Prove that $a b \leq \frac{a^{3}}{3}+\frac{b^{3 / 2}}{3 / 2}$
85039
Purely "algebraic" proof of Young's Inequality
78144
How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?
69673
Prove that $\left (\frac{a^2 + b^2 +c^2}{a+b+c} \right) ^ {(a+b+c)} > a^a b^b c^c$
56315
Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
41558
Prove that $\frac4{abcd} \geq \frac a b + \frac bc + \frac cd +\frac d a$
41293
Prove the inequalities $|z_1 +z_2| \geq \frac{1}{2}(|z_1|+|z_2|)|\frac {z_1} {|z_1|} +\frac {z_2} {|z_2|}|$
44705
Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
56101
Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
44233
How to prove this inequality $ x + \frac{1}{x} \geq 2 $
136403
Prove that $\sqrt{a}\leq\frac{1+a}{2}$
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
56166
Prove that $\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.$
81618
Prove $||a|-|b| | \leq |a-b|$
107494
prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$
51653
Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$
124868
Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
45619
Prove: $|a\sin x+b \cos x|\leq \sqrt{a^2+b^2}$