query: Exponentials inequality: $|e^x + e^y| \leq |x - y|$ for $x,y<0$.
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: 95293
negative_passages.text: Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
negative_passages.docid: 32407
negative_passages.text: Prove $\exp(x+y) = \exp(x) \exp(y)$ for $\exp(x) = \sum_{n=0}^\infty \frac {x^n}{n!}$
negative_passages.docid: 72081
negative_passages.text: $\forall x,y>0, x^x+y^y \geq x^y + y^x$
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: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
negative_passages.docid: 118613
negative_passages.text: Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
negative_passages.docid: 62605
negative_passages.text: How to find $\mathbb{E}(X\mid\mathbf{1}_{X<Y})$ where $X,Y$ are i.i.d exponential variables?
negative_passages.docid: 156669
negative_passages.text: Trying to prove a simple (?) inequality
negative_passages.docid: 148451
negative_passages.text: Polynomial inequality proof
negative_passages.docid: 140988
negative_passages.text: Prove $\exp(x)\exp(y)=\exp(y)\exp(x)$ if $[x,y]=0$.
negative_passages.docid: 116387
negative_passages.text: How to prove that $\Gamma(x) \leq x^x$ for $x \geq 1$
negative_passages.docid: 19078
negative_passages.text: Prove an interpolation inequality
negative_passages.docid: 105316
negative_passages.text: Prove : $|x - y| \leqslant \epsilon$ $\forall \epsilon > 0$ iff $x = y$
negative_passages.docid: 101499
negative_passages.text: Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series
negative_passages.docid: 23264
negative_passages.text: Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $
negative_passages.docid: 135550
negative_passages.text: Inequality question­