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The very first question : If z1 and z2...

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Asked by vishakhachandan026 27th April 2020, 3:09 PM
Answered by Expert
Answer:
Q: if z1 and z2 are two complex numbers such that |z1|<1<|z2| then prove that |(1-z1 z2 bar)/(z1-z2)|<1
Solution:
Let z1=a+ib and z2=c+id
 
 rightwards double arrow z with bar on top subscript 2 equals c minus i d
G i v e n colon space vertical line z subscript 1 vertical line less than 1 less than vertical line z subscript 2 vertical line
rightwards double arrow a squared plus b squared less than 1 less than c squared plus d squared space... space left parenthesis i right parenthesis
T o space p r o v e colon space
open vertical bar fraction numerator 1 minus z subscript 1 z with bar on top subscript 2 over denominator z subscript 1 minus z subscript 2 end fraction close vertical bar less than 1
i. e. space vertical line 1 minus z subscript 1 z with bar on top subscript 2 vertical line less than vertical line z subscript 1 minus z subscript 2 vertical line
i. e. space vertical line 1 minus open parentheses a plus i b close parentheses open parentheses c minus i d close parentheses vertical line less than vertical line open parentheses a plus i b close parentheses minus open parentheses c plus i d close parentheses vertical line
i. e. space vertical line 1 minus open parentheses a c minus i a d plus i b c minus i squared b d close parentheses vertical line less than vertical line open parentheses a minus c close parentheses plus i open parentheses b minus d close parentheses vertical line
i. e. space open vertical bar 1 minus open parentheses a c plus b d close parentheses minus i open parentheses b c minus a d close parentheses close vertical bar less than vertical line open parentheses a minus c close parentheses plus i open parentheses b minus d close parentheses vertical line
i. e. space open square brackets 1 minus open parentheses a c plus b d close parentheses close square brackets squared plus open parentheses b c minus a d close parentheses squared less than open parentheses a minus c close parentheses squared plus open parentheses b minus d close parentheses squared
S o l v i n g space t h i s space w e space g e t
1 plus open parentheses a squared plus b squared close parentheses open parentheses c squared plus d squared close parentheses less than a squared plus b squared plus c squared plus d squared space w h i c h space i s space t r u e space f r o m space left parenthesis i right parenthesis
H e n c e comma space open vertical bar fraction numerator 1 minus z subscript 1 z with bar on top subscript 2 over denominator z subscript 1 minus z subscript 2 end fraction close vertical bar less than 1
 
Answered by Expert 28th April 2020, 12:14 PM
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