An inequality between the James type constant and the modulus of smoothness

Recently, Takahashi has introduced the James type constant. In this study, we have presented an inequality between the James type constant and modulus of smoothness, and shown that the equalities in the inequality can hold for some concrete spaces. In particular, for any τ ≥ 0 , we have calculated the exact values of the James type constant, J X , t ( τ ) of l ∞ − l p , for 1 ≤ t ≤ p and p ≥ 2 .