Suppose $a \in \mathbb{C}$, $|a| < 1$, and $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. How to prove dependence of $|f(z)|$ on $|z|$?
88801
Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?
63860
Measure of Dependence Between Two $\sigma$-algebras
35556
$f(z)=\frac{z}{1+|z|} $ Is 1-1?
20955
Let $f(z)$ be analytic on $\mathbb{D}$ = {${z\in\mathbb{C}:|z-1|<1}$} such that $f(1) = 1$, if $f(z) = f(z^2)$ for all $z\in\mathbb{D}$,
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
92376
Let $f:A \rightarrow B$ and C ⊂A. Define f[C]={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
68409
Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
61640
Prove c exists for f:[0,1] -> R
141115
$f$ be an analytic function defined on $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$
126736
Show that $\lim_{\epsilon\to0^{+}}\frac{1}{2\pi i}\int_{\gamma_\epsilon}\frac{f(z)}{z-a} \, dz=f(a)$
13270
Prove $\exists \xi \in (a,b):f'(\xi)<1+f^2(\xi)$
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
106284
Establishing a Basis for $F[x]/(f(x))$
117640
Prove that $\lim \inf \dfrac{x_{n+1}}{x_n}\leq \lim \inf x_n^{1\over n} \leq \lim \sup x_n^{1\over n} \leq \lim \sup \dfrac{x_{n+1}}{x_n}. $
80255
How to prove that $|a-b|<\epsilon$ implies $|b|-\epsilon<|a|<|b|+\epsilon$?
74392
Show that a function satisfies the Cauchy-Riemann function if and only if $\frac{df}{d\overline{z}}=0$
158561
Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is