query: Prove inequality with $e^x$ and $\ln$ on the same side
positive_passages.docid: 155232
positive_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 148451
negative_passages.text: Polynomial inequality proof
negative_passages.docid: 2708
negative_passages.text: Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$
negative_passages.docid: 108995
negative_passages.text: Prove $|e^{i\theta} -1| \leq |\theta|$
negative_passages.docid: 87567
negative_passages.text: How to prove $1+x \leq e^x~\forall x \in \mathbb{R}?$
negative_passages.docid: 80556
negative_passages.text: Demonstrate by induction the inequality: $\ln(1+n)\leq\sum_{i=1}^{n}\frac{1}{i}\leq1+\ln(n)$
negative_passages.docid: 118613
negative_passages.text: Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
negative_passages.docid: 45742
negative_passages.text: Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
negative_passages.docid: 78174
negative_passages.text: How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
negative_passages.docid: 142922
negative_passages.text: How to prove that $e^x$ is convex?
negative_passages.docid: 6449
negative_passages.text: Prove logarithmic inequality with greatest integer function.
negative_passages.docid: 125175
negative_passages.text: convexity proof of a function including ln and sums
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 20626
negative_passages.text: Prove that $x^y < y^x$
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 90796
negative_passages.text: Inequality Regarding e
negative_passages.docid: 142785
negative_passages.text: How can I prove Holder inequality for $0<p<1$?