P-convexity of Musielak Orlicz function spaces of Bochner type

Abstract

Abstract• It is proved that tSe Musielak-Orhicz function space L 4, (p, X) of Bochner type 1v P-convex if and only if l>oth spaces t4(p, It) and X are P-convex. In particular, the Lebesgue-Bochner space L~(p, 2<) is P-convex uf X is P-convex. Introduct ionRelationships between various kinds of convexities of Banach spaces and the reflexivity of them were developed by many authors. The earliest result concerning that problem was obtained by D. Milman in 1938 (seo 1 17j). Milman proved that every uniformly convex Banacb space is reflexive. D. Giesy ¡31 and RC. James [9] raised the question wbether Banach spaces which are uniformly non-4 with some positive integer n =2 (i.e. B-convex spaces) are reflexive. Although there were sorne aflirmative results in particular cases, the answer in general case was negative ¡10]. In 1970 C.A. Kottman [14] introduced a sligbtly stronger than B-convexity geometric property implying reflexivity and called it P -convexity. Ye Yining, He Miaohong and It Pluciennik [22] proved that for Orliez spaces reflexivity is equivalent to P-convexity. The same result for Musie]ak-Orhicz sequence and funetion spaces were obtained by Ye Yining and Huang Yafeng ¡231 and by P. Kolwicz and It Pluciennik [lii respectively.

P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type

“…Yining, et.al [5,6] in their paper entitled "P-convexity and reflexivity of Orlicz spaces" proved that for Orlicz spaces reflexivity is equivalent to Pconvexity. The same result for the MusielakOrlicz sequence and function spaces were obtained by Kolwicz and Pluciennik [7][8][9]. Given f extended real valued function on measurable set X .…”

“…the modular convex generates the Musielak-Orlicz function spaces of Bochner-type, which is also the Banach space [3,4].…”

Comparison of P-convexity, O-convexity and other geometrical properties

“…In [7], Kolwicz and Pluciennik proved that the Musielak-Orlicz sequence space ℓ φ (X) of Bochner type is P-convex if and only if both spaces ℓ φ (R) and X are P-convex, which implies that if X is P-convex, the same holds for ℓ p (X) if 1 < p < ∞.…”

On a modified version of Jensen inequality

“…4(2b 2 2b + 1)(5b 2 2b + 1)(1 + bZ)(5b 2 4b + 1) b4(5b 4 9b 4 + lOb 4b + 6b + b 2 + 5b 2 8b 2 9b 2 b 2 + 2b 4b + 6b + 1)/(4(2b 2 2b + 1)(5b 2 2b + 1)…”

“…It can be done in different ways. One of the posibilities is to be found in a paper of Kolwicz and Ptuciennik [1] in connection with some functionals on…”