... of the linear transformation in Banach spaces,” Journal of the MathematicalSociety of Japan, vol. 2, pp. 64–66, 1950.7 D. G. Bourgin, “Classes of transformations and bordering transformations,” ... 6f2y2.1 for all x, y ∈ X. Interchanging x with y in 1.9,weobtainf2y xf2y − x 4fy x fy − x 2f2y− 4fy− 6fx2.2 for all x, y ∈ X. Since ... T∞i1Tρ2i−1x,2i−1x23i1t,Tρ2i−1x,2·2i−1x23it4,ρ0,2i−1x23it33.22 for all x ∈ X and all t>0.Proof. Putting y x in 3.20,weobtainμf3x−6f 2x15fxt≥ ρx,xt3.23Hindawi Publishing CorporationJournal of Inequalities and...