Limit and diffirentiability of a function

[lmedin02]

Homework Statement

For complex numbers f and g, and for 1<p<\infty we have \lim_{t\rightarrow 0}\dfrac{|f+tg|^p-|f|^p}{t}=|f|^{p-2}(\bar{f}g+f\bar{g}); i.e., |f+tg|^p is differentiable.

I would like to show that the above statement is true.

Homework Equations

The Attempt at a Solution

I have try several attempts in the direction of manipulating the convex function |x|^p. But no reasonable conclusions yet.

 

[lmedin02]

So, I have made some progress by rewriting the problem and using L'hopitals rule. But I am still off by a factor of \dfrac{p}{2}.

rewriting: |f+tg|^2=f\bar{f}+tf\bar{g}+t\bar{f}g+t^2g\bar{g}

When I apply L'Hopitals rule I get \dfrac{p}{2}|f|^{p-2}(\bar{f}g+f\bar{g})