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

Abstract

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X ± λ ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ± λ f ≥ 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 R 2 {{\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 = ρ 2 \lambda ={\rho }^{2} , ρ > 0 \rho \gt 0 , then X λ ( Ω ) {X}_{\lambda }\left(\Omega ) (resp. X − λ ( Ω ) {X}_{-\lambda }\left(\Omega ) ) reduces to the class of functions f ∈ C 2 ( I ) f\in {C}^{2}\left(I) such that f f is trigonometrically ρ \rho -convex (resp. hyperbolic ρ \rho -convex) on I I .