If-a-b-c-d-0-prove-that-a-3-bc-b-3-cd-c-3-da-d-3-ab-a-b-c-d-

Question Number 153412 by mathdanisur last updated on 07/Sep/21

If a; b; c; d \in (0; \infty) prove that :

\frac{a^{3}}{bc} + \frac{b^{3}}{cd} + \frac{c^{3}}{da} + \frac{d^{3}}{ab} ⩾ a + b + c + d

Answered by puissant last updated on 07/Sep/21

a ⩾ b ⩾ c ⩾ d \Rightarrow a^{3} ⩾ b^{3} ⩾ c^{3} ⩾ d^{3}

a n d t h e n, \frac{1}{b c} ⩾ \frac{1}{c d} ⩾ \frac{1}{d a} ⩾ \frac{1}{a b}

u s i n g r e a r r a n g e m e n t i n e q u a l i t y, w e g e t :

f (a, b, c, d) = \frac{a^{3}}{b c} + \frac{b^{3}}{c d} + \frac{c^{3}}{d a} + \frac{d^{3}}{a b} ⩾ \frac{a^{2}}{d} + \frac{b^{2}}{a} + \frac{c^{2}}{b} + \frac{d^{2}}{c}

\Rightarrow f (a, b, c, d) ⩾ a^{2} \times \frac{1}{d} + b^{2} \times \frac{1}{a} + c^{2} \times \frac{1}{b} + d^{2} \times \frac{1}{c} .

W e h a v e,

a ⩾ b ⩾ c ⩾ d \Rightarrow \frac{1}{d} ⩾ \frac{1}{a} \Rightarrow \frac{a^{2}}{d} ⩾ a

b ⩾ c \Rightarrow \frac{1}{c} ⩾ \frac{1}{b} \Rightarrow \frac{b^{2}}{c} + \frac{c^{2}}{b} ⩾ \frac{b^{2}}{b} + \frac{c^{2}}{c} = b + c

c ⩾ d \Rightarrow \frac{1}{d} ⩾ \frac{1}{c} \Rightarrow \frac{c^{2}}{d} + \frac{d^{2}}{c} ⩾ \frac{c^{2}}{c} + \frac{d^{2}}{d} = c + d

H e n c e,

∴∵ \frac{a^{3}}{b c} + \frac{b^{3}}{c d} + \frac{c^{3}}{d a} + \frac{d^{3}}{a b} ⩾ a + b + c + d .

Commented by mathdanisur last updated on 07/Sep/21

ThankYou Ser