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Strict inequalities for the derivatives of functions satisfying certain boundary conditions

Authors:

A. I. Zvyagintsev

Affiliation:

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

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:

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

wheren≥2, 0≤1≤n−2,−1≤m≤1, m+1≤n−3, and1≤p,q,r≤∞. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 712–724, November, 1997. Translated by N. K. Kulman

Keywords:

functions of a single variable boundary conditions L _p-estimates of derivatives

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