Let $I$ be an interval of $\mathbf { R }$. Let $f : I \rightarrow \mathbf { R }$ be a convex function. Show that, for all $p \in \mathbf { N } ^ { \star }$, for all $\left( \lambda _ { 1 } , \ldots , \lambda _ { p } \right) \in \left( \mathbf { R } _ { + } \right) ^ { p }$ such that $\sum _ { i = 1 } ^ { p } \lambda _ { i } = 1$ and for all $\left( x _ { 1 } , \ldots , x _ { p } \right) \in I ^ { p }$, we have: $$f \left( \sum _ { i = 1 } ^ { p } \lambda _ { i } x _ { i } \right) \leq \sum _ { i = 1 } ^ { p } \lambda _ { i } f \left( x _ { i } \right)$$ Hint: You may proceed by induction on $p$.
Let $I$ be an interval of $\mathbf { R }$. Let $f : I \rightarrow \mathbf { R }$ be a convex function. Show that, for all $p \in \mathbf { N } ^ { \star }$, for all $\left( \lambda _ { 1 } , \ldots , \lambda _ { p } \right) \in \left( \mathbf { R } _ { + } \right) ^ { p }$ such that $\sum _ { i = 1 } ^ { p } \lambda _ { i } = 1$ and for all $\left( x _ { 1 } , \ldots , x _ { p } \right) \in I ^ { p }$, we have:
$$f \left( \sum _ { i = 1 } ^ { p } \lambda _ { i } x _ { i } \right) \leq \sum _ { i = 1 } ^ { p } \lambda _ { i } f \left( x _ { i } \right)$$
Hint: You may proceed by induction on $p$.