On the Fenchel–Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces

In this paper, we investigate the properties of the Fenchel–Moreau conjugate of G-function with respect to the coupling function \(c(x,A)=|A[x]^2|\) ${c(x,A)=|A[x]}^2|$ . We provide conditions that guarantee that the conjugate is also a G-function. We also show that if a G-function G is twice differentiable and its second derivative belongs to the Orlicz space generated by the Fenchel–Moreau conjugate of G then the modular generated by G is twice differentiable on the Orlicz space generated by G. We also investigate second-order differentiability of action functionals on anisotropic Orlicz–Sobolev spaces.