<s>[INST] Proving Example 1.1.15 of secrets in inequalities 
if $a,b,c,d$ are positive real numbers,Prove:$$\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)^2\ge\frac{1}{a^2}+\frac{4}{a^2+b^2}+\frac{9}{a^2+b^2+c^2}+\frac{16}{a^2+b^2+c^2+d^2}$$

I was reading the solution of it from book and something was not understandable for me.
I have problem in understanding when equality occurs in inequalities that I highlighted them.
for example the first one:$$a^2+c^2 \ge 2 ca$$ $$b^2+c^2\ge 2bc$$ $$a^2+b^2+2c^2 \ge 2(ac+bc)$$ $$4a^2+4b^2+8c^2\ge 8ac+ 8bc$$
but $8a^2+8b^2+8c^2 > 4a^2+4b^2+8c^2$ when $a,b,c$ are positive real numbers.So If i am wrong,where I made mistake?
Also about the third inequality I highlighted,I have problem at proving it.I would appreciate if someone helps me there.
 [/INST] For the first:


*

*$\displaystyle \frac1{ac+bc}\ge\frac1{a^2+b^2+c^2}$ or $a^2+b^2+c^2\ge ac+bc$


*

*By AM_GM:
$$\frac{a^2+b^2}2\ge\sqrt{a^2b^2}=ab,\frac{b^2+c^2}2\ge bc,\frac{c^2+a^2}2\ge ca$$

*Since $ab\ge0$, Adding all:
$$a^2+b^2+c^2\ge ab+bc+ca\ge ac+bc$$



For the second:


*

*$\displaystyle \frac{4}{b^2+c^2}\ge\frac1{a^2+b^2+c^2}$ or $4a^2+4b^2+4c^2\ge b^2+c^2$ or $4a^2+3b^2+3c^2\ge0$ which is true as $x^2\ge0,\forall x\in\mathbb R$


For the third:


*

*$\displaystyle \frac{18}{ab+bd+cd}\ge\frac{16}{a^2+b^2+c^2+d^2}$ or $\displaystyle a^2+b^2+c^2+d^2\ge\frac{8}{9}(ab+bd+cd)$

*

*Similiar to first we can prove $a^2+b^2+c^2+d^2\ge ab+bc+cd+da\ge ab+bd+cd\ge \frac{8}{9}(ab+bd+cd)$ since $\frac89<1$


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