𝑃-convexity of Orlicz-Bochner spaces

Abstract

Abstract. A characterization of P -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space L Φ (µ, X)is P-convex if and only if both spaces L Φ (µ) and X are P -convex. In particular, the Lebesgue-Bochner space L p (µ, X) with 1 < p < ∞ is P -convex iff X is P -convex.

I‐convexity and Q‐convexity in Orlicz–Bochner function spaces endowed with the Orlicz norm

“…Since B ‐convexity (B‐C for short) and uniform nonsquareness (U‐NS) were given by A. Beck and R. C. James respectively , these convexities were widely used in probability theory, fixed point theory and any other fields . Some revelent convexities, such as P ‐convexity (P‐C), O ‐convexity (O‐C), Q ‐convexity (Q‐C) and I ‐convexity (I‐C) were raised and researched by many mathematicians . Let X be a Banach space.…”

“…Some convexities including (U‐NS), (P‐C) and (B‐C) in Orlicz space, Lebesgue–Bochner space and Orlicz–Bochner space were characterized by G. Alhert, H. Hudzik, A. Kamińska, P. Kolwicz, R. Płuciennik, M. A. Smith and B. Turett etc. . The characterization of Q ‐convexity in Orlicz space was given in .…”

“…Before this, in part 2 of this paper we will give some criteria for a Banach space being I ‐convex as well as being Q ‐convex. Some ideas and methods come from .…”

Musielak–Orlicz–Bochner function spaces which are uniformly noncreasy

“…As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given. The topic of this paper is related to the topic of and .…”

Modified version of Jensen equation and orthogonal additivity

“…We start with a functional inequality that occurs in some considerations concerning Orlicz spaces. For example P. Kolwicz and R. P luciennik [2] deal with the inequality…”