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I am trying to prove the following inequality:

let [tex]z\in \mathbb{D}[/tex] then

[tex]\left| \frac{z}{\lambda} +1-\frac{1}{\lambda}\right|<1[/tex] if and only if [tex]\lambda\geq1.[/tex]

The direction if [tex]\lambda>1[/tex] is pretty easy, but I am wondering about the other direction.

Thanks in advance

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# Inequality in the unit disc

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