We consider the Banach space $X=C([0,1])$ endowed with the norm $\|v\|_{\infty}=\max _{t \in[0,1]}|v(t)|$ and, we define the cone
$\mathcal{C}=\{v \in X: v$ is concave and $v(0)=v(1)=0\}$. We can prove the following result:

Given a function $v$ in the cone $\mathcal{C}$ and a point $p \in(0,1),$ the following estimates hold:
$(i)$
$$
v(t) \geq\left\{\begin{array}{ll}
\frac{t}{p} v(p) & t<p \\
\frac{1-t}{1-p} v(p) & t>p
\end{array}\right.
$$
and
$(i i)$
$$
v(t) \leq\left\{\begin{array}{ll}
\frac{t}{p} v(p) & t>p \\
\frac{1-t}{1-p} v(p) & t<p
\end{array}\right.
$$
Moreover, for all $0<t_{0}<t_{1}<1,$ we have
$$
\min _{t \in\left[t_{0}, t_{1}\right]} v(t) \geq c_{t_{0}, t_{1}}\|v\|_{\infty}  \,\,\,\,\,\,\,\,\,\, (1)  
$$
where $c_{t_{0}, t_{1}}:=\min \left\{t_{0}, 1-t_{1}\right\}$.

It is possible to obtain a generalization of (1) for some cone of functions $u$ defined in a $\Omega$ subset of $\mathbb{R}^n$ ($n\geq 2$) satisfying $u|_{\partial\Omega}=0$?.