Advertisement

Entire functions of bounded index

  • S. M. Shah
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 599)

Keywords

Entire Function Linear Differential Equation Exponential Type Entire Solution Manuscripta Math 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. P. Boas, Entire Functions, Academic Press, New York, 1954.zbMATHGoogle Scholar
  2. 2.
    Anil K. Bose, A note on entire functions of bounded index, Proc. Amer. Math. Soc. 21 (1969), 257–262.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. L. Cartwright, Integral Functions, Camb. Univ. Press, 1962.Google Scholar
  4. 4.
    C. S. Davis, Entire functions of bounded absolute index, Doctoral dissertation, Catholic Univ. of America, Washington, D.C., 1973.Google Scholar
  5. 5.
    K. A. Ekblaw, The functions of bounded index as a subspace of a space of entire functions, Doctoral dissertation, Univ. of Kentucky, Lexington, 1970.zbMATHGoogle Scholar
  6. 6.
    K. A. Ekblaw, The functions of bounded index as a subspace of a space of entire functions, Pacific J. Math. 37 (1971), 353–355.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    K. A. Ekblaw, Polynomials of maximal index, Aequationes math. 10 (1974), 34–39.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Günter Frank, Picardsche Ausnahmewerte bei Lösungen linearer Differentialgleichungen, Dissertation, Universitat Karlsruhe (1969).Google Scholar
  9. 9.
    Günter Frank, Über den Index einer ganzen Funktion, Arch. der Math. 22 (1971), 175–180.CrossRefzbMATHGoogle Scholar
  10. 10.
    Günter Frank, Über ganze Funktionen, die einer linearen Differentialgleichung genügen, manuscripta math. 4 (1971), 225–253.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Günter Frank and Erwin Mues, Über den Index der Lösungen linearer Differentialgleichungen, manuscripta math. 5 (1971), 155–163.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Günter Frank and Erwin Mues, Über das Wachstum des Index ganzer Funktionen, Math. Ann. 195 (1972), 114–120.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Günter Frank and Erwin Mues, Über lokale Maximalbeträge und den Index ganzer Funktionen, Arch. der Math. 23 (1972), 269–277.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Günter Frank, Zur lokalen Werteverteilung der Lösungen linearer Differentialgleichungen, manuscripta math. 6 (1972), 381–404.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    G. H. Fricke and R. E. Powell, Some product theorems on functions of bounded index, Indian J. Pure Appl. Math. 3 (1972), 795–800.MathSciNetzbMATHGoogle Scholar
  16. 16.
    Gerd H. Fricke, A characterization of functions of bounded index, Indian J. Math. 14 (1972), 207–212.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gerd H. Fricke, A note on multivalence of functions of bounded index, Proc. Amer. Math. Soc. 40 (1973), 140–142.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gerd H. Fricke, Functions of bounded index and their logarithmic derivatives, Math. Ann. 206 (1973), 215–223.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G. H. Fricke, S. M. Shah and W. C. Sisarcick, A characterization of entire functions of exponential type and M-bounded index, Indiana Univ. Math. J. 23 (1973), 405–412.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gerd H. Fricke, Entire functions having positive zeros, Indian J. Pure and Appl. Math. 5 (1974), 478–485.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Gerd H. Fricke, Entire functions of locally slow growth, J. d'Analyse Math. 28 (1975), 101–122.CrossRefzbMATHGoogle Scholar
  22. 22.
    Gerd H. Fricke, A characterization of functions having positive zeros, Indian J. Math., to appear.Google Scholar
  23. 23.
    G. H. Fricke and S. M. Shah, Entire functions satisfying a linear differential equation, Indag. Math. 37 (1975), 39–41.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    G. H. Fricke and R. E. Powell, Bounded index and summability methods, J. Austral. Math. Soc. Ser. A 21 (1976), 79–87.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gerd H. Fricke, Entire functions with prescribed asymptotic behavior, Rocky Mountain J. Math. 6 (1976), 237–246.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. W. Goodman, The valence of sums and products, Canad. J. Math. 20 (1968), 1173–1177.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Fred Gross, Entire functions of bounded index, Proc. Amer. Math. Soc. 18 (1967), 974–980.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Fred Gross, Entire functions of exponential type, J. Res. Nat. Bur. Standards Section B 74B (1970), 55–59.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    W. K. Hayman, Research problems in Function Theory, Athlone Press, London, 1967.zbMATHGoogle Scholar
  30. 30.
    W. K. Hayman, Differentialgleichungen und lokale Wurzelen, Meh. spolšn, Sredy rodstv. Probl. Analiz., 591–595 (Russian), (1972).Google Scholar
  31. 31.
    W. K. Hayman, Differential inequalities and local valency, Pacific J. Math. 44 (1973), 117–137.MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    L. F. Heath, Vector-valued entire functions of bounded index satisfying a differential equation, to appear.Google Scholar
  33. 33.
    V. G. Iyer, On the space of integral functions, I, J. Indian Math. Soc. 12 (1948), 13–30.MathSciNetzbMATHGoogle Scholar
  34. 34.
    A. C. King, A class of entire functions of bounded index and radii of univalence of some functions of zero order, Doctoral dissertation, Univ. of Kentucky, Lexington, 1970.Google Scholar
  35. 35.
    A. C. King and S. M. Shah, Indices of Lindelöf functions and their derivatives, Rocky Mountain J. Math. 2 (1972), 579–594.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    O. Knab, Wachstumsordnung und Index der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten, manuscripta math. 18 (1976), 299–316.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    J. G. Krishna and S. M. Shah, Functions of bounded indices in one and several complex variables, Macintyre Memorial Volume, Athens, Ohio, 223–235, (1970).Google Scholar
  38. 38.
    T. V. Lakshminarasimhan, A note on entire functions of bounded index, J. Indian Math. Soc. 38 (1974), 43–49.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Boo-sang Lee and S. M. Shah, The type of an entire function satisfying a linear differential equation, Arch. der Math. Basel 20 (1969), 616–622.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Boo-sang Lee and S. M. Shah, An inequality involving the Bessel function and its derivatives, J. Math. Anal. and Appl. 30 (1970), 144–155.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Boo-sang Lee and S. M. Shah, Inequalities satisfied by entire functions and their derivatives, Trans. Amer. Math. Soc. 149 (1970), 109–117.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Boo-sang Lee and S. M. Shah, On the growth of entire functions of bounded index, Indiana Univ. Math. J. 20 (1970/71), 81–87.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    B. Lepson, Differential equations of infinite order, hyperdirichlet series and entire functions of bounded index, Proc. Sympos. Pure Math. vol. XI, Amer. Math. Soc., Providence, R.I., (1968), 298–307.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    J. S. J. MacDonnell, Some convergnece theorems for Dirichlet series whose coefficients are entire functions of bounded index, Catholic Univ. Press, Washington, D.C., 1957.Google Scholar
  45. 45.
    V. Maric and S. M. Shah, Entire functions defined by gap power series and satisfying a differential equation, Tohoku Math. J. (2) 21 (1969), 621–631.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    J. Nikolaus, Über ganze Losungen linearer Differentialgleichungen, Archiv d. Math. 18 (1967), 618–626.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    J. Nikolaus, Lineare Differentialgleichungen mit gegebener ganzer Lösung, Math. Zeit 103 (1968), 30–36.MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    G. Polya and G. Szegö, Problems and Theorems in Analysis, Vol. II, Springer-Verlag, New York, 1976.CrossRefzbMATHGoogle Scholar
  49. 49.
    Walter Pugh, Sums of functions of bounded index, Proc. Amer. Math. Soc. 22 (1969), 319–323.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    W. J. Pugh and S. M. Shah, On the growth of entire functions of bounded index, Pacific J. Math. 83 (1970), 191–201.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Q. I. Rahman and L. Stankiewicz, Differential inequalities and local valency, Pacific J. of Math. LIV, No. 2, (1974), 165–182.MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable, vol. I, P. Noordhoff, Gronigen, 1960.zbMATHGoogle Scholar
  53. 53.
    S. M. Shah, Entire functions of bounded index, Proc. Amer. Math. Soc. 19 (1968), 1017–1022.MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    S. M. Shah, Entire functions satisfying a linear differential equation, J. Math. Mech. 18 (1968/69), 131–136.MathSciNetzbMATHGoogle Scholar
  55. 55.
    S. M. Shah, Entire functions of unbounded index and having simple zeros, Math. Zeit. 118 (1970), 193–196.MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    S. M. Shah, On entire functions of bounded index whose derivatives are of unbounded index, J. London Math. Soc. (2) 4 (1971), 127–139.MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    S. M. Shah, Analytic functions with univalent derivatives and entire functions of exponential type, Bull. Amer. Math. Soc. 78 (1972), 154–171.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    S. M. Shah, Functions of bounded index and M-bounded index, J. of Math. and Physical Sciences 7 (1973), Silver Jubilee Issue, S81–S86.zbMATHGoogle Scholar
  59. 59.
    S. M. Shah, Entire functions whose Fourier transforms vanish outside a finite interval, J. Math. Anal. and Applications 53 (1976), 174–185.MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    S. M. Shah and S. N. Shah, A new class of functions of bounded index, Trans. Amer. Math. Soc. 173 (1972), 363–377.MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    S. M. Shah and W. C. Sisarcick, On entire functions of exponential type, J. Res. Nat. Bur. Standards, Sect. B 75B (1971), 141–147.MathSciNetzbMATHGoogle Scholar
  62. 62.
    S.M. Shah and S.Y. Trimbte, Entire functions with univalent derivatives, J. Math. Anal. and Appl. 33(1971), 220–229.MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Shantilal N. Shah, Functions of exponential type are differences of functions of bounded index, Canad. J. Math., to appear.Google Scholar
  64. 64.
    W.C. Sisarcick, Variations of the definition of a function of bounded index, Doctoral dissertation, Univ. of Kentucky, Lexington, 1971Google Scholar
  65. 65.
    W.C. Sisarcick, Metric spaces of entire functions, Indian J. of Pure and Applied Math., to appear.Google Scholar
  66. 66.
    V. Sreenivasulu, Entire functions of bounded index over non-Archimedean field, Annales Polonici Math. 28(1973), 163–173.MathSciNetzbMATHGoogle Scholar
  67. 67.
    Hsiung Tang, Entire functions related to a Bessel function, Tamkang J. of Math., Taiwan, to appear.Google Scholar
  68. 68.
    Georges Valiron, Lectures on the General Theory of Integral Functions, Chelsea, New York, 1949.Google Scholar
  69. 69.
    J.M. Whittaker, The lower order of integral functions, J. London Math. Soc. 8(1933), 20–27.MathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Hans Wittich, Neuere Untersuchungen über Eindeutige Analytische Funktionen, Springer-Verlag, Berlin, 1955.CrossRefzbMATHGoogle Scholar
  71. 71.
    Hans Wittich, Zur Kennzeichung linearer Differentialgleichungen mit konstanten Koeffizienten, Festband Zum, 70 Geburtstag von R. Nevanlinna, Springer, Berlin, 128–134, 1966.CrossRefGoogle Scholar
  72. 72.
    J.C. Clunie and W.K. Hayman, Symposium on Complex Analysis Canterbury, Lond. Math. Soc. Lecture Notes, 1974.Google Scholar
  73. 73.
    R. Tijdeman, On the distribution of values of certain functions, Doctoral dissertation, Univ. of Amsterdam, 1969.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • S. M. Shah
    • 1
  1. 1.University of KentuckyLexington

Personalised recommendations