 # fa.useful evaluation – Inequalities in particular duty cones Answer

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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 aim{array}privilege.$$

and
$$(i i)$$
$$v(t) leqleft{commence{array}{ll} frac{t}{p} v(p) & t>p frac{1-t}{1-p} v(p) & t
Moreover, for all $$0 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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