Prove the inequality: $ | 1 - K | < 1 $

Question

Given

\begin{gather*} Q \geq 0 \\ R > 0 \\ P > 0 \\ QR - S^2 > 0 \\ P = \frac{-(2S - Q) + \sqrt{(2S-Q)^2+4(QR-S^2)}}{2} \\ K = \frac{S+P}{R+P} \end{gather*} prove $$ |1-K|<1. $$

Answer 1

We have $|1-K|<1$ iff $|R-S|<R+P$. Since $P>Q/2-S$, it suffices to show that $|R-S|\le R+Q/2-S.$ If $R\ge S$ then we are done, so we assume $R<S$. Then we have to show that $4(S-R)\le Q$. Since $Q>S^2/R$, it suffices to show that $4(S-R)\le S^2/R$, that is $0\le (S-2R)^2$.