Relation between Two Operator Inequalities <img src="http://latex.codecogs.com/gif.latex?f(B^{\frac{1}{2}}AB^{\frac{1}{2}})\geq&space;B^{-1}" title="f(B^{\frac{1}{2}}AB^{\frac{1}{2}})\geq B^{-1}" /> and <img src="http://latex.codecogs.com/gif.latex?A^{-1}\geq&space;g(A^{\frac{1}{2}}BA^{\frac{1}{2}})" title="A^{-1}\geq g(A^{\frac{1}{2}}BA^{\frac{1}{2}})" />