query: Show that if $|f(z)| \leq M$ for $z \in \partial D$ for $z \in \mathbb{C}$ and M being a constant, then $|f(z)| \leq M$ for all $z \in D$
positive_passages.docid: 148470
positive_passages.text: bound of the function
negative_passages.docid: 106767
negative_passages.text: $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$
negative_passages.docid: 20955
negative_passages.text: 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 $z\in\mathbb{D}$,
negative_passages.docid: 72856
negative_passages.text: Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
negative_passages.docid: 84103
negative_passages.text: Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem
negative_passages.docid: 45676
negative_passages.text: Showing that $f$ is linear function if $\forall z \in \mathbb{C}$, $|f(z)| \leq 1 + |z|$.
negative_passages.docid: 106290
negative_passages.text: Prove a function is constant. (complex analysis)
negative_passages.docid: 153855
negative_passages.text: Complex analysis, showing a function is constant
negative_passages.docid: 143000
negative_passages.text: Show that there exists a $c\in \mathbb{C}$ such that $f(z)=c \bar{z}$ for all $\vert z\vert=1$.
negative_passages.docid: 74202
negative_passages.text: Show that $|f(z_1)|=|f(z_2)|≤|z_1|^2=|z_2|^2$.
negative_passages.docid: 131754
negative_passages.text: Prove $ \forall c \in \mathbb{R^{+}}, \forall B \in \mathbb{N}, \exists n \in \mathbb{N}, (n \geq B) \wedge (\sqrt{2n} > c \ln(n+2)) $?
negative_passages.docid: 15441
negative_passages.text: Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
negative_passages.docid: 63526
negative_passages.text: Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
negative_passages.docid: 80826
negative_passages.text: To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
negative_passages.docid: 107518
negative_passages.text: Entire function $f$ such that $|f(z)| ≤ K |z|^3$ for $|z|\ge1$ and $f(z) = f(iz)$ for every $z∈C$
negative_passages.docid: 48275
negative_passages.text: $f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
negative_passages.docid: 90343
negative_passages.text: Proving the inequality $|e^z-1|\leq e^{|z|}-1$