A nonvanishing holomorphic function on a disc has a holomorphic logarithm

Given: A disc $D(a,r)$ with $r>0$ and a nowhere-zero holomorphic function $h$ on it. For $z,w\in D(a,r)$ and $t\in [0,1]$ the triangle inequality gives $\mathrm{\mid }((1-t)z+tw)-a\mathrm{\mid }\le (1-t)\mathrm{\mid }z-a\mathrm{\mid }+t\mathrm{\mid }w-a\mathrm{\mid }<r$, so every segment between two points of the disc stays in it; taking $z=a$ makes the disc star-shaped with respect to $a$ in the sense of Complex star-shaped and convex domains are the published Euclidean notions under the identification $\mathbb{C}={\mathbb{R}}^2$ and Star-shaped open subsets of Euclidean space, and [L4] makes the disc a connected, hence a complex, domain. Also $h^{\mathrm{\prime }}$ is itself holomorphic on the disc, because a holomorphic function has complex derivatives of all orders locally (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle), so the quotient rule makes $h^{\mathrm{\prime }}\mathrm{/}h$ holomorphic there, $h$ being nowhere zero (Linearity, product, reciprocal, and quotient rules for complex derivatives); and the complex chain rule, the derivative of $\exp$, and the exponential addition law are supplied by The chain rule for complex derivatives, The complex exponential is entire and its complex derivative is itself, and $\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential.

Every holomorphic function on an open set star-shaped with respect to $a$ has a primitive there (Every holomorphic function on a star-shaped domain has a primitive).

A holomorphic function whose derivative vanishes everywhere on a complex domain is constant (A holomorphic function with zero derivative on a domain is constant).

For every nonzero complex number $z$, the solutions of $\exp w=z$ are exactly $\mathrm{Log}z+2\pi ik$ with $k\in \mathbb{Z}$ (All logarithms of $z\mathrm{\ne }0$ are $\mathrm{Log}z+2\pi ik$, $k\in \mathbb{Z}$).

A segment $t\mapsto (1-t)v_0+t{v}_1$ that lies in a subset $A$ is a continuous path in $A$, and a path-connected subset of a topological space is a connected subset (A finite concatenation of straight segments in ${\mathbb{R}}^n$ is a continuous path, Every path-connected space is connected, and every path component lies inside a component, claim 2).