let f be analytic and not constant on a domain G. show that the area of...

let f be analytic and not constant on a domain G. show that the area of f(G)is strictly positive

(complex variables)

Solutions

Expert Solution

Since is analytic and not constant on a domain G so obeys open mapping theorem which states that ,

Open mapping theorem : If is a non-constant analyic open map on a domain G then image of a open set is open that is if is open then is open .

Let , since G is a domain so there exist a open set U containing such that .

By open mapping theorem is open and .

Since is open and so there exist a such that the open ball .......(i)

Now , ........(ii)

From (i) and (ii) we get ,

Hence area of is strictly positive .

If you have any doubt or need more clarification at any step please comment .

Colby Messinger answered 1 year ago

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