In this entry, we address the following question
Let be a derivable function, with a continuous derivative on . Prove that if , then
Solution. The best possible constant comes from the -norm of . Since
this constant is indeed . We get equality if and only if is a real multiple of .
Define . The LHS is
and the RHS becomes
In the language of inner products, we want to show that
The standard C-S inequality gives
and we are done.
Source: RMO 2008, Grade 12, Problem 2.