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Limit and diffirentiability of a function

  1. Mar 30, 2012 #1
    1. The problem statement, all variables and given/known data
    For complex numbers [itex]f[/itex] and [itex]g[/itex], and for [itex]1<p<\infty[/itex] we have [itex]\lim_{t\rightarrow 0}\dfrac{|f+tg|^p-|f|^p}{t}=|f|^{p-2}(\bar{f}g+f\bar{g})[/itex]; i.e., [itex]|f+tg|^p[/itex] is differentiable.

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

    2. Relevant equations

    3. The attempt at a solution

    I have try several attempts in the direction of manipulating the convex function [itex]|x|^p[/itex]. But no reasonable conclusions yet.
  2. jcsd
  3. Mar 30, 2012 #2
    So, I have made some progress by rewriting the problem and using L'hopitals rule. But I am still off by a factor of [itex]\dfrac{p}{2}[/itex].

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

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