Show that the inequality $\frac{s}{t} + \frac{t}{r} + \frac{r}{s} >= 3$, assuming s,t,r >0.

Question

For this question, I made an assumption that $r<=s<=t$. If I manage to proof it under this assumption, is the result still hold true when $r<=t<=s$ or $ t<=s<=r$, if it is true, why?

Sorry, I didn't learn about AM-GM inequality. Would you mind to explain how AM-GM inequality works in this question here? My knowledge of AM-GM is limited to $a^2+b^2>=2ab $....

Thank you very much for your reply. This question has been confusing me for a long period of time.

@Peter Szilas Thank you very much... But I don't know AM-GM's application in this way.. sorry. The way I did it is manipulating the variables — Henry Cai, Jan 08 '20 at 21:20
@Peter Szilas Would you mind to explain it in more detail? Thank you very much — Henry Cai, Jan 08 '20 at 21:21
Henry. For these problems AM-GM is handy.It is worth looking it up. For 2 variables $(a+b \ge 2√a√b) $ quite easy to prove. Look it up in Wiki, for example. — Peter Szilas, Jan 08 '20 at 21:32
@Peter Szilas Thank you veery much, I finally got it. Using Jensen's inequalities to prove AM-GM; then, this works. Again, much thanks — Henry Cai, Jan 08 '20 at 21:33
Very good. AM-GM proof for 3 variables you also find at Stack exchange.Greetings. — Peter Szilas, Jan 08 '20 at 21:36

Michael Rozenberg · Accepted Answer · 2020-01-08T21:33:50.243

2

$$\sum_{cyc}\left(\frac{s}{t}-\frac{s}{r}\right)=\frac{\sum\limits_{cyc}(s^2r-s^2t)}{str}=\frac{(s-t)(t-r)(r-s)}{str}\geq0$$ for $r\leq s\leq t$, which says that it's enough to prove our inequality for $r\leq t\leq s$.

But it's better to use AM-GM: $$\sum_{cyc}\frac{s}{t}\geq3\sqrt[3]{\frac{s}{t}\cdot\frac{t}{r}\cdot\frac{r}{s}}=3.$$

I use AM-GM for three numbers:

for positives $x$, $y$ and $z$ we have: $$\frac{x+y+z}{3}\geq\sqrt[3]{xyz}$$

or $$x+y+z\geq3\sqrt[3]{xyz}.$$ Now, take $x=\frac{s}{t},$ $y=\frac{t}{r}$ and $z=\frac{r}{s}.$

edited Jan 08 '20 at 21:33

answered Jan 08 '20 at 21:17

Michael Rozenberg

194,933

Thank you very much... would you mind to explain $\sum_{cyc}$ for me? I didn't learn that... – Henry Cai Jan 08 '20 at 21:18
@Henry Cai $\sum\limits_{cyc}\frac{s}{t}=\frac{s}{t}+\frac{t}{r}+\frac{r}{s}.$ – Michael Rozenberg Jan 08 '20 at 21:19
Thank you, but why $\sqrt[3]{\frac{s}{t} \cdot \frac{t}{r} \cdot \frac{r}{s}} >=1 $... – Henry Cai Jan 08 '20 at 21:23
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@Henry Cai Because $\frac{s}{t}\cdot\frac{t}{r}\cdot\frac{r}{s}=\frac{str}{trs}=1.$ – Michael Rozenberg Jan 08 '20 at 21:29
Thank you very much... I still don't understand how does cyc works in AM-GM inequality. Sorry for any inconvenience I made. I will search on it . – Henry Cai Jan 08 '20 at 21:30
I get it thank you – Henry Cai Jan 08 '20 at 21:32
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@Henry Cai You are welcome! – Michael Rozenberg Jan 08 '20 at 21:34
I got the AM-GM part.. thank you. How does the cycling variables part work? Would you mind to eexplain more, I really need it. – Henry Cai Jan 08 '20 at 21:35
@Henry Cai Just $s\rightarrow t\rightarrow r\rightarrow s$. For example, $\sum\limits_{cyc}s=s+t+r$ or $\sum\limits_{cyc}st=st+tr+rs$. See also my previous example, – Michael Rozenberg Jan 08 '20 at 21:38
Let us continue this discussion in chat. – Henry Cai Jan 08 '20 at 21:41

Show that the inequality $\frac{s}{t} + \frac{t}{r} + \frac{r}{s} >= 3$, assuming s,t,r >0.

1 Answers1