Linear 𝑞-difference, difference and differential operators preserving some 𝒜-entire functions

Huang, Jiaxing; Ng, Tuen-Wai

doi:10.1090/proc/16321

Linear $q$-difference, difference and differential operators preserving some $\mathcal {A}$-entire functions
HTML articles powered by AMS MathViewer

by Jiaxing Huang and Tuen-Wai Ng;

Proc. Amer. Math. Soc. 151 (2023), 3469-3479

DOI: https://doi.org/10.1090/proc/16321

Published electronically: April 20, 2023

HTML | PDF | Request permission

Abstract:

We apply Rossi’s half-plane version of Borel’s theorem to study the zero distribution of linear combinations of $\mathcal {A}$-entire functions (Theorem 1.2). This provides a unified way to study linear $q$-difference, difference and differential operators (with entire coefficients) preserving subsets of $\mathcal {A}$-entire functions, and hence obtain several analogous results for the Hermite-Poulain theorem to linear finite ($q$-)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomials of functions in certain sub-classes of $\mathcal {A}$-entire functions.

References

Carlos Berenstein, Der-Chen Chang, and Bao Qin Li, A note on Wronskians and linear dependence of entire functions in $\textbf {C}^n$, Complex Variables Theory Appl. 24 (1994), no. 1-2, 131–144. MR 1269838, DOI 10.1080/17476939408814706
W. Bergweiler, A. Eremenko, and J. K. Langley, Real entire functions of infinite order and a conjecture of Wiman, Geom. Funct. Anal. 13 (2003), no. 5, 975–991. MR 2024413, DOI 10.1007/s00039-003-0437-4
W. Bergweiler, A. Eremenko, and J. K. Langley, Zeros of differential polynomials in real meromorphic functions, Proc. Edinburgh Math. Soc. 48 (2005), 279–293.
W. Bergweiler and W. H. J. Fuchs, On the zeros of the second derivative of real entire functions, J. Anal. 1 (1993), 73–79. MR 1230507
Walter Bergweiler, Katsuya Ishizaki, and Niro Yanagihara, Meromorphic solutions of some functional equations, Methods Appl. Anal. 5 (1998), no. 3, 248–258. MR 1659151, DOI 10.4310/MAA.1998.v5.n3.a2
Julius Borcea and Petter Brändén, Pólya-Schur master theorems for circular domains and their boundaries, Ann. of Math. (2) 170 (2009), no. 1, 465–492. MR 2521123, DOI 10.4007/annals.2009.170.465
E. Borel, Lecons sur les fonctions entieres, Gauthier-Villars, Paris, 1921.
Petter Brändén, Ilia Krasikov, and Boris Shapiro, Elements of Pólya-Schur theory in the finite difference setting, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4831–4843. MR 3544533, DOI 10.1090/proc/13115
David A. Brannan and James G. Clunie (eds.), Aspects of contemporary complex analysis, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR 623462
Yik-Man Chiang and Shao-Ji Feng, On the Nevanlinna characteristic of $f(z+\eta )$ and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105–129. MR 2407244, DOI 10.1007/s11139-007-9101-1
George Csordas and Richard S. Varga, Integral transforms and the Laguerre-Pólya class, Complex Variables Theory Appl. 12 (1989), no. 1-4, 211–230. MR 1040921, DOI 10.1080/17476938908814366
G. Csordas and R. S. Varga, Problems and theorems in the theory of multiplier sequences, Serdica Math J 22 (1996), no. 4, 515–524.
N. G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17 (1950), 197–226. MR 37351, DOI 10.1215/S0012-7094-50-01720-0
Alexandre Eremenko and L. A. Rubel, On the zero sets of certain entire functions, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2401–2404. MR 1322920, DOI 10.1090/S0002-9939-96-03294-7
W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. MR 164038
Simon Hellerstein and Jack Williamson, Derivatives of entire functions and a question of Pólya, Trans. Amer. Math. Soc. 227 (1977), 227–249. MR 435393, DOI 10.1090/S0002-9947-1977-0435393-4
Jiaxing Huang and Tuen-Wai Ng, Hypertranscendency of perturbations of hypertranscendental functions, J. Math. Anal. Appl. 491 (2020), no. 2, 124390, 11. MR 4126404, DOI 10.1016/j.jmaa.2020.124390
Olga Katkova, Mikhail Tyaglov, and Anna Vishnyakova, Linear finite difference operators preserving the Laguerre-Pólya class, Complex Var. Elliptic Equ. 63 (2018), no. 11, 1604–1619. MR 3847101, DOI 10.1080/17476933.2017.1400539
Olga Katkova, Mikhail Tyaglov, and Anna Vishnyakova, Hermite-Poulain theorems for linear finite difference operators, Constr. Approx. 52 (2020), no. 3, 357–393. MR 4181781, DOI 10.1007/s00365-020-09507-0
Ilpo Laine, Nevanlinna theory and complex differential equations, De Gruyter Studies in Mathematics, vol. 15, Walter de Gruyter & Co., Berlin, 1993. MR 1207139, DOI 10.1515/9783110863147
J. K. Langley, Non-real zeros of linear differential polynomials, J. Anal. Math. 107 (2009), 107–140. MR 2496401, DOI 10.1007/s11854-009-0005-4
J. K. Langley, Postgraduate notes on complex analysis, http://www.maths.nottingham.ac.uk/personal/jkl/pgl.pdf, 2014.
B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, RI, 1964. MR 156975, DOI 10.1090/mmono/005
B. Ja. Levin and I. V. Ostrovskiĭ, The dependence of the growth of an entire function on the distribution of zeros of its derivatives, Sibirsk. Mat. Ž. 1 (1960), 427–455 (Russian). MR 130979
Rolf Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Chelsea Publishing Co., New York, 1974 (French). Reprinting of the 1929 original. MR 417418
Tuen-Wai Ng and Chung-Chun Yang, On the zeros of $\sum a_i\exp g_i$, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 7, 137–139. MR 1487577
Nikola Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 164003
G. Pólya, Über Annäherung durch polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo 36 (1913), 279–295.
G. Pólya, Bemerkung Über die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48 (1926), no. 3-4, 305–317 (German). MR 1555227, DOI 10.1007/BF02565336
J. Schur and G. Pólya, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144 (1914), 89–113 (German). MR 1580897, DOI 10.1515/crll.1914.144.89
Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, London Mathematical Society Monographs. New Series, vol. 26, The Clarendon Press, Oxford University Press, Oxford, 2002. MR 1954841
John Rossi, A half-plane version of a theorem of Borel, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 111–118. MR 955813, DOI 10.1007/978-1-4613-9602-4_{9}
T. Sheil-Small, On the zeros of the derivatives of real entire functions and Wiman’s conjecture, Ann. of Math. (2) 129 (1989), no. 1, 179–193. MR 979605, DOI 10.2307/1971490
Masatsugu Tsuji, On Borel’s directions of meromorphic functions of finite order. I, Tohoku Math. J. (2) 2 (1950), 97–112. MR 41224, DOI 10.2748/tmj/1178245639
Peter Walker, Separation of zeros of translates of polynomials and entire functions, J. Math. Anal. Appl. 206 (1997), no. 1, 270–279. MR 1429291, DOI 10.1006/jmaa.1997.5232

AMS Website Down

Proceedings of the American Mathematical Society

Linear $q$-difference, difference and differential operators preserving some $\mathcal {A}$-entire functions
HTML articles powered by AMS MathViewer

Abstract: