Geodesic
M. Riesz-Schur-type inequalities for entire functions of exponential type
Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 24-32 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove a general M. Riesz-Schur-type inequality for entire functions of exponential type. If $f$ and $Q$ are two functions of exponential types $\sigma > 0$ and $\tau \geqslant 0$, respectively, and if $Q$ is real-valued and the real zeros of $Q$, not counting multiplicities, are bounded away from each other, then $$ |f(x)|\le (\sigma+\tau) (A_{\sigma+\tau}(Q))^{-1/2}\|Q f\|_{\mathrm C(\mathbb R)},\qquad x\in \mathbb R, $$ where $$ A_s(Q) \stackrel{\mathrm{def}}{=}\inf_{x\in\mathbb R} \bigl([Q'(x)]^2+s^2 [Q(x)]^2\bigr). $$ We apply this inequality to the weights $Q(x)\stackrel{\mathrm{def}}{=} \sin (\tau x)$ and $Q(x) \stackrel{\mathrm{def}}{=} x$ and describe the extremal functions in the corresponding inequalities. Bibliography: 7 titles.
Keywords: M. Riesz-Schur-type inequalities, Duffin-Schaeffer inequality, entire functions of exponential type. 
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Michael I. Ganzburg; Paul Nevai; Tamás Erdélyi. M. Riesz-Schur-type inequalities for entire functions of exponential type. Sbornik. Mathematics, Tome 206 (2015) no. 1, pp. 24-32. http://geodesic.mathdoc.fr/item/SM_2015_206_1_a2/

[1] P. Nevai, “The Bernstein inequality and the Schur inequality are equivalent”, J. Approx. Theory, 182 (2014), 103–109 | DOI | MR | Zbl

[2] P. Nevai, The true story of $n$ vs. $2n$ in the Bernstein inequality (to appear)

[3] M. Riesz, “Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome”, Jahresber. Deutsch. Math.-Ver., 23 (1914), 354–368 | Zbl

[4] Q. I. Rahman, G. Schmeißer, Analytic theory of polynomials, London Math. Soc. Monogr. (N.S.), 26, The Clarendon Press, Oxford Univ. Press, Oxford, 2002, xiv+742 pp. | MR | Zbl

[5] I. Schur, “Über das Maximum des absoluten Betrages eines Polynoms in einem gegebenen Intervall”, Math. Z., 4:3-4 (1919), 271–287 | DOI | MR | Zbl

[6] R. J. Duffin, A. C. Schaeffer, “Some inequalities concerning functions of exponential type”, Bull. Amer. Math. Soc., 43:8 (1937), 554–556 | DOI | MR | Zbl

[7] V. È. Kacnel'son, “Equivalent norms in spaces of entire functions”, Math. USSR-Sb., 21:1 (1973), 33–55 | DOI | MR | Zbl