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# Inequalities for entire functions of exponential type satisfying $$f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)}$$

Acta Mathematica Hungarica
Authors: N. Govil and M. Qazi

## Abstract

Let f be an entire function of exponential type satisfying the condition
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$f(z) \equiv e^{i\gamma } e^{i\tau z} \overline {f(\bar z)}$$ \end{document}
for some real γ. Lower and upper estimates for ∫−∞ |f′(x)|p dx in terms of ∫−∞ |f(x)|p dx, for such a function f belonging to L p(R), have been known in the case where p ∊ [1, ∞) and γ = 0. In this paper, these estimates are shown to hold for any p ∊ (0, ∞) and any real γ.
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