Useful properties in inequalities

by henderson, Sep 26, 2016, 3:46 PM

$\bf\color{blue} Useful\ properties \ in \ inequalities$ $\color{green}\blacktriangleright$ $ab+bc+ca+abc=4\iff \frac{a}{a+2}+\frac{b}{b+2}+\frac{c}{c+2}=1.$

$\color{green}\blacktriangleright$ $(x-y)(y-z)(z-x)= (xy^2 + yz^2 + zx^2)-(x^2y + y^2z + z^2x).$

$\color{green}\blacktriangleright$ $(x-y)(y-z)(z-x)(x + y + z) = (xy^3 + yz^3 + zx^3)-(x^3y + y^3z + z^3x).$

$\color{green}\blacktriangleright$ $\frac{(a^2+bc)(b^2+ca)}{(c+a)(b+c)}+\frac{(b^2+ca)(c^2+ab)}{(a+b)(c+a)}+\frac{(c^2+ab)(a^2+bc)}{(b+c)(a+b)}=a^2+b^2+c^2.$

$\color{green}\blacktriangleright$ $\sum_{cyc} \frac{a^2}{a+b} = \sum_{cyc} \frac{b^2}{a+b}.$

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Useful properties in inequalities

by henderson, Sep 26, 2016, 3:46 PM

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