Discussion
Let \(f(z)\) and \(g(z)\) be two non-constant analytic functions on a region G. Then which of the following is possible ?
Answer: C
The possible case is \(\bar{f} + \bar{g}\) is analytic.
\(\bar{f} + \bar{g} = \overline{f + g}\). A function and its conjugate are both analytic only when the function is constant, so this is analytic exactly when \(f + g\) is constant, for example \(g = -f + 1\) with \(f\) non-constant. So it can happen.
The others cannot. If \(f - \bar{g}\) were analytic, then \(\bar{g} = f - (f - \bar{g})\) would be analytic, forcing \(g\) constant; the same argument rules out \(f + \bar{g}\). And \(fg \equiv 0\) on a region would force \(f \equiv 0\) or \(g \equiv 0\) (the ring of analytic functions on a region has no zero divisors), contradicting non-constancy.
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