10 ανισότητες από το mathematica

Nα αποδειχθούν οι ανισότητες:

1) ${\displaystyle 3(a^4+a^2+1)\ge (a^2+a+1{)}^2,a>0}$

2) ${\displaystyle (\frac{1}{|a|}+\frac{1}{|b|}+\frac{1}{|c|})(|a|+|b|+|c|)\ge 9,a,b,c\ne 0}$

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

4) ${\displaystyle \sqrt{a}+\sqrt{b}\le \sqrt{\frac{a^2}{b}}+\sqrt{\frac{b^2}{a}},a,b>0}$

5) $a^4+b^4+c^4+d^4\ge 4abcda,b,c,d\in R$

6) ${\displaystyle a^2{b}^2+b^2{c}^2+c^2{a}^2\ge (a+b+c)abca,b,c\in R}$

7) ${\displaystyle \frac{a^3}{b^2}+\frac{b^3}{c^2}+\frac{c^3}{a^2}\ge \frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a},a,b,c>0}$

8) $2a^4+1\ge 2a^3+a^2$

9) $3(a+b{)}^2+8b^2\ge 2(a-b{)}^2+4ab$

 10) ${\displaystyle \{\begin{array}{l}{\displaystyle (a+b+c)+2(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca})+\frac{8}{abc}\ge \frac{121}{12}}\\ \\ \alpha \nu a,b,c>0:\{\begin{array}{l}ab\ge 12\\ bc\ge 8\end{array}\end{array}}$

Πηγή: mathematica