query: Proof that $| e^x-e^y|\leq | x-y|$ for $\Re(x), \Re(y) < 0$
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: 117680
negative_passages.text: Prove that $|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$
negative_passages.docid: 14677
negative_passages.text: Proving the exponential inequality: $x^y+y^x\gt1$
negative_passages.docid: 31846
negative_passages.text: Does $\neg(x > y)$ imply that $y \geq x$?
negative_passages.docid: 149287
negative_passages.text: Prove that $A,B>0$ and $A_{ij}0$
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 142922
negative_passages.text: How to prove that $e^x$ is convex?
negative_passages.docid: 73367
negative_passages.text: On $E[E[Y|X]|X]= E[Y|X]$
negative_passages.docid: 49240
negative_passages.text: $\forall x,y \in \mathbb{R}:\vert\cos x- \cos y\vert\leq\vert x-y\vert$
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: 105047
negative_passages.text: Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 149889
negative_passages.text: Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$
negative_passages.docid: 138931
negative_passages.text: Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
negative_passages.docid: 81618
negative_passages.text: Prove $||a|-|b| | \leq |a-b|$
negative_passages.docid: 129584
negative_passages.text: How prove $E|X|^p<\infty$ and $E|Y|^p<\infty$, if $E|X+Y|^p<\infty$
negative_passages.docid: 98921
negative_passages.text: Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?