Skip to main content
    • Aa
    • Aa

Entire functions with two radially distributed values


We study entire functions whose zeros and one-points lie on distinct finite systems of rays. General restrictions on these rays are obtained. Non-trivial examples of entire functions with zeros and one-points on different rays are constructed, using the Stokes phenomenon for second order linear differential equations.

Hide All
[1] Azarin V. S. Asymptotic behavior of subharmonic functions of finite order (Russian). Mat. Sb. (N.S.) 108 (150) (1979), no. 2, 147167, 303. (English transl.: Math. USSR, Sb. 36 (1980), 135–154.)
[2] Baker I. N. Entire functions with linearly distributed values. Math. Z. 86 (1964), 263267.
[3] Baker I. N. Entire functions with two linearly distributed values. Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 2, 381386.
[4] Bender C. M., Boettcher S. and Meisinger P. N. PT-symmetric quantum mechanics. J. Math. Phys. 40 (1999), no. 5, 22012229.
[5] Bergweiler W. and Eremenko A. Goldberg's constants. J. Anal. Math. 119 (2013), no. 1, 365402.
[6] Blondel V. Simultaneous Stabilization of Linear Systems (Springer, Berlin, 1994).
[7] Bieberbach L. Über eine Vertiefung des Picardschen Satzes bei ganzen Funktionen endlicher Ordnung. Math. Z. 3 (1919), 175190.
[8] Biernacki M. Sur le déplacement des zéros des fonctions entières par leur dérivation. Comptes Rendus 175 (1922), 1820.
[9] Biernacki M. Sur la théorie des fonctions entières. Bulletin de l'Académie polonaise des sciences et des lettres, Classe des sciences mathématiques et naturelles, Série A (1929), 529–590.
[10] Boas H. P. and Boas R. P. Short proofs of three theorems on harmonic functions. Proc. Amer. Math. Soc. 102 (1988), no. 4, 906908.
[11] Dorey P., Dunning C. and Tateo R. On the relation between Stokes multipliers and the T-Q systems of conformal field theory. Nuclear Physics B 563 (1999), 573602.
[12] Dorey P., Dunning C. and Tateo R. The ODE/IM correspondence. J. Phys. A 40 (2007), no. 32, R205R283.
[13] Drasin D. Value distributions of entire functions in regions of small growth. Ark. Mat. 12 (1974), 281296.
[14] Drasin D. and Shea D. F. Pólya peaks and the oscillation of positive functions. Proc. Amer. Math. Soc. 34 (1972), 403411.
[15] Edrei A. Meromorphic functions with three radially distributed values. Trans. Amer. Math. Soc. 78 (1955), 276293.
[16] Eremenko A. Value distribution and potential theory. Proceedings of the ICM, Vol. II (Beijing, 2002) (Higher Ed. Press, Beijing, 2002), pp. 681690.
[17] Eremenko A. Simultaneous stabilisation, avoidance and Goldberg's constants. arXiv: 1208.0778.
[18] Eremenko A. Entire functions, PT-symmetry and Voros's quantization scheme. arXiv: 1510.02504.
[19] Goldberg A. A. and Ostrovskii I. V. Distribution of values of meromorphic functions (Amer. Math. Soc., Providence, RI, 2008).
[20] Hörmander L. The Analysis of Linear Partial Differential Operators I. 2nd ed. (Springer, Berlin, 1990.
[21] Hörmander L. Notions of Convexity (Birkhäuser, Boston, 1994).
[22] Kobayashi T. An entire function with linearly distributed values. Kodai Math. J. 2 (1979), no. 1, 5481.
[23] Lehto O. and Virtanen K. I. Quasiconformal Mappings in the Plane (Springer, New York – Heidelberg, 1973).
[24] Levin B. Ya. Distribution of zeros of entire functions. Amer. Math. Soc. (Providence, RI, 1970).
[25] Milloux H. Sur la distribution des valeurs des fonctions entières d'ordre fini, à zéros reels. Bull. Sci. Math. (2) 51 (1927), 303319.
[26] Nevanlinna R. Über die Konstruktion von meromorphen Funktionen mit gegebenen Wertzuordnungen. Festschrift zur Gedächtnisfeier für Karl Weierstraß (Westdeutscher Verlag, Köln – Opladen, 1966), pp. 579582.
[27] Ozawa M. On the zero-one set of an entire function. Kodai Math. Sem. Rep. 8 (1977), no. 4, 311316.
[28] Pólya G. and Szegő G. Problems and Theorems in Analysis. Vol. I: Series, Integral Calculus, Theory of Functions (Springer, New York, 1972).
[29] Ransford T. Potential Theory in the Complex Plane (Cambridge University Press, Cambridge, 1995).
[30] Rubel L. A. and Yang C.-C. Interpolation and unavoidable families of meromorphic functions. Michigan Math. J. 20 (1974), no. 4, 289296.
[31] Shin K. The potential (iz) m generates real eigenvalues only, under symmetric rapid decay boundary conditions. J. Math. Phys. 46 (2005), no. 8, 082110, 17pp.
[32] Sibuya Y. Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient (North–Holland, Amsterdam, 1975).
[33] Sibuya Y. Non-trivial entire solutions of the functional equation f(λ) + f(ωλ)f−1λ) = 1. Analysis 8 (1998), 271295.
[34] Sibuya Y. and Cameron R. An entire solution of the functional equation f(λ) + f(ωλ)f−1λ) = 1. Lecture Notes Math. 312 (Springer, Berlin, 1973), pp. 194202.
[35] Tabara T. Asymptotic behavior of Stokes multipliers for y″ - (x σ + λ)y = 0, (σ ⩾ 2) as λ → ∞. Dynam. Contin. Discrete Impuls. Systems 5 (1999), 93105.
[36] Winkler J. Zur Existenz ganzer Funktionen bei vorgegebener Menge der Nullstellen und Einsstellen. Math. Z. 168 (1979), 7786.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 27 *
Loading metrics...

Abstract views

Total abstract views: 138 *
Loading metrics...

* Views captured on Cambridge Core between 4th April 2017 - 19th October 2017. This data will be updated every 24 hours.