A ≥ B ≥ 0 Ensures $${\left( {{B^{\frac{r}{2}}}{A^p}{B^{\frac{r}{2}}}} \right)^{\frac{1}{q}}} \geqslant {\left( {{B^{\frac{r}{2}}}{A^p}{B^{\frac{r}{2}}}} \right)^{\frac{1}{q}}}$$ for R ≥ 0, P ≥ 0, G ≥ 1 with (1 + R)Q ≥P + R and its Recent Appications