# $a,b,c > 0$ and $a+b+c=1$. Prove that $\frac{1}{a+b^2}+\frac{1}{b+c^2}+\frac{1}{c+a^2}\geqslant \frac{13}{2(1-abc)}$

$$a,b,c > 0$$ and $$a+b+c=1$$. Prove that

$$\frac{1}{a+b^2}+\frac{1}{b+c^2}+\frac{1}{c+a^2}\geqslant \frac{13}{2(1-abc)}$$

This inequality can be solved by computer ( essentially, remove the constrain through homogenizing, clear denominator, expanding and proving positive terms). Since this mechanical strategy is very simple to do but impractical during math contest, I am looking forward to solution using classical inequalities approach.

• Where did that inequality come from? Feb 18, 2021 at 17:19

I would not call this an answer, but perhaps could be helpful. Observe that: \begin{align*} (1-a)(1-b)(1-c)&=\color{red}{1-a-b-c}+ab+ac+bc-abc\\ &=ab+ac+bc-abc \\ &=\frac{(a+b+c)^2-(a^2+b^2+c^2)}{2}-abc \\ &=\frac{1-(a^2+b^2+c^2)}{2}-abc \\ &=1-abc-\frac{1+a^2+b^2+c^2}{2} \\ &=1-abc-\frac{a+b+c+a^2+b^2+c^2}{2}. \end{align*} Thus, $$a+b+c+a^2+b^2+c^2=2[(1-abc)-(1-a)(1-b)(1-c)].$$ By Cauchy Schwartz inequality, $$(a+b+c+a^2+b^2+c^2)\left(\frac{1}{a+b^2}+\frac{1}{b+c^2}+\frac{1}{c+a^2}\right)\geq3^2=9.$$ As a result, $$\frac{1}{a+b^2}+\frac{1}{b+c^2}+\frac{1}{c+a^2}\geq\frac{9}{2[(1-abc)-(1-a)(1-b)(1-c)]}.$$ If we can show $$\frac{9}{2[(1-abc)-(1-a)(1-b)(1-c)]}\geq\frac{13}{2(1-abc)},$$ then we are done. Suppose this is true. It gives $$13[(1-abc)-(1-a)(1-b)(1-c)]\leq 9(1-abc),$$ or equivalently, $$(1-abc)\leq\frac{13}{4}(1-a)(1-b)(1-c).$$ If this inequality is true under the constraint $$a+b+c=1$$ and $$a,b,c>0$$, then we are done. I checked that the equality precisely holds when $$a=b=c=1/3$$. Hope anyone can finish it.
• That last inequality doesn't hold for $a=b=\frac{1}{4}, c = \frac{1}{2}$. The LHS would be $\frac{124}{128}$, while the RHS would be $\frac{117}{128}$ Dec 31, 2020 at 1:40
• The method reached a dead end because of the application of CS. IE the inequality $9 / ( \sum a + b^2 ) >= 13/ 2 ( 1- abc)$ isn't true. The re-write of $1-abc$ hasn't quite been used as yet. Dec 31, 2020 at 18:40