Question Number 142906 by liberty last updated on 07/Jun/21
$${If}\:{abc}=\mathrm{1}\:{and}\:{a},{b},{c}>\mathrm{0}\:{prove} \\ $$ $${that}\:\frac{{a}}{{b}^{\mathrm{2}} \left({c}+\mathrm{1}\right)}+\frac{{b}}{{c}^{\mathrm{2}} \left({a}+\mathrm{1}\right)}+\frac{{c}}{{a}^{\mathrm{2}} \left({b}+\mathrm{1}\right)}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$
Answered by Snail last updated on 07/Jun/21
$${Let}\:{us}\:{recall}\:{Titu}'{s}\:{Lemma} \\ $$ $$\Sigma\frac{{x}_{{i}} ^{\mathrm{2}} }{{y}_{{i}} }\geqslant\:\frac{\left(\Sigma{x}_{{i}} \right)^{\mathrm{2}} }{\Sigma{y}_{{i}} } \\ $$ $${Denote}\:{give}\:\:{ineauality}\:{by}\:{E} \\ $$ $${As}\:{abc}=\mathrm{1}\:\:{Multiplying}\:{the}\:{inequality}\:{both}\:{side} \\ $$ $${by}\:{a}^{\mathrm{2}} {b}^{\mathrm{2}} {c}^{\mathrm{2}} \:{and}\:{using}\:{its}\:{valud}\:\mathrm{1}\:{we}\:{ca}\:{transform} \\ $$ $${it}\:{and}\:{get}\:... \\ $$ $$\frac{{a}^{\mathrm{3}} {c}^{\mathrm{2}} }{{c}+\mathrm{1}}+\frac{{b}^{\mathrm{3}} {a}^{\mathrm{2}} }{{a}+\mathrm{1}}+\frac{{c}^{\mathrm{3}} {b}^{\mathrm{2}} }{{b}+\mathrm{1}}\geqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$ $${now}\:{let}\:{x}_{\mathrm{1}} ={a}^{\mathrm{2}} {c}^{\mathrm{2}} \:{x}_{\mathrm{2}} ={b}^{\mathrm{2}} {a}^{\mathrm{2}} \:{x}_{\mathrm{3}} ={c}^{\mathrm{2}} {b}^{\mathrm{2}} \\ $$ $${y}_{\mathrm{1}} =\frac{{c}+\mathrm{1}}{{a}}\:\:\:{y}_{\mathrm{2}} =\frac{{a}+\mathrm{1}}{{b}}\:\:{y}_{\mathrm{3}} =\frac{{b}+\mathrm{1}}{{c}} \\ $$ $$ \\ $$ $${E}\:\geqslant\frac{\left({a}^{\mathrm{2}} {c}^{\mathrm{2}} +{b}^{\mathrm{2}} {c}^{\mathrm{2}} +{c}^{\mathrm{2}} {a}^{\mathrm{2}} \right)^{\mathrm{2}} }{\frac{{c}+\mathrm{1}}{{a}}+\frac{{b}+\mathrm{1}}{{c}}+\frac{{a}+\mathrm{1}}{{b}}}\:\:={Z}\left({Let}\right) \\ $$ $${Now}\:{usi}\:{ng}\:{the}\:{fact}\:{that}\:\:\:\:\left({a}^{\mathrm{2}} {c}^{\mathrm{2}} +{b}^{\mathrm{2}} {c}^{\mathrm{2}} +{a}^{\mathrm{2}} {b}^{\mathrm{2}} \right)^{\mathrm{2}} \geqslant\mathrm{9} \\ $$ $${and}\:{by}\:{doing}\:{LCM}\:{of}\:{denominator}\:{we}\:{can}\:{get}\:{that}\: \\ $$ $${Z}\:=\frac{\mathrm{3}}{\mathrm{2}\:} \\ $$ $${So}\:{E}\geqslant{Z}\:{equality}\:{holds}\:{when}\:{all}\:\:\frac{{x}_{{i}} }{{y}_{{i}} }\:{are}\:{same}\: \\ $$ $${Proved}.... \\ $$ $$ \\ $$ $$ \\ $$