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Convolution inequalities, regular variation and exceptional sets

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References

  1. J. M. Anderson,The behaviour of integral and subharmonic functions, J. Analyse Math.17 (1966), 185–208.

    MATH  MathSciNet  Google Scholar 

  2. P. D. Barry,On a theorem of Besicovitch, Quart. Jour. Math. (Oxford) (2)14 (1963), 293–302.

    MATH  MathSciNet  Google Scholar 

  3. P. D. Barry,On a theorem of Kjellberg, Quart. Jour. Math. (Oxford) (2)15 (1964), 179–191.

    MATH  MathSciNet  Google Scholar 

  4. P. D. Barry,Some theorems related to the cos πρ theorem, Proc. London Math. Soc. (3)21 (1970), 334–360.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. D. Barry,On the growth of increasing functions, Bull. London Math. Soc.2 (1970), 29–33.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. P. Boas,Entire Functions, Academic Press, New York, 1954.

    MATH  Google Scholar 

  7. M. L. Cartwright,On the minimum modulus of integral functions, Proc. Cambridge Phil. Soc.30 (1934), 412–420.

    MATH  Google Scholar 

  8. A. Denjoy,Sur un théorème de Wiman, C. R. Acad. Sci. Paris193 (1931), 828–830.

    MATH  Google Scholar 

  9. D. Drasin,Tauberian theorems and slowly varying functions, Trans. Amer. Math. Soc.133 (1968), 333–356.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Drasin,A flexible proximate order, Bull. London Math. Soc.6 (1974), 129–135.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Drasin and D. F. Shea,Asymptotic properties of entire functions extremal for the cos πρ theorem, Bull. Amer. Math. Soc.75 (1969), 119–122.

    MATH  MathSciNet  Google Scholar 

  12. D. Drasin and D. F. Shea,Pólya peaks and the oscillation of positive functions, Proc. Amer. Math. Soc.34 (1972), 403–411.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Edrei,Sums of deficiencies of meromorphic functions, J. Analyse Math.14 (1965), 79–107.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Edrei,Locally tauberian theorems for meromorphic functions of lower order less than one, Trans. Amer. Math. Soc.140 (1969), 309–332.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Edrei,Extremal problems of the cos πρ-type (to appear).

  16. A. Edrei and W. H. J. Fuchs,Tauberian theorems for a class of meromorphic functions with negative zeros and positive poles, inContemporary Problems in the Theory of Analytic Functions, Nauka, 1966, pp. 339–358.

  17. M. Essén,Note on “A theorem on the minimum modulus of entire functions” by Kjellberg, Math. Scand.12 (1963), 12–14.

    MATH  MathSciNet  Google Scholar 

  18. M. Essén,Studies on a convolution inequality, Arkiv för Math.5 (1963), 113–152.

    Article  Google Scholar 

  19. M. Essén and T. Ganelius,Asymptotic properties of entire functions of order less than 1 inProc. Sympos. Pure Math., Vol. 11, Amer. Math. Soc., Providence, 1968, 193–201.

    Google Scholar 

  20. W. Feller,Introduction to Probability Theory and its Applications (II), New York, 1966.

  21. W. Feller,On regular variation and local limit theorems, inBerkeley Symposium on Mathematical Statistics and Probability, Berkeley, 1967, pp. 373–388.

  22. W. H. J. Fuchs,Proof a conjecture of G. Polya concerning gap series, Ill. J. Math.7 (1963), 661–667.

    MATH  Google Scholar 

  23. W. H. J. Fuchs,Topics in Nevanlinna theory, inProceedings of NRL Conference on Classical Function Theory, Washington, 1970, pp. 1–32.

  24. T. Ganelius,Regularly varying functions and Poincaré’s theorem on difference equations, inSymposia on Theoretical Physics and Mathematics, Vol. 10, New York, 1970, pp. 7–17.

  25. G. H. Hardy,Divirgent Series, Oxford, 1949.

  26. W. K. Hayman,On the characteristic of functions meromorphic in the plane and of their integrals, Proc. London Math. Soc. (3)14A (1965), 93–128.

    Article  MathSciNet  Google Scholar 

  27. W. K. Hayman,Some examples related to the cos πρ theorem, inMathematical Essays Dedicated to A. J. Macintyre, Athens, Ohio, 1970, pp. 149–170.

  28. M. Heins,Entire functions with bounded minimum modulus; subharmonic function analogues, Ann. of Math. (2)49 (1948), 200–213.

    Article  MathSciNet  Google Scholar 

  29. U. Hellsten, B. Kjellberg and F. Norstad,Subharmonic functions in a circle, Arkiv för Math.8 (1970), 185–193.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Inoue,Sur le module minimum des fonctions sousharmoniques et des fonctions entires d’ordre < 1/2, Mem. Fac. Sci. Kyushu Univ.A4 (1949), 185–193.

    Google Scholar 

  31. G. S. Jordan,Regular varying functions and convolutions with real kernels, Trans. Amer. Math. Soc.194 (1974), 177–194.

    Article  MATH  MathSciNet  Google Scholar 

  32. B. Kjellberg,On the minimum modulus of entire functions of lower order less than one, Math. Scand.8 (1960), 189–197.

    MATH  MathSciNet  Google Scholar 

  33. B. Kjellberg,A Theorem on the minimum modulus of entire functions, Math. Scand.12 (1963), 5–11.

    MATH  MathSciNet  Google Scholar 

  34. J. Korevaar, T. van Aardenne-Ehrenfest, and N. G. DeBrujn,A note on slowly oscillating functions, Nieuw Arch. Wisk.23 (1949), 77–86.

    Google Scholar 

  35. J. J. Levin and D. F. Shea,On the asymptotic behavior of the bounded solutions of some integral equations, J. Math. Anal. Appl.37 (1972), 42–82.

    Article  MathSciNet  Google Scholar 

  36. J. E. Littlewood,On the asymptotic approximation to integral functions of zero order, Proc. London Math. Soc. (2)5 (1907), 361–410.

    Article  Google Scholar 

  37. I. V. Ostrovskii,On the deficiencies of meromorphic functions of lower order less than one, Dokl. Akad. Nauk SSSR150 (1963), 32–35.

    MathSciNet  Google Scholar 

  38. V. P. Petrenko,The growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR Ser. Mat.33 (1969), 414–454.

    MATH  MathSciNet  Google Scholar 

  39. D. F. Shea,On a complement to Valiron’s tauberian theorem for the Stietjes transform, Proc. Amer. Math. Soc.21 (1969), 1–9.

    Article  MATH  MathSciNet  Google Scholar 

  40. E. C. Titchmarsh,Theory of Fourier Integrals, Oxford, 1948.

  41. M. Tsuji,Potential Theory in Modern Function Theory, Tokyo, 1958.

  42. G. Valiron,Sur les fonctions entières d’ordre fini et d’ordre null, et en particulier les fonctions à correspondance régulière, Ann. Fac. Sci. Univ. Toulouse (3)5 (1913), 117–257.

    MathSciNet  Google Scholar 

  43. A. Weitsman,Asymptotic behavior of meromorphic functions with extremal deficiencies, Trans. Amer. Math. Soc.140 (1969), 333–352.

    Article  MATH  MathSciNet  Google Scholar 

  44. D. Widder,The Laplace Transform, Princeton, 1941.

  45. E. T. Whittaker and G. M. Watson,Modern Analysis, Cambridge, 1927.

  46. J. Williamson,Asymptotic behavior of a class of entire functions of extremal growth, J. London Math. Soc. (2)7 (1973).

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Authors and Affiliations

  1. Purdue University, Lafayette, Ind., U.S.A.

    David Drasin & Daniel F. Shea

  2. University of Wisconsin, Madison, Wisc., U.S.A.

    David Drasin & Daniel F. Shea

Authors
  1. David Drasin
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  2. Daniel F. Shea
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The authors were supported in part by grants from the National Science Foundation.

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Drasin, D., Shea, D.F. Convolution inequalities, regular variation and exceptional sets. J. Anal. Math. 29, 232–293 (1976). https://doi.org/10.1007/BF02789980

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  • DOI: https://doi.org/10.1007/BF02789980

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