Some estimates of differentiable functions

Abstract

Suppose thatx(t) ∈ C ^(n)]_[a,b and has n zeros at the pointsa and b. It is shown that if x⁽ⁿ⁾(t) preserves sign on [a, b], then

$$\left| {x\left( t \right)} \right| \geqslant \frac{{p_0 }}{{n - 1}}\mathop {\left[ {\mathop {\sup }\limits_{\tau \in \left( {a, b} \right)} \frac{{\left| {x\left( \tau \right)} \right|}}{{\left( {\tau - a} \right)^{p - 1} \left( {b - \tau } \right)^{q - 1} }}} \right]}\limits_{\left( {a< t< b} \right),} \left( {t - a} \right)^p \left( {b - t} \right)^q $$

where p and q are the multiplicities of the zeros of x(t) ata and b, respectively, and p_o=min{p,q}. Two-sided estimates of the Green's function for a two-point interpolation problem for the operator Lx ≡ 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.