Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

In this paper, the connection between the functional inequalities $$f\Big(\frac{x+y}{2}\Big)\leq\frac{f(x)+f(y)}{2}+\alpha_J(x-y) \qquad (x,y\in D)$$ and $$\int_0^1f\big(tx+(1-t)y\big)\rho(t)dt \leq\lambda f(x)+(1-\lambda)f(y) +\alpha_H(x-y) \qquad (x,y\in D)$$ is investigated, where $D$ is a convex subset of a linear space, $f:D\to\R$, $\alpha_H,\alpha_J:D-D\to\R$ are even functions, $\lambda\in[0,1]$, and $\rho:[0,1]\to\R_+$ is an integrable nonnegative function with $\int_0^1\rho(t)dt=1$.