query: $|\mathrm{e}^{z}- \mathrm{e}^{w}| \leq |z-w|$ for $\{z \in \mathbb{C} : \mathrm{Re}z \leq 0 \}$
positive_passages.docid: 16852
positive_passages.text: $|e^a-e^b| \leq |a-b|$ for complex numbers with non-positive real parts
negative_passages.docid: 125209
negative_passages.text: Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
negative_passages.docid: 63124
negative_passages.text: Show that $ \lVert A \rVert_2^2 \leq \lVert A \rVert _1 \lVert A \rVert _ \infty $
negative_passages.docid: 95626
negative_passages.text: $\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
negative_passages.docid: 155935
negative_passages.text: Examples where $H\ne \mathrm{Aut}(E/E^H)$
negative_passages.docid: 152982
negative_passages.text: Show that the range of $p(z)e^{q(z)}$ for nonconstant polynomials $p$ and $q$ is all of $\mathbb{C}$.
negative_passages.docid: 83713
negative_passages.text: If $z,w\in \mathbb{C}^{\times}$, then $|z+w|=|z|+|w|$ iff $w=tz$ for some $t\gt 0$
negative_passages.docid: 111056
negative_passages.text: Harmonic conjugate of $\ln|z|$ in $\mathbb C \setminus \{0\}$
negative_passages.docid: 155311
negative_passages.text: How to get a set $E$ such that $0<\mu(E\cap I)<\mu(E)$ for all interval $I$?
negative_passages.docid: 61262
negative_passages.text: Is $\cot(z)-\frac1z$ bounded on a given circle?
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 69656
negative_passages.text: Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
negative_passages.docid: 68484
negative_passages.text: Revision of Borel-Cantelli, $(X_n)_{n \geq 1}$ sqn of $\geq 0$ identical RVs with $E(X) < \infty$ then $X_n/n \to 0$ a.s., are my arguments correct?
negative_passages.docid: 127047
negative_passages.text: How to show that $\sum_{i=1}^n | \langle f, f_i\rangle |^2 \leq \Vert f \Vert^2$
negative_passages.docid: 49061
negative_passages.text: Show that $\int_{\partial P}z\frac {f'(z)} {f(z)} dz $ is on the lattice $\Lambda$
negative_passages.docid: 153107
negative_passages.text: Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
negative_passages.docid: 105527
negative_passages.text: Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$