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Transactions of the American Mathematical Society

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$ L\sp p$ inequalities for entire functions of exponential type


Authors: Qazi I. Rahman and G. Schmeisser
Journal: Trans. Amer. Math. Soc. 320 (1990), 91-103
MSC: Primary 30D15
DOI: https://doi.org/10.1090/S0002-9947-1990-0974526-4
MathSciNet review: 974526
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Abstract: Let $ f$ be an entire function of exponential type $ \tau$ belonging to $ {L^p}$ on the real line. It has been known since a long time that

$\displaystyle \int_{ - \infty }^\infty {{{\left\vert {f'(x)} \right\vert}^p}dx ... ...}^\infty {{{\left\vert {f(x)} \right\vert}^p}dx\quad {\text{if}}\;p \geq 1.} } $

We prove that the same inequality holds also for $ 0 < p < 1$. Various other estimates of the same kind have been obtained.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0974526-4
Keywords: Bernstein's inequality, $ {L^p}$ inequalities, entire functions of exponential type
Article copyright: © Copyright 1990 American Mathematical Society

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