Inequality with complex numbers and a power

I am working on a problem in nonlinear analysis, and I would like to estimate a term that I can write, in abstract form, as follows: $$\left||z|^{2p-2}(\Re (z \overline{h}))^2 - |w|^{2p-2} (\Re (w \overline{h}))^2 \right|, \tag{1}$$ where $z$, $w$ and $h$ are complex numbers and $p>0$. My question is: does there exist a constant $C>0$ such that (1) is bounded by $$C |z-w|^{2p} |h|^2?$$ I am pretty sure it is possible to prove this upper bound, by it seems I can't.

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