1007/s10474-012-0234-3
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DOI: https://doi.org/10.1007/s10474-012-0234-3