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fa.useful evaluation – Inequalities in particular duty cones

We deem the Banach area $X=C([0,1])$ endowed with the norm $|v|_{infty}=max _{t in[0,1]}|v(t)|$ and, we outline the cone

$mathcal{C}={u in X mid u mbox{ is concave down, } u geq 0, u(0)=u(1)=0}$. We can show the next outcome:

Given a duty $v$ within the cone $mathcal{C}$ and some extent $p in(0,1),$ the next estimates maintain:

$(i)$

$$

v(t) geqleft{commence{array}{ll}

frac{t}{p} v(p) & t<p

frac{1-t}{1-p} v(p) & t>p

aim{array}privilege.

$$

and

$(i i)$

$$

v(t) leqleft{commence{array}{ll}

frac{t}{p} v(p) & t>p

frac{1-t}{1-p} v(p) & t<p

aim{array}privilege.

$$

Moreover, for all $0<t_{0}<t_{1}<1,$ now we have

$$

min _{t inleft[t_{0}, t_{1}right]} v(t) geq c_{t_{0}, t_{1}}|v|_{infty} ,,,,,,,,,, (1)

$$

the place $c_{t_{0}, t_{1}}:=min left{t_{0}, 1-t_{1}privilege}$.

It is workable to secure a generalization of (1) for some cone of capabilities $u$ outlined in a subset $Omega$ of $mathbb{R}^n$ ($ngeq 2$) satisfying $u|_{partialOmega}=0$?.

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