Let \(U\subset\mathbb C\) be open and \(P\in U\). Let \(f\) be holomorphic on \(U\) and nonconstant on the component containing \(P\), with \(f(P)=f'(P)=0\). Use the argument principle to prove that there is \(\delta>0\) such that \(f^{-1}(Q)\) contains at least two distinct points whenever \(0<|Q|<\delta\).
The nonconstancy qualification is necessary: the function \(f=0\) satisfies \(f(P)=f'(P)=0\) but has no preimage of any nonzero value. Nonconstancy on an unrelated component of \(U\) does not supply the local hypothesis at \(P\).
Solution
1. In a sufficiently small disk, \(P\) is the only zero and the only critical point.
The Taylor expansion and the identity theorem give \[ f(z)=(z-P)^m h(z),\qquad m\geq2,\quad h(P)\ne0, \] with \(h\) holomorphic near \(P\) [@SS03]. Differentiating gives \[ f'(z)=(z-P)^{m-1}\bigl(mh(z)+(z-P)h'(z)\bigr). \] Both \(h\) and the bracketed factor are nonzero at \(P\). Choose \(r>0\) with \(\overline{D(P,r)}\subset U\) so small that these two factors are nonzero on a neighborhood of the closed disk. Then \(f\) has exactly \(m\) zeros in the disk counted with multiplicity, all at \(P\), and \(f'\) has no zero there except \(P\). Set \(\delta=\min_{|z-P|=r}|f(z)|>0\).
2. The argument-principle count stays equal to \(m\) under a small constant perturbation.
Fix \(0<|Q|<\delta\). For \(0\leq t\leq1\) and \(|z-P|=r\), \[ |f(z)-tQ|\geq\delta-|Q|>0. \] Thus the argument principle applies to \(f-tQ\) on this circle and gives its number of zeros, with multiplicity, as \[ N(t)=\frac1{2\pi i}\int_{|z-P|=r} \frac{f'(z)}{f(z)-tQ}\,dz\in\mathbb Z_{\geq0} \] [@SS03]. The uniform positive denominator bound makes this integral continuous in \(t\). A continuous integer-valued function on \([0,1]\) is constant, so \(N(1)=N(0)=m\).
3. These \(m\) preimages are distinct.
None of the zeros of \(f-Q\) is \(P\), since \(Q\ne0=f(P)\). Step 1 shows that the derivative is nonzero at each of them. Each zero is therefore simple by its Taylor expansion. Consequently the multiplicity count \(m\geq2\) from step 2 counts distinct points of \(D(P,r)\subset U\). All of them belong to \(f^{-1}(Q)\), proving the assertion for every \(0<|Q|<\delta\).