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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1977, Volume 21, Issue 5, Pages 653–664 (Mi mz7997)

Some estimates of differentiable functions

Yu. V. Pokornyi

Voronezh State University

Abstract: Suppose that $x(t)\in C_{[a,b]}^{(n)}$ and has $n$ zeros at the points $a$ and $b$. It is shown that if $x^{(n)}(t)$ preserves sign on $[a,b]$, then
$$|x(t)|\ge\frac{p_0}{(n-1)}[\sup\limits_{\tau\in(a,b)}\frac{|x(\tau)|}{(\tau-a)^{p-1}(b-\tau)^{q-1}}](t-a)^p(b-t)^q\quad(a<t<b),$$
where $p$ and $q$ are the multiplicities of the zeros of $x(t)$ at $a$ and $b$, respectively, and $p_0=\min\{p,q\}$. Two-sided estimates of the Green's function for a two-point interpolation problem for the operator $Lx\equiv x^{(n)}$ are established in the proof. As an application, new conditions for the solvability of de la Vallée Poussin's two-point boundary problems are obtained.

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English version:
Mathematical Notes, 1977, 21:5, 366–373

Bibliographic databases:

UDC: 517.5

Citation: Yu. V. Pokornyi, “Some estimates of differentiable functions”, Mat. Zametki, 21:5 (1977), 653–664; Math. Notes, 21:5 (1977), 366–373

Citation in format AMSBIB
\Bibitem{Pok77} \by Yu.~V.~Pokornyi \paper Some estimates of differentiable functions \jour Mat. Zametki \yr 1977 \vol 21 \issue 5 \pages 653--664 \mathnet{http://mi.mathnet.ru/mz7997} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=445047} \zmath{https://zbmath.org/?q=an:0414.34012} \transl \jour Math. Notes \yr 1977 \vol 21 \issue 5 \pages 366--373 \crossref{https://doi.org/10.1007/BF01788233}