parmenides51 wrote: Let $a \ge b \ge c > 0$. Prove that $$(a-b+c)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right) \ge 1$$ https://artofproblemsolving.com/community/c6h299899p1890035 https://artofproblemsolving.com/community/c6h1366870p7516619 https://artofproblemsolving.com/community/c4h555146p3225813

Let $a \ge b \ge c > 0$. Prove that $$(1.1a-b+c)\left(\frac{1}{a}-\frac{1}{2b}+\frac{1}{c}\right) \ge \frac{21}{20}+\frac{1}{\sqrt 5}$$$$(1.1a-b+c)\left(\frac{1}{a}-\frac{1}{3b}+\frac{1}{c}\right) \ge \frac{2}{15}\left(8+\sqrt {15}\right)$$