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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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The majority of the formerly considered interpolatory problems depending on a parameterh and dealing, on the basis of certain assigned elements, with the construction of the set of all entire functionsF(z) from a given classK, exhibits a stability property with respect to small changes of the parameterh involved. The present paper contains an example of such an interpolatory problem (as well as its complete solution), which depends on a real parameterh and is posed on a certain class of functions of exponential type, and for which the uniqueness class is not stable for any irrationalh. The problem mentioned is, in a certain sense, a modification of the known Lidstone problem concerning entire functions of exponential type.

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The following result is proved. LetΛn} be a sequence of complex numbers with ¦Reλ n¦≧δ¦λ n ¦, δ>0, and letg be an entire function of exponential type with a sequence of zeros which satisfies the same condition. There exists an entire function of exponential typef≠0 such thatf(λ)=0 and ¦f(iy)¦≦¦g(iy)¦,yR, if and only if there exists a constantM such that for all numbersr andR, 0<r<R<+∞, we have
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\max \left\{ {\mathop \sum \limits_{\mathop {r \leqq |\lambda _n |< R}\limits_{\operatorname{Re} \lambda _n< 0} } - \operatorname{Re} \frac{1}{{\lambda _n }},\mathop \sum \limits_{\mathop {r \leqq |\lambda _n |< R}\limits_{\operatorname{Re} \lambda _n > 0} } \operatorname{Re} \frac{1}{{\lambda _n }}} \right\} \leqq \frac{1}{{2\pi }}\mathop \smallint \limits_r^R \frac{{\ln |g(iy)g( - iy)|}}{{y^2 }}dy + M.$$ \end{document}
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Some properties of functions of exponential type Bull. Amer. Math. Soc. 44 236 – 240 10.1090/S0002-9904-1938-06725-0 . [8

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Abstract  

LetB σ be the class of entire functions of exponential type σ, real valued and bounded in modulus by 1 in the real line. A setG of functions defined on the segment [-T-r, T+r], wherer is a fixed positive number, is called an (&, δ)-net of the classB σ on the segment [-т, т] if for any f∃B σ there existsgG such that for anyx∃[-T,T]

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left| {f(x) - g(x)} \right| \leqq \frac{\varepsilon }{{2r}}\int\limits_{x - r}^{x + r} {\left| {f(t)} \right|dt + \delta .}$$ \end{document}
The main result consists in the following: For any positive σ, r, &≦1, δ≦1 and sufficiently largeT we have
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_{\varepsilon ,\delta } (B_\sigma ,T) \leqq \frac{{2\sigma T}}{\pi }\log \frac{{c(\sigma r)}}{{\max (\varepsilon ,\delta )}},$$ \end{document}
where c(σr) depends only on the product σr. The main tool of the proof of this inequality is the following estimate of the derivative of a polynomialP(x) with real coefficients:
\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\| {P'(x)} \right\|_{L_p ( - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2},{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}) \leqq } c\left( {q + 1 + \sum\limits_{i = 1}^{n - q} {\frac{1}{{\left| {a_i } \right|^2 }}} } \right)\left\| {P(x} \right\|_{L_p ( - 1,1)} ,$$ \end{document}
whereq is the number of roots of the polynomialP(x) lying in the disk z<1; a1, ..., an−g are the other roots, с is an absolute constant, and 1≦p≦∞.

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. Functions of exponential type . Ann. Math ., 65 : 582 – 592 , 1957 . [31] B . Sz.-Nagy . Uber Integralungleichungen zwischen einer Function und ihrer Ableitung . Acta Sci. Math ., 10 : 64 – 74 , 1941 . [32] V. M . Tikhomirov and G. G . Magaril

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