@ACuriousMind Define $$B=C(1+\epsilon^{-1})\rho^{-2}-4\lambda\sigma +(2(n-1)^2+\epsilon)R,$$ $R,\lambda,\rho>0$, $n\ge 2$, $0<\epsilon<2$, $0\le \sigma\le 1$. The claim is that $$\frac{B+\sqrt{B^2-4(2-\epsilon)\lambda^2 (2\sigma^2-\epsilon)}}{4-2\epsilon}\le \frac{(4(n-1)^2+2\epsilon)R}{4-2\epsilon}+C'((1+\epsilon^{-1})\rho^{-2}+\lambda)‌​$$

edited: @ACuriousMind Define $$B=C(1+\epsilon^{-1})\rho^{-2}-4\lambda\sigma +(2(n-1)^2+\epsilon)R,$$ $R,\lambda,\rho>0$, $n\ge 2$, $0<\epsilon<2$, $0\le \sigma\le 1$. The claim is that $$\frac{B+\sqrt{B^2-4(2-\epsilon)\lambda^2 (2\sigma^2-\epsilon)}}{4-2\epsilon}\le \frac{(4(n-1)^2+2\epsilon)R}{4-2\epsilon}+C'((1+\epsilon^{-1})\rho^{-2}+\lambda)$$

said: @ACuriousMind Define $$B=C(1+\epsilon^{-1})\rho^{-2}-4\lambda\sigma +(2(n-1)^2+\epsilon)R,$$ $R,\lambda,\rho>0$, $m\ge 2$, $0<\epsilon<2$, $0\le \sigma\le 1$. The claim is that $$\frac{B+\sqrt{B^2-4(2-\epsilon)\lambda^2 (2\sigma^2-\epsilon)}}{4-2\epsilon}\le \frac{(4(n-1)^2+2\epsilon)R}{4-2\epsilon}+C'((1+\epsilon^{-1})\rho^{-2}+\lambda)$$