On the Factor Opposing the Lebesgue Norm in Generalized Grand Lebesgue Spaces

We prove that if 1 < p< ∞ and δ:] 0 , p- 1] →] 0 , ∞[is continuous, nondecreasing, and satisfies the Δ 2 condition near the origin, then [Figure not available: see fulltext.] This result permits to clarify the assumptions on the increasing function against the Lebesgue norm in the definition of generalized grand Lebesgue spaces and to sharpen and simplify the statements of some known results concerning these spaces