Chủ đề này có 39 trả lời

[DOTOANNANG]

$$\frac{x}{y+ z}\,+ \,\frac{y}{z+ x}\,+ \,\frac{z}{x+ y}\,\geq \, \frac{3}{2}$$

 

$$\frac{x}{y+ z}\,+ \,\frac{y}{z+ x}\,+ \,\frac{z}{x+ y}\,+ \frac{x^{\,2}}{y^{\,2}+ z^{\,2}}\,+ \,\frac{y^{\,2}}{z^{\,2}+ x^{\,2}}+ \frac{z^{\,2}}{x^{\,2}+ y^{\,2}}\,\geq \, \frac{3}{2}$$

 

 

[DOTOANNANG]

$$\frac{x}{\sqrt{2\,  (\,  y^{\,  2}\,  + \,  z^{\,  2}\,  )}}\,  + \,  \frac{y}{z+ x}\,  + \,  \frac{z}{x+ y}\,  \geq\,  \frac{3}{2}$$

[DOTOANNANG]

$$\frac{x}{y\,+\,z}\,+\,\frac{y}{z\,+\,x}\,+\,\frac{z}{x\,+\,y}\,\leq\, \frac{3}{2}\,+\,\frac{\max\,\left \{ \,(\,x\,-\,y\,)^{\,2}\,, \,(\,y\,-\,z\,)^{\,2}\,,\, (\,z\,-\,x\,)^{\,2}\, \right \}}{x\,y\,+\,y\,z\,+\,z\,x}$$

[DOTOANNANG]

$$\sum  \,\frac{x}{y\,+\,z}\,+\, \prod\, \frac{x}{y\,+\,z}\,\geq \, \frac{1\,3}{8}$$

[DOTOANNANG]

$$\sum  2\, \,\frac{x}{y\,+\,z}\,\leq  \, \sum\, \frac{x}{\sqrt{\,y\,z\,}}\,\leq\,  \sum \,\frac{x^{\,2}}{y\,z}$$

[DOTOANNANG]

$$\sum  \, \,\frac{x}{y\,+\,z}\,\geq\, \frac{x^{\,3}\,+\,y^{\,3}\,+\,z^{\,3}\,-\, 3\,xyz}{2\,(\,x+y\,)\,(\,y+z\,)\,(\,z+x\,)}\,+\, \frac{3}{2}$$

 

$<\,=\,>$

 

$$\sum \,x\,(\,x\,-\,y\,)\,(\,x\,-\,z\,)\,\geq \,0$$

[DOTOANNANG]

$$\frac{x}{y}\,+\,\frac{y}{z}\,+\,\frac{z}{x}\,-\,\frac{3}{2}\,-\,\frac{x}{y\,+\,z}\,-\,\frac{y}{z\,+\,x}\,-\,\frac{z}{x\,+\,y}\,\geq \,\frac{(\,x\,-\,y\,)\,(\,y\,-\,z\,)}{x\,z}$$

 

 

https://diendantoanh...cc-aa-bcb-le-0/

[DOTOANNANG]

$$\frac{x}{y\,+\,z}\,+\,\frac {y}{z\,+\,x}\,+\,\frac {z}{x\,+\,y} \,\geq\, \frac{3}{2}\,+\,\frac{\sqrt{\,13\,+\,16\sqrt{\,2}}}{2}\,\left(\,\frac{x\,-\,y}{x\,+\,y}\,+\,\frac{y\,-\,z}{y\,+\,z}\,+\,\frac{z\,-\, x}{z\,+\,x}\,\right)$$

[Khoa Linh]

Mạnh hơn BĐT NESBIT của thầy Cẩn:

$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3(a^2+b^2+c^2)}{(a+b+c)^2}+\frac{1}{2}$

[DOTOANNANG]

$$a\,\frac{x}{y\,+\, z}\,+\, b\,\frac{y}{z\,+\, x}\,+\, c\,\frac{z}{x\,+ \,y}\,\geq \, \frac{a+b+c}{2}\,-\, \frac{1}{2}\, \sum \,(\sqrt{\,a}\,-\, \sqrt{\,b})^{\,2} $$

[DOTOANNANG]

$x\,y\,+\, y\,z\,+ \,z\,x\,>\,  0$

 

$$\frac{x}{y+ z}\,+ \,\frac{y}{z+ x}\,+ \,\frac{z}{x+ y}\,\geq \, \frac{3}{2}$$

[DOTOANNANG]

$$\frac{x}{y+ z}\,+ \,\frac{y}{z+ x}\,+ \,\frac{z}{x+ y}\,\geq \, \frac{5}{4}\,+\, \frac{3\,(x^{\,3}\,+\,y^{\,3}\,+\,z^{\,3})}{4\,(\,x\,+\,y\,+\,z\,)\,(\,xy\,+\,yz\,+\,zx\,)}$$

[tr2512]

$x,y,z\ge0$

$\frac{x}{{y + z}} + \frac{y}{{x + z}} + \frac{z}{{x + y}} \le \frac{{\left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2}} \right)}}{{xy\left( {x + y} \right) + yz\left( {y + z} \right) + z{\rm{x}}\left( {z + x} \right)}}$

[DOTOANNANG]

$$\sum \,\frac{x}{y\,+\,z}\,\geq \,1\,+\,\frac{x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}}{2\,(\,xy\,+\,yz\,+\,zx)}\,+\,\frac{xyz\,-\,(\,x+y-z\,)\,(\,y+z-x\,)\,(\,z+x-y\,)}{2\,(\,x+y+z\,)\,(\,xy+yz+zx\,)}$$

[DOTOANNANG]

$$\frac{x}{y\,+\,z}\,+\,\frac{y}{z\,+\,x}\,+\,\frac{z}{x\,+\,y}\,+\,16\,\frac{x\,y\,+\,y\,z\,+\,z\,x}{x^{\,2}\,+\,y^{\,2}\,+\,z^{\,2}}\,\geq\, 8$$

 

 

 

 

 

 

 

 

 

Đẳng thức xảy ra khi $x\,=\, 2\,+\,\sqrt{\,3}\,,\, y\,=\, 1\,,\, z\,=\, 0$ cùng các hoán vị.

[DOTOANNANG]

\[2\,{\frac {x}{y+z}}+{\frac {y}{z+x}}+{\frac {z}{x+y}}\,\geq\, -\,1\,+\,2\,\sqrt {2}\]

[DOTOANNANG]

$$\sum_{cyc}^{ }\frac{x}{y+ z}+ \frac{\sum_{cyc}^{ }xy}{\sum_{cyc}^{ }(x-y)^{2}}\geq \frac{5}{2}$$

[DOTOANNANG]

$x\geqq y\geqq z$

 

$$\frac{x}{x+ y}+ \frac{y}{y+ z}+ \frac{z}{z+ x}> \frac{3}{2}+ \frac{x^{2}y}{2\,(x+y)^{3}}$$

[hoangkimca2k2]

$a,b,c>0$

 $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{3}{2}+\frac{(a+b)^{2}+(b+c)^{2}+(c+a)^{2}}{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}$$

Bài viết đã được chỉnh sửa nội dung bởi hoangkimca2k2: 25-04-2018 - 09:31

[hoangkimca2k2]

$a,b,c>0$;$a+b+c=3$

$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq \frac{21}{16}+\frac{27(a^{3}+b^{3}+c^{3})}{16(a+b+c)^{3}}$$