suppose that :

f(z)= u(x,y) + i v(x,y) is an analytic function on a domain D . prove that if thee are real constants a,b,c (not all zeros ) with :

a u(x,y) + b v(x,y) =c for all (x,y) D then f is constant on D

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my ideas:

f costant implise f prime is zero ?!