query: How can we prove $(a_1+a_2+\cdots+a_n)^r\le a_1^r+ a_2^r+\cdots+a_n^r$ where each $a_i>0$ and $0<r<1$.
positive_passages.docid: 18477
positive_passages.text: $(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$ for $n$ integer, $n\geq 2$, $W\gt 0$ and $a$ constant, real, $0\lt a\lt 1$
negative_passages.docid: 7201
negative_passages.text: Prove that $x_1^2+x_2^2+x_3^2=1$ yields $ \sum_{i=1}^{3}\frac{x_i}{1+x_i^2} \le \frac{3\sqrt{3}}{4} $
negative_passages.docid: 24469
negative_passages.text: Prove $ 2<(1+\frac{1}{n})^{n}$
negative_passages.docid: 22666
negative_passages.text: Proving that if $A^n = 0$, then $I - A$ is invertible and $(I - A)^{-1} = I + A + \cdots + A^{n-1}$
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: 54900
negative_passages.text: How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$
negative_passages.docid: 68771
negative_passages.text: Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.
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: 78309
negative_passages.text: On an expansion of $(1+a+a^2+\cdots+a^n)^2$
negative_passages.docid: 25981
negative_passages.text: Prove that $\sum_{k=0}^n a_k x^k = 0$ has at least $1$ real root if $\sum_{k=0}^n \frac{a_k}{k+1} = 0$
negative_passages.docid: 76267
negative_passages.text: Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.
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: 130281
negative_passages.text: Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
negative_passages.docid: 21457
negative_passages.text: Prove $\sqrt{2\sqrt{3\sqrt{\cdots\sqrt{n}}}}<3$ by induction.
negative_passages.docid: 146546
negative_passages.text: Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$
negative_passages.docid: 133228
negative_passages.text: Prove $(|x| + |y|)^p \le |x|^p + |y|^p$ for $x,y \in \mathbb R$ and $p \in (0,1]$.
negative_passages.docid: 139503
negative_passages.text: How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?