Why does this inequality hold? ("Complex Analysis" by Kunihiko Kodaira.)

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I am reading "Complex Analysis" by Kunihiko Kodaira.

I cannot understand why the following inequality holds.

Please tell me the reason why this inequality holds.

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Expand the product inside the modulus, cancel the terms inside the modulus. Then use the property of modulus to get the sum of absolute values and then again write it as a difference of the given two terms at the end.