Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q^2}= 1$, then $$\frac{1}{p (p +2)} +\frac{1}{q (q +2)}\geq\frac{21\sqrt{21}-71}{80} $$Equality holds when $p=2+2\sqrt{\frac{7}{3}},q=\frac{\sqrt{21}+1}{5}.$

parmenides51 wrote: Prove that if the positive real numbers $p$ and $q$ satisfy $\frac{1}{p}+\frac{1}{q}= 1$, then a) $\frac{1}{3} \le \frac{1}{p (p + 1)} +\frac{1}{q (q + 1)} <\frac{1}{2}$ b) $\frac{1}{p (p - 1)} + \frac{1}{q (q - 1)} \ge 1$ Let $ a,b>0 $ and $ \frac1{a}+\frac1{b}=1. $ Prove that $$\frac13\le \frac1{a(a+1)}+\frac1{b(b+1)}\le\frac12$$

Let $ a,b>0 $ and $ \frac1{a}+\frac1{b}=1. $ Prove that $$ \frac{a}{b(b+1)}+ \frac {b}{a(a+1)}\ge\frac 23$$$$\frac{a+1}{b(b+2)}+\frac{b+1}{a(a+2)} \ge\frac 34$$$$\frac1{a(b^2+a+1)}+\frac1{b(a^2+b+1)}\le\frac 17$$$$\frac1{a(ab+b+1)}+\frac1{b(ab+a+1)}\le\frac 17$$