How to show $\frac{2x}{2+x}<\log(x+1)$
39504
Why does $ \frac{2x}{2+x}$ provide a particularly tight lower bound for $\ln(1+x)$ for small positive values of $x$?
13902
Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
81831
Evaluate $\int_0^{\infty}\frac{\log x}{1+e^x}\,dx$
82961
Show that $\left| \frac{1}{x}-\frac{1}{z} \right| \le \left|\frac{1}{x} - \frac{1}{y} \right| + \left|\frac{1}{y} - \frac{1}{z} \right|$.
133485
How to solve $\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$?
122971
how to prove that $\ln(1+x)< x$
39270
Showing that $e^x > 1+x$
142360
How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
18016
Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.
100995
For what $x$ : $2\sqrt{x}+\dfrac{1}{\sqrt{x+1}} \leq 2\sqrt{x+1}$
132452
Integral $\int_{\frac{1}{2}}^1 \frac{\ln x}{\sqrt{x^2+1}}dx$
26962
Proving an inequality involving the logarithm function: $\frac{1}{n+1} \leq \ln \left(1+\frac 1n\right) \leq \frac 1n$
38448
Show that a simple inequality holds: $a<b$ implies $\frac ab < \frac{a+1}{b+1}$
117262
Proving a logarithm inequality
150581
How do I show that $\frac a{1 - a^2} + \frac b{1 - b^2} + \frac c{1 - c^2} \ge \frac {3 \sqrt 3}2$
44233
How to prove this inequality $ x + \frac{1}{x} \geq 2 $
155005
How to show $\sum_{k=n}^\infty{\frac{1}{k!}} \leq \frac{2}{n!}$