Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X±λ(Ω)={f∈C2(Ω):Δf±λf≥0}{X}_{\pm \lambda }\left(\Omega )=\{f\in {C}^{2}\left(\Omega ):\Delta f\pm \lambda f\ge 0\}, where λ>0\lambda \gt 0 and Ω\Omega is an open subset of R2{{\mathbb{R}}}^{2}. We also obtain a characterization of the set X−λ(Ω){X}_{-\lambda }\left(\Omega ). Notice that in the one-dimensional case, if Ω=I\Omega =I (an open interval of R{\mathbb{R}}) and λ=ρ2\lambda ={\rho }^{2}, ρ>0\rho \gt 0, then Xλ(Ω){X}_{\lambda }\left(\Omega ) (resp. X−λ(Ω){X}_{-\lambda }\left(\Omega )) reduces to the class of functions f∈C2(I)f\in {C}^{2}\left(I) such that ff is trigonometrically ρ\rho -convex (resp. hyperbolic ρ\rho -convex) on II.