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For $\alpha = 0,1,2,3$, does this inequality always hold for any complex number $z_1, z_2$? $$ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq ( |z_1 - z_2 |^2 + |z_2| ^2 )|z_1 - z_2|$$ $\overline{z}$ denotes the complex conjugate of $z$.

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The inequality does not hold in the reals. Let $z_1=2$ and $z_2=1$.

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  • $\begingroup$ It think the case $z_1 = 2, z_2 = 0$ hold.. $\endgroup$
    – Luma
    Apr 27, 2013 at 2:42
  • $\begingroup$ Sorry, typo. It was $z_2=1$. $\endgroup$ Apr 27, 2013 at 2:52
  • $\begingroup$ Thank you for the answer! p.s. I wrote a new question for weaker version: $ | \;\overline{z_1}^{3-\alpha} z_1^\alpha - \overline{z_2}^{3-\alpha} z_2 ^\alpha | \leq C ( |z_1 - z_2 |^2 + |z_2| ^2 )|z_1 - z_2|$ $\endgroup$
    – Luma
    Apr 27, 2013 at 3:03

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