query: If $f(\frac{t_1+t_2}{2}) \leq \frac{f(t_1)+f(t_2)}{2}$, show that $f(\frac{t_1+t_2+ \cdots +t_n}{n})\leq \frac{f(t_1)+f(t_2)+\cdots f(t_n)}{n}$
positive_passages.docid: 16943
positive_passages.text: Midpoint-Convex and Continuous Implies Convex
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 110153
negative_passages.text: How to prove this inequality?$\frac{1}{n+1} + \frac{1}{n+2} + \cdots+\frac{1}{n+n} + \frac{1}{4n} > \ln 2$
negative_passages.docid: 9044
negative_passages.text: Convergence of the sequence $(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})$
negative_passages.docid: 14952
negative_passages.text: Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$
negative_passages.docid: 27304
negative_passages.text: Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction
negative_passages.docid: 72600
negative_passages.text: Proof that $n!\leq {(\frac{n+1}{2})}^{n}$
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: 121455
negative_passages.text: $f(x)$ if $f(xy)=f(x) +f(y) +\frac{x+y-1}{xy}$
negative_passages.docid: 86209
negative_passages.text: Show this inequality $\frac{n}{a_1 - a_0} + \frac{n - 1}{a_2 - a_1} + \cdots + \frac{1}{a_n - a_{n-1}} \ge \sum_{k=1}^n \frac{k^2}{a_k}$
negative_passages.docid: 103947
negative_passages.text: Inequality : $\sum_{k=1}^n x_k\cdot \sum_{k=1}^n \frac{1}{x_k} \geq n^2$
negative_passages.docid: 33097
negative_passages.text: Proving an inequality without an integral: $\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}$
negative_passages.docid: 95266
negative_passages.text: Show $|t|\leq \pi \Rightarrow |e^{it}-1|\geq 2|t|/ \pi$
negative_passages.docid: 36837
negative_passages.text: Prove $\frac{1+a^{2}}{1+b+c^{2}} +\frac{1+b^{2}}{1+c+a^{2}} +\frac{1+c^{2}}{1+a+b^{2}} \geq 2$
negative_passages.docid: 124654
negative_passages.text: Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac1{2^n - 1} \le n$
negative_passages.docid: 101918
negative_passages.text: Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.
negative_passages.docid: 104853
negative_passages.text: Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$