Help me with this problem Plz

[LightLucifer]

$a,b,c>0$ Prove That
$$\sum_{cyc}(\frac{a+1}{b+1})^{\frac{1}{a}}\leqslant \sum_{cyc}(\frac{a}{b})^{\frac{1}{a+1}}$$

[nooonuii]

Show that

$\Big(\dfrac{a+1}{b+1}\Big)^{\frac{1}{a}}\leqslant \Big(\dfrac{a}{b}\Big)^{\frac{1}{a+1}}$

by using weighted AM-GM inequality

[LightLucifer]

I've tried to figure it out for an hour.But I couldn't get how weighted AM-GM work with that. Could you give me more hint plz.

[Onasdi]

$\displaystyle{(a)^{\frac{1}{a+1}}(b)^{\frac{a}{a+1}}\le a\bigg(\frac{b+1}{a+1}\bigg)}$

[LightLucifer]

Oh!! Why did I doesn't notice that??

Thank you for all posts.

[nooonuii]

จริงๆแล้วผมทำไว้แบบนี้ครับ ได้อีกแบบแต่ก็คล้ายๆกัน

$\Big(\dfrac{b}{a}\Big)^{\frac{a}{a+1}}\cdot \Big(1\Big)^{\frac{1}{a+1}}\leq \dfrac{b+1}{a+1}$

วิธีใช้ weighted AM-GM สำหรับผมขั้นแรกจะพยายามจัดรูปอสมการให้กำลัง $\leq 1$ ก่อนเลย