Let , then for all ,
If , then
Proof. We consider two cases separately.
Case . The strict convexity of implies that for any
Writing instead of we obtain
or
Using Clarkson inequality
we arrive at
or
Repeating this procedure we ca replace the constant
by
By iteration one obtains the constant
Case . Fix and expand the real function
Using Taylor formula
Then, provided for all ,
At the same time
Schwartz inequality yields
Now,
Lastly