Let \(\varepsilon =(\varepsilon _{o},\dots,\varepsilon _{k})\in [ 0,\infty [ ^{k+1}, p=(p_{o},\dots,p_{k})\in [ 0,1[^{k+1}\) and \(t\in ]0,1[\) be fixed parameters. A real valued function \(f\) defined on an open convex set \(D\) is called \((\varepsilon ,p,t)-\) convex if it satisfies \[ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\sum_{i=0}^{k}\varepsilon _{i}\left| x-y\right| ^{p_{i}}\;\text{for }x,y\in D. \] The authors prove that if \(f\) is \((\varepsilon ,p,t)-\) convex and locally bounded from above at a point of \(D\) then it satisfies the convexity-type inequality \[ f(sx+(1-s)y)\leq sf(x)+(1-s)f(y)+\sum_{i=0}^{k}\varepsilon _{i}\Phi_{p_{i},t}(s)\left| x-y\right| ^{p_{i}} \] for \(x,y\in D\), \(s\in [ 0,1],\) where \(\Phi _{p_{i},t}:[0,1]\rightarrow \mathbb{R}\) is defined by \[ \Phi _{p_{i},t}(s)=\max \left\{ \frac{1}{\left( 1-t\right) ^{p_{i}}-(1-t)}; \frac{1}{t^{p_{i}}-t}\right\} (s(1-s))^{p_{i}}. \]