Contour integrals around concentric circles agree, by deformation

Let $f$ be analytic on $\mathbf C-\{0\}$ and let $\nu_1,\nu_2:[0,1]\to\mathbf C-\{0\}$ be defined by $\nu_1(t)=e^{2\pi it}$ and $\nu_2(t)=10e^{2\pi it}$. Suppose $I_1=\int_{\nu_1}f(z)dz$ and $I_2=\int_{\nu_2}f(z)dz$. Then: