On the zeros of the successive derivatives of integral functions II

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


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
$$\delta \geqslant \pi ,$$
and the function
$$f(z) = CoS \pi z$$
shows that π is the best constant, because cos πz satisfies all conditions.


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    I. J. Schoenberg, On the zeros of successive derivatives of integral functions, Trans. Amer. Math. Soc. 40 (1936), 12–23.MathSciNetCrossRefzbMATHGoogle Scholar
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    _____, Norm inequalities for a certain class of C functions, Israel J. of Math. 10 (1971), 364–372.MathSciNetCrossRefzbMATHGoogle Scholar
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    _____, The elementary cases of Landau's problem of inequalities between derivatives, Amer. Math. Monthly 80 (1973), 121–158.MathSciNetCrossRefzbMATHGoogle Scholar
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    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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