Let Γ andψ = Γ′/Γ denote the Gamma function and the Psi function respectively. Let furtherλ 1,⋯,λ n ∈ ℝ+ denote weights,λ 1 +⋯+λ = 1. The following pair of inequalities is proved: $$\begin{gathered} \Gamma (x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } ) \leqslant \Gamma ^{\lambda _1 } (x_1 ) \cdot \cdot \cdot \Gamma ^{\lambda _n } (x_n )(x_1 ,...,x_n \geqslant \alpha ), \hfill \\ \Gamma (x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } ) \geqslant \Gamma ^{\lambda _1 } (x_1 ) \cdot \cdot \cdot \Gamma ^{\lambda _n } (x_n )(0< x_1 ,...,x_n \leqslant \alpha ) \hfill \\ \end{gathered} $$ where α is the unique positive root of the equationψ(α) + αψ′(α) = 0. The first of the above inequalities is also valid for allx 1,⋯,x n ∈ ℝ+ under the restraint $$x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } \geqslant \beta $$ whereβ is the unique positive root of the equationβψ(β) = −1. This result is extended to convex solutionsf: ℝ+ → ℝ of the difference equationf(x + 1) − f(x) = ϕ(x) for certain functionsϕ: ℝ+ → ℝ. Under suitable conditions the inequality $$\lambda _1 f(x_1 ) + \cdot \cdot \cdot + \lambda _n f(x_n ) \geqslant f(x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } )$$ is obtained for allx 1,⋯, x n ∈ ℝ+ satisfying $$x_1^{\lambda _1 } \cdot \cdot \cdot x_n^{\lambda _n } \geqslant 1$$ .