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Strict inequalities for the derivatives of functions satisfying certain boundary conditions

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Abstract

For functions satisfying the boundary conditions

$$f(0) = f'(0) = \cdot \cdot \cdot = f^{(m)} (0) = 0, f(1) = f'(1) = \cdot \cdot \cdot = f^{(1)} (1) = 0$$

, the following inequality with sharp constants in additive form is proved:

$$\left\| {f^{(n - 1)} } \right\|_{L_q (0,1)} \leqslant A\left\| f \right\|_{L_p (0,1)} + B\left\| {f^{(n)} } \right\|_{L_r (0,1)}$$

wheren≥2, 0≤1≤n−2,−1≤m≤1, m+1≤n−3, and1≤p,q,r≤∞.

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Authors and Affiliations

  1. St. Petersburg City Hall Higher Management School, St. Petersburg, Russia

    A. I. Zvyagintsev

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  1. A. I. Zvyagintsev
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Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 712–724, November, 1997.

Translated by N. K. Kulman

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Zvyagintsev, A.I. Strict inequalities for the derivatives of functions satisfying certain boundary conditions. Math Notes 62, 596–606 (1997). https://doi.org/10.1007/BF02361298

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  • DOI: https://doi.org/10.1007/BF02361298

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