$$x^2+y^2\geq 2xy$$ $$x^3y+xy^3\leq x^4+y^4$$ $$(x+y)(y+z)(z+x)\geq 8xyz$$ $$4(x^3+y^3)\geq (x+y)^3$$ $$(x+y+z)^3\geq 27xyz$$ $$xyz\geq (x+y-z)(y+z-x)(z+x-y)$$ $$x^2(1+y^2)+y^2(1+z^2)+z^2(1+x^2)\geq 6xyz$$ $$8(x^3+y^3+z^3)^2\geq 9(x^2+yz)(y^2+zx)(z^2+xy)$$ $$4(x^5+y^5+z^5+w^5)\geq (x^3+y^3+z^3+w^3)(x^2+y^2+z^2+w^2)$$ $$(b+c+d)(c+d+a)(d+a+b)(a+b+c)\geq 81abcd$$ $$(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\geq 9x^2y^2z^2$$ $$6xyz\leq xy(x+y)+yz(y+z)+zx(z+x)$$ $$x^2y^2+y^2z^2+z^2x^2\geq xyz(x+y+z)$$ $$(x+y+z)^3\geq (x+y-z)(y+z-x)(z+x-y)$$ $$27(x^4+y^4+z^4)\geq (x+y+z)^4$$ $$(x+y+z+w)(x^3+y^3+z^3+w^3)\geq (x^2+y^2+z^2+w^2)^2$$ $$x(x-y)(x-z)+y(y-z)(y-x)+z(z-x)(z-y)\geq 0$$ $$x^2(x-y)(x-z)+y^2(y-z)(y-x)+z^2(z-x)(z-y)\geq 0$$ $$x^5(x-y)(x-z)+y^5(y-z)(y-x)+z^5(z-x)(z-y)\geq 0$$ $$9(x^6+y^6+z^6)\geq (x^3+y^3+z^3)(x^2+y^2+z^2)(x+y+z)$$ $$16(x^3+y^3+z^3+w^3)\geq (x+y+z+w)^3$$ $$(x+y-z)^2+(y+z-x)^2+(z+x-y)^2\geq xy+yz+zx$$ $$x^3+y^3+z^3+3xyz\geq x^2(y+z)+z^2(x+y)+y^2(z+x)$$ $$6xyz\leq x^2(y+z)+y^2(z+x)+z^2(x+y)$$ $$x^2(y+z)+y^2(z+x)+z^2(x+y)\leq 2(x^3+y^3+z^3)$$ $$abcd\geq (b+c+d-2a)(c+d+a-2b)(d+a+b-2c)(a+b+c-2d)$$ $$3(x^2y+y^2z+z^2x)(xy^2+yz^2+zx^2)\geq xyz(x+y+z)^3$$ $$(x^3+y^3)^2\leq (x^2+y^2)^3$$ $$(x^7+y^7)^3\leq (x^3+y^3)^7$$ $$1/x+1/y+1/z\geq 9/(x+y+z)$$ $$x/(y+z)+y/(z+x)+z/(x+y)\geq 3/2$$ $$x^{-5}(x-y)(x-z)+y^{-5}(y-z)(y-x)+z^{-5}(z-x)(z-y)\geq 0$$ $${1\over x}+{1\over y}+{1\over z}\leq {x^8+y^8+z^8\over x^3y^3z^3}$$ $${1\over x+y+z}+{1\over y+z+w}+{1\over z+w+x}+{1\over w+x+y}\geq {16/3\over x+y+z+w}$$ $$(x^2+y^2)/(x+y)+(y^2+z^2)/(y+z)+(z^2+x^2)/(z+x)\geq x+y+z$$ $$(a^3+b^3+c^3+d^3+e^3)(1/a+1/b+1/c+1/d+1/e)\geq 5(a^2+b^2+c^2+d^2+e^2)$$ $$3/(b+c+d)+3/(c+d+a)+3/(d+a+b)+3/(a+b+c)\geq 16/(a+b+c+d)$$