query: Show that $\mathrm{min}(\frac{a}{b},\frac{a'}{b'}) \leq \frac{a+a'}{b+b'} \leq \mathrm{max}(\frac{a}{b},\frac{a'}{b'})$
positive_passages.docid: 20040
positive_passages.text: shorter proof of generalized mediant inequality?
negative_passages.docid: 87303
negative_passages.text: Show that $\frac{a}{c} + \frac{b}{d} +\frac{c}{a} + \frac{d}{b}\le-12$
negative_passages.docid: 80350
negative_passages.text: $\limsup \frac{a_n}{b_n}0$ s.t $a_n\le Mb_n$
negative_passages.docid: 149700
negative_passages.text: Prove that $\frac{a_1^2}{a_1+a_2}+\frac{a_2^2}{a_2+a_3}+ \cdots \frac{a_n^2}{a_n+a_1} \geq \frac12$
negative_passages.docid: 102817
negative_passages.text: $A,B\in M_{n}(\mathbb{R})$ so that $A>0, B>0$, prove that $\det (A+B)>\max (\det(A), \det(B))$
negative_passages.docid: 56315
negative_passages.text: Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
negative_passages.docid: 99165
negative_passages.text: How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?
negative_passages.docid: 7974
negative_passages.text: lim sup inequality $\limsup ( a_n b_n ) \leq \limsup a_n \limsup b_n $
negative_passages.docid: 137864
negative_passages.text: Prove that $ \left(1+\frac a b \right) \left(1+\frac b c \right)\left(1+\frac c a \right) \geq 2\left(1+ \frac{a+b+c}{\sqrt[3]{abc}}\right)$.
negative_passages.docid: 88687
negative_passages.text: Proof of the inequality $||a+b|^p - |a|^p| \leq \epsilon |a|^p + C_{\epsilon} |b|^p$
negative_passages.docid: 62173
negative_passages.text: How do I show that $\int_0^\infty \frac{\sin(ax) \sin(bx)}{x^{2}} \, \mathrm dx = \pi \min(a,b)/2$
negative_passages.docid: 41293
negative_passages.text: Prove the inequalities $|z_1 +z_2| \geq \frac{1}{2}(|z_1|+|z_2|)|\frac {z_1} {|z_1|} +\frac {z_2} {|z_2|}|$
negative_passages.docid: 15441
negative_passages.text: Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
negative_passages.docid: 137791
negative_passages.text: Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
negative_passages.docid: 122326
negative_passages.text: Proving $|P(A\cap B)-P(A)P(B)|\leq \frac{1}4$
negative_passages.docid: 72470
negative_passages.text: Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
negative_passages.docid: 136403
negative_passages.text: Prove that $\sqrt{a}\leq\frac{1+a}{2}$