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On the zeros of the successive derivatives of integral functions II

  • I. J. Schoenberg
Conference paper
  • 381 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 599)

Abstract

In 1936 the author proved [2] the following: Theorem 1. If f(z) (≢ 0) is entire of exponential type δ such that each f(v)(x) (v=0, 1, …) vanishes somewhere in the interval I1=[0,½] of the real axis, then
$$\delta \geqslant \pi ,$$
(1)
and the function
$$f(z) = CoS \pi z$$
(2)
shows that π is the best constant, because cos πz satisfies all conditions.

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References

  1. 1.
    J. D. Buckholtz, The Whittaker constant and successive derivatives of entire functions, J. Approximation Theory 3 (1970), 194–212.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    I. J. Schoenberg, On the zeros of successive derivatives of integral functions, Trans. Amer. Math. Soc. 40 (1936), 12–23.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    _____, Norm inequalities for a certain class of C functions, Israel J. of Math. 10 (1971), 364–372.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    _____, The elementary cases of Landau's problem of inequalities between derivatives, Amer. Math. Monthly 80 (1973), 121–158.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. M. Whittaker, Interpolatory function theory, Cambridge Tracts in Math. and Math. Phys. No. 33, Cambridge, 1935.Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • I. J. Schoenberg
    • 1
    • 2
  1. 1.Mathematics Research CenterUniversity of WisconsinMadison
  2. 2.University of PittsburghGermany

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