If $f$ is holomorphic on the unit disk and $f(0)=0$, then $\lvert f(z) + f(-z) \rvert \le 2 \lvert z \rvert^2$
51659
If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$
39583
ii) Show that $u$ is not a real part of the function which analytic on $\mathbb{C} \backslash \lbrace 0 \rbrace$
108846
Estimate of analytic function in a unit disk
110219
Bound on the derivative of a holomorphic function at $0$
103706
If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
74051
No analytic function in $\{|z|<1\}$ such that $f(0)=1$ and $|f(z)|\ge 1+|z|^2$. Proof strategy
103125
Show that a non-constant holomorphic function has zeroes in a compact set
154742
If $f(z)$ analytic in the domain $D$ and $f'(z)=0$ so $f(z)=$constant?
106175
A function is analytic if and only if it is holomorphic
143920
zeroes of non-zero holomorphic function form an atmost countable set
93694
Boundary behaviour of holomorphic function on unit disk
19750
If a function $f$ is holomorphic on the closed unit disk centered at the origin and is real valued whenever $|z| = 1$, then $f$ is constant.
149634
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
123467
Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$.
43440
Complex Analysis: Showing analytic function is zero
18877
$f(z)$ and $\overline{f(\overline{z})}$ simultaneously holomorphic
95632
Where is this function holomorphic?