We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$ $$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$ Let $p$ and $q$ be in $\mathbb { R } ^ { 2 }$.
[III.B.1)] Verify $$F ( q ) - F ( p ) = \int _ { 0 } ^ { 1 } d F _ { p + t ( q - p ) } ( q - p ) \mathrm { d } t$$
[III.B.2)] Show $$\langle F ( q ) - F ( p ) , q - p \rangle \geqslant \alpha \| q - p \| ^ { 2 }$$
We consider $\alpha \in \mathbb { R } _ { + } ^ { * }$ and $F \in \mathcal { C } ^ { 1 } \left( \mathbb { R } ^ { 2 } , \mathbb { R } ^ { 2 } \right)$. We assume that for all $( p , h ) \in \mathbb { R } ^ { 2 } \times \mathbb { R } ^ { 2 }$
$$\left\langle d F _ { p } ( h ) , h \right\rangle \geqslant \alpha \| h \| ^ { 2 }$$
Let $p$ and $q$ be in $\mathbb { R } ^ { 2 }$.
\begin{enumerate}
\item[III.B.1)] Verify
$$F ( q ) - F ( p ) = \int _ { 0 } ^ { 1 } d F _ { p + t ( q - p ) } ( q - p ) \mathrm { d } t$$
\item[III.B.2)] Show
$$\langle F ( q ) - F ( p ) , q - p \rangle \geqslant \alpha \| q - p \| ^ { 2 }$$
\end{enumerate}