In what follows, let \(D\) be a nonempty, open, convex subset of a linear space \(X\); further, let \(H\subset[0,1]\) be a nonempty set and \(s\in]0,1]\). As a generalization of the know concept, the authors introduce the following concept: a function \(f:D\to\mathbb R\) is called \((H,s)\)-convex if, for all \(x,y\in D\) and \(\lambda\in H\), the following inequality holds: \[ f(\lambda x+(1-\lambda)y)\leq{\lambda}^sf(x)+(1-\lambda)^sf(y). \] The particular case when \(H=[0,1]\) reduces to the notion of \(s\)-convexity due to Breckner. If \(H=\{\lambda\}\), then the \((H,s)\)-convexity is termed briefly by \((\lambda,s)\)-convexity. The authors first investigate some elementary properties of \((H,s)\)-convex functions and then prove Bernstein-Doetch-type results in the case when the underlying space \(X\) is a normed space. Two representative theorems of the paper are the following. Theorem 1: Let \(D\) be a nonempty, open, convex subset of a normed space and \(\lambda\in\,]0,1[\) be fixed. If \(f:D\to\mathbb R\) is \((\lambda,s)\)-convex and locally bounded from above at a point of \(D\), then \(f\) is locally bounded from above on \(D\). Theorem 2: Let \(D\) be a nonempty, open, convex subset of \(\mathbb R^n\) and \(f:D\to\mathbb R\) is \((\lambda,s)\)-convex function with fixed \(\lambda\in]0,1[\). Assume that there exist a set \(S\subset D\) of positive Lebesgue measure (respectively, of second Baire category) and a Lebesgue measurable (respectively, Baire measurable) function \(g:S\to\mathbb R\) such that \(f\leq g\), then \(f\) is locally bounded from above on \(D\).