query: Proving $ \left| \sum_{k=1}^n a_k b_k \right| \leq \dfrac{1}{\alpha} \sum_{k=1}^n a_k^2 + \dfrac{\alpha}{4} \sum_{k=1}^n b_k^2 $ for any $\alpha > 0$
positive_passages.docid: 27796
positive_passages.text: inequality of absolute value problem
negative_passages.docid: 98768
negative_passages.text: Prove $\left(1+\frac{x}{n}\right)^n < e^x$, where $x$ is any positive real number and $n$ is any positive integer.
negative_passages.docid: 153226
negative_passages.text: Proof of $\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$
negative_passages.docid: 6100
negative_passages.text: Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
negative_passages.docid: 97119
negative_passages.text: Prove that $\prod_{i=1}^n(1+x_i)\leq \sum_{i=0}^n\frac{S^i}{i!}$, where $x_i\in\mathbb{R^+}$.
negative_passages.docid: 15259
negative_passages.text: Proving $\prod \limits_{k=0}^{n}(1-a_k) \geq1- \sum\limits_{k=0}^{n}a_k$
negative_passages.docid: 81842
negative_passages.text: How to prove by induction that $\frac{a^n+b^n}{2}\geq\left(\frac{a+b}{2}\right)^n$?
negative_passages.docid: 154716
negative_passages.text: Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
negative_passages.docid: 14197
negative_passages.text: show that $\sum_{i=0}^l \dfrac{{l\choose i}}{{n\choose k + i}} = \dfrac{n+1}{(n-l+1) {n-l\choose k}}$
negative_passages.docid: 69653
negative_passages.text: Inequality $\left(a-1+\frac{1}b\right)\left(b-1+\frac{1}c\right)\left(c-1+\frac{1}a\right)\leq1$
negative_passages.docid: 105641
negative_passages.text: Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
negative_passages.docid: 139503
negative_passages.text: How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
negative_passages.docid: 150565
negative_passages.text: Prove $(a+b)^\frac{1}{n} \le a^\frac{1}{n} + b^\frac{1}{n}$
negative_passages.docid: 30386
negative_passages.text: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$
negative_passages.docid: 60673
negative_passages.text: To prove $\sum_{cyc}\frac{1}{a^3+b^3+abc} \le \frac{1}{abc}$
negative_passages.docid: 130553
negative_passages.text: How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?
negative_passages.docid: 90902
negative_passages.text: Prove $\frac{a+b+c}{abc} \leq \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}$.