\(\left(a^5+b^5\right)\left(a+b\right)\ge\left(a^4+b^4\right)\left(a^2+b^2\right)\) (a.b>0)

\(\frac{a^2+3}{\sqrt{a^2+2}}>2\)

\(\frac{1}{1+a^2}+\frac{1}{1+b^2}\ge\frac{2}{1+ab}\)

\(a^4+b^4\le\frac{a^6}{b^2}+\frac{b^6}{a^2}\)

\(a^3+b^3+c^3\ge3abc\)

\(\frac{a^3+b^3}{2}\ge\left(\frac{a+b}{2}\right)^{^{ }3}\)

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}>\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\)

\(a^2+b^2+c^2+d^2+e^{^{ }2}\ge a\left(b+c+d+e\right)\)