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In this paper, we study the class of Banach spaces with James constant . It is shown that, for a Banach space of three or more dimensions, the James constant becomes if and only if the norm is induced by an inner product. Moreover, the symmetric absolute norms on with James constant are completely characterized in terms of convex functions on the unit interval, which provides many new examples of such norms other than the Euclidean or regular octagonal norms. However, it is also shown that there exist two-dimensional normed spaces with James constant outside of the family of symmetric absolute norms.

In this paper, we introduce new geometric properties as generalizations of p-uniform smoothness and q-uniform convexity of Banach spaces. Furthermore, using generalized Beckner's inequality, we characterize the properties in terms of norm inequalities. As an application, we consider the duality relation.

A silica-supported triphenylphosphane (Silica-3p-TPP) with a Ph3P-type core, immobilized on a silica surface, was synthesized and characterized by nitrogen-absorption measurements and solid-state NMR spectroscopy. The tripodal immobilization constrains the mobility of the phosphane molecule and causes the lone pair on the phosphorus atom to face in the direction perpendicular to the support, resulting in the selective formation of a 1:1 metal-phosphane species that is free from unfavorable steric repulsions caused by the silica surface. Heterogeneous Pd catalysts created in this manner enabled room-temperature Suzuki-Miyaura cross-coupling reactions with unactivated chloroarenes, despite the moderate electronic and steric nature of the Ph3P-based ligands. These catalysts also showed potential in reactions with more challenging substrates under mild conditions. Tripodally immobilized and well-dispersed phosphanes on the silica surface were crucial for high catalytic activity.

The presented system is applicable to one-pot biaryl synthesis of biphenyl derivative (V) via Miyaura borylation—Suzuki coupling sequence.

The new heterogeneous phosphane (POS), which is tripodally immobilized on a silica surface, selectively forms a 1:1 palladium- phosphane molecule.