Parametrized Furuta inequality and its application

As generalizations of the Furuta inequality and grand Furuta inequality, we give parametrized forms of them; if A greater than or equal to B > 0, then [GRAPHICS] for u less than or equal to 0, 0 less than or equal to 3 less than or equal to p and [GRAPHICS] for t is an element of [0, 1], 0 less than or equal to t < p less than or equal to beta, u less than or equal to 0 and delta is an element of [0, beta]. Applying these results, we can easily show the monotone properties of an operator function for A greater than or equal to B defined by [GRAPHICS] it is increasing for u less than or equal to 0 and decreasing for beta greater than or equal to p where t is an element of [0, 1], 0 less than or equal to t < p less than or equal to beta, u less than or equal to 0 and delta is an element of [0, beta].