Some Properties Of Generalized Gao's Constant And Modulus Of Smoothness

This thesis is devoted to study several properties of generalized James constant and generalized modulus of smoothness. The paper organized as follows:Firstly, several geometric properties of generalized Gao's constant is discussed. More-over, We get two sufficient conditions for space X to have uniformly normal structure(?) If exist a∈[0,1] satisfy E(a,X)< 2(1+a)2(1+(1-a)/(J(a,X)+2a)2, therefore X have uniformly normal structure.(?) If exist a a∈[0,1] satisfy E(a,X)< 2(1+4a+(1-a)/(μ(X)))2, therefore X have uniformly normal structure. These results generalized the results of J.Gao.Next, Similar to modulus of convexityδ(λ)(ε), we introduce the generalized modulus of smoothnessρ(λ)(ε). Using the modulus, this paper get two necessary and sufficient conditions for uniformly smoothness. Specifically, it calculates the Hilbert spaces H, space lp(p>1) and renormed endless sequence spaces Xp(p>1), namely In addition, using generalized modulus of convexity and generalized modulus of smoothness, we give the difinition of generalized modulus of deformation and the generalized deformation on spaces, these two definition can depict the deformation degree of spaces X under the generalized modulus.Finally, the with parameter t constant named Boronti constant A2(t,X) and with parameterξ,ηconstant named Alonso-Llonso-Fuster constant Tξ,η(X) are given. These two constants are based on the with parameter t constant E(t,X) and with parameterξ,ηconstant Eξ,η(X) who introduced by J.Gao. Using the two constants, we get several sufficient conditions for Banach space X to have uniformly normal structure, these con-clusions promotion the results of J.Gao. Specifically, ingenious use Hanner inequality this paper calculates the value of constant Tξ,η(X) on some specific space, when 1≤p≤2, it did not give a precise value of Tξ,η(X) on spaces Xp,λ.