Prove Complex Inequality: Re z < 0

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SUMMARY

The discussion focuses on proving the inequality \(\left| e^z - 1 \right| < \left| z \right|\) for complex numbers \(z\) where \(\text{Re } z < 0\). Participants suggest using the identities \(\left| z \right|^2 = z \bar{z}\) and \(\bar{e^z} = e^{\bar{z}}\) to manipulate the inequality. Squaring both sides is recommended to simplify the proof. The conversation emphasizes the importance of ensuring both sides remain positive during the proof process.

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  • Familiarity with exponential functions of complex variables
  • Knowledge of complex conjugates and their applications
  • Ability to manipulate inequalities involving complex magnitudes
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smyroosh
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How to prove the following inequality: for complex z such that Re z < 0 :

[tex]\left| e^z-1\right| < \left| z\right|[/tex] ?
 
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use the identities

[tex]\left|z\right|^{2}=z\bar{z}[/tex]

and

[tex]\bar{e^{z}}=e^{\bar{z}}[/tex]

since both sides of the inequality are positive
you can square it up
 
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