Question Number 156573 by MathSh last updated on 12/Oct/21
$$\mathrm{if}\:\:\:\mathrm{a};\mathrm{b};\mathrm{c};\mathrm{d}>\mathrm{1}\:\:\:\mathrm{then}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\sum}\:\mathrm{log}_{\boldsymbol{\mathrm{a}}} \:\left(\frac{\mathrm{b}^{\mathrm{3}} \:+\:\mathrm{c}^{\mathrm{3}} \:+\:\mathrm{d}^{\mathrm{3}} }{\mathrm{b}^{\mathrm{2}} \:+\:\mathrm{c}^{\mathrm{2}} \:+\:\mathrm{d}^{\mathrm{2}} }\right)\:\geqslant\:\mathrm{4} \\ $$