Inequality $ab\le \frac{a^p}{p}+\frac{b^q}{q}$
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$
91617
Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
68389
Proving $ \frac{(a-b)^2}{8a}\leqslant\frac{a+b}{2} - \sqrt{ab} \leqslant\frac{(a-b)^2}{8b}$ when $ 0<b\le a$
52204
Proof of one inequality $a+b+c\leq\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}$
73987
Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
126557
Proving the inequality $\sum_{\text{cyc}} \frac{a}{a^2+b^3+c^3}\le\frac1{5abc}$
72470
Prove that $2|ab| \leq a^2 + b^2$ and $|a|+|b| \leq \sqrt {2}(a^2+b^2)^{1/2}$
72313
Show that for all real numbers $a$ and $b$, $\,\, ab \le (1/2)(a^2+b^2)$
157416
Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$
26876
How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?
154716
Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
140599
Inequality Proof in Real Numbers: If $p+q=1$ then $pq\le\frac14$
15116
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
5465
This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$
91131
Question regarding an inequality
136403
Prove that $\sqrt{a}\leq\frac{1+a}{2}$
156513
Prove that inequality $\sum _{cyc}\frac{a^2+b^2}{a+b}\le \frac{3\left(a^2+b^2+c^2\right)}{a+b+c}$