Let (${Omega}$, $mathcal{S}$, ${mu}$) be a probabilistic measure space, ${varepsilon}{in}mathbb{R}$, ${delta}{geq}0$, p > 0 be given numbers and let $P{subset}mathbb{R}$ be an open interval. We consider a class of functions $f:P{ ightarrow}mathbb{R}$, satisfying the inequality $$f(EX){leq}E(f{circ}X)+{varepsilon}E({mid}X-EX{mid}^p)+{delta}$$ for each $mathcal{S}$-measurable simple function $X:{Omega}{ ightarrow}P$. We show that if additionally the set of values of ${mu}$ is equal to [0, 1] then $f:P{ ightarrow}mathbb{R}$ satisfies the above condition if and only if $$f(tx+(1-t)y){leq}tf(x)+(1-t)f(y)+{varepsilon}[(1-t)^pt+t^p(1-t)]{mid}x-y{mid}^p+{delta}$$ for $x,y{in}P$, $t{in}[0,1]$. We also prove some basic properties of such functions, e.g. the existence of subdifferentials, Hermite-Hadamard inequality.