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As a continuation of our previous work with the same title, the construction of grand Furuta inequality (GFI) is improved in order to give a constructive proof of Uchiyama's result (2003). Afterwards the generalization of Uchiyama's result is obtained by mathematical induction which implies Furuta's recent results. (2010) : 47A63, 47B15, 47B65. Mathematics subject classificationdoi:10.7153/mia-13-56 fatcat:4bs4faux3vf3fmvfsj5lba6igq