Hello. I need a bit of help or a tip maybe..

I am to show that if f is a holomorphic function that is of the form f(x+iy) = u(x) + i*v(y) where u and v are real functions,

then f(z) = λz+c where λ is a real number and c is a complex one.

How would I begin to prove this?

Thanks to everyone in advance.