If $f: \mathbb{C_+} \to \mathbb{C_+}$ is analytic and $a\in \mathbb{C_+}$, is $|f'(a)| \leq \frac{\text{Im} (f(a))}{\text{Im}(a)}$?
132431
Proving $|f'(z)|\leq\frac{\text{Re}(f(z))}{\text{Re}(z)}$
51934
Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
135772
how to show image of a non constant entire function is dense in $\mathbb{C}$?
52305
Is there a general relation between $a/b$ and $(a+c)/(b+c)$ where $a,b,c > 0 $?
27772
Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant.
125566
The Identity Theorem for real analytic functions
51395
$f\in L^1(\mathbb{R}) \cap C^1(\mathbb{R})$ and $f'\in L^1(\mathbb{R})$
128658
Analyticity of $\overline {f(\bar z)}$ given $f(z)$ is analytic
154742
If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
127000
When does $V = \ker(f) \oplus \operatorname{im}(f)$?
148474
on the boundary of analytic functions
39583
ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
136818
There is a $C > 0$ such that $\vert f(z)-f(w)\vert \leq C\vert z-w\vert$ for a holomorphic map into $B(0, R)$.
23801
A reflection question about a complex function
9844
Proof composition of analytic functions is analytic
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
62901
Bound for Analytic Function on Unit Disk