Inequality with positive reals

Question

For each $x,y,z>0$, define $$ f(x,y,z)=\frac{3x^2(y+z)+2xyz}{(x+y)(y+z)(z+x)}. $$

Fix also $a,b,c,d>0$ with $a\le b$. How can we prove "reasonably" the following inequality? $$ \frac{f(a,b,d)+f(c,b,d)}{f(b,a,d)+f(c,a,d)} \le \frac{a+c}{b+c}. $$

Answer 1

The expression
$$\frac{f(a,b,d)+f(c,b,d)}{f(b,a,d)+f(c,a,d)} \le \frac{a+c}{b+c}$$ is equivalent to $$(b+c)(f(a,b,d)+f(c,b,d))\le(a+c)(f(b,a,d)+f(c,a,d))$$ The $LHS$ is equal to $$\frac{3a^2(b+c)}{(a+b)(a+d)}+\frac{2adb^2+2abcd}{(a+b)(a+d)(b+d)}+\frac{3c^2}{c+d}+\frac{2bcd}{(b+d)(c+d)}$$ The $RHS$ is equal to $$\frac{3b^2(a+c)}{(a+b)(b+d)}+\frac{2bda^2+2abcd}{(a+b)(a+d)(b+d)}+\frac{3c^2}{c+d}+\frac{2acd}{(a+d)(c+d)}$$ so we have to prove that $$\frac{3a^2(b+c)}{(a+b)(a+d)}+\frac{2adb^2}{(a+b)(a+d)(b+d)}+\frac{2bcd}{(b+d)(c+d)}$$ is less or equal than $$\frac{3b^2(a+c)}{(a+b)(b+d)}+\frac{2bda^2}{(a+b)(a+d)(b+d)}+\frac{2acd}{(a+d)(c+d)}$$ Multiplying by $(a+b)(a+d)(b+d)(c+d)$ one has respectively $$(b+d)(c+d)(b+c)(3a^2)+(c+d)(2adb^2)+(a+b)(a+d)(2bcd)$$ and $$(a+d)(a+c)(c+d)(3b^2)+(c+d)(2bda^2)+(a+b)(b+d)(2acd)$$

What precedes becomes after simplification $$a^2(b+c)(b+d)\le b^2(a+c)(a+d)$$ where $a,b,c,d\gt 0$ and $a\le b$.

Finally we have the evident inequality $$\color{red}{0\le ab(b-a)(c+d)+(b^2-a^2)cd}$$

Answer 2

we get $$\frac{a+c}{b+c}-\frac{f(a,b,d)+f(c,b,d)}{f(b,a,d)+f(c,a,d)}=-{\frac { \left( 3\,c+d \right) \left( a-b \right) \left( abc+abd+ca d+cbd \right) \left( c+a \right) }{ \left( 3\,{a}^{2}{b}^{2}c+3\,{a}^ {2}{b}^{2}d+3\,{a}^{2}b{c}^{2}+4\,{a}^{2}bcd+2\,{a}^{2}b{d}^{2}+3\,{a} ^{2}{c}^{2}d+2\,{a}^{2}c{d}^{2}+6\,a{b}^{2}{c}^{2}+8\,a{b}^{2}cd+3\,a{ b}^{2}{d}^{2}+8\,ab{c}^{2}d+4\,abc{d}^{2}+3\,a{c}^{2}{d}^{2}+6\,{b}^{2 }{c}^{2}d+3\,{b}^{2}c{d}^{2}+3\,b{c}^{2}{d}^{2} \right) \left( c+b \right) }} $$ and this is positive since we have $$a\le b$$