Implications between approximate convexity properties and approximate Hermite-Hadamard inequalities

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

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