query: Sup of holomorphic function $f$ is $\geq 2|f'(0)|$
positive_passages.docid: 149634
positive_passages.text: $ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$
negative_passages.docid: 48222
negative_passages.text: Composition of a harmonic function with a holomorphic function is still harmonic
negative_passages.docid: 134195
negative_passages.text: Holomorphic function $f:B(0,1) \to B(1,4)$ with $f(0)=3$ and $f(\frac{1}{2})=1$
negative_passages.docid: 94680
negative_passages.text: Holomorphic function $f$ such that $f'(z_0) \neq 0$
negative_passages.docid: 93806
negative_passages.text: $\|f\|_{L^{p}}=\sup\left\{\int_{{\bf R}^{n}}|f(x)g(x)|~dx:\|g\|_{L^{q}}\le1\right\}$
negative_passages.docid: 93694
negative_passages.text: Boundary behaviour of holomorphic function on unit disk
negative_passages.docid: 133099
negative_passages.text: Decomposition of holomorphic function as a sum of entire functions
negative_passages.docid: 43136
negative_passages.text: Branch of the n-th root of a holomorphic function is holomorphic
negative_passages.docid: 59745
negative_passages.text: Behavior of holomorphic functions on the boundary of the unit disk
negative_passages.docid: 130537
negative_passages.text: Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
negative_passages.docid: 20192
negative_passages.text: Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
negative_passages.docid: 136818
negative_passages.text: There is a $C > 0$ such that $\vert f(z)-f(w)\vert \leq C\vert z-w\vert$ for a holomorphic map into $B(0, R)$.
negative_passages.docid: 74891
negative_passages.text: sum of holomorphic functions
negative_passages.docid: 115824
negative_passages.text: Holomorphic functions which preserve orientation
negative_passages.docid: 51659
negative_passages.text: If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$
negative_passages.docid: 44458
negative_passages.text: Uniform limit of holomorphic functions
negative_passages.docid: 4084
negative_passages.text: Does there exist a holomorphic function with the following property?