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The connection between the functional inequalities $$f\left(\frac{x+y}{2}\right)⩽\frac{f\left(x\right)+f\left(y\right)}{2}+{\alpha }_J\left(x-y\right),x,y\in D,$$ and $${\int }_0^{1}f\left(tx+\left(1-t\right)y\right)\rho \left(t\right)dt⩽\lambda f\left(x\right)+\left(1-\lambda \right)f\left(y\right)+{\alpha }_{\mathrm{H}}\left(x-y\right),x,y\in D,$$ is investigated, where D is a convex subset of a linear space, f: D → ℝ, α H;α J: D-D → ℝ are even functions, λ ∈ [0; 1], and ρ: [0; 1] →ℝ+ is an integrable nonnegative function with ∫01 ρ(t) dt = 1.