The authors give some inequalities of Hadamard type for \(h\)-convex functions. The function \(f\:I\rightarrow \mathbb R\) is called an \(h\)-function if, for all \(x,y\in I\) and \(\alpha \in (0,1)\), \(f(\alpha x+(1-\alpha)y)\leq h(\alpha)f(x)+h(1-\alpha)f(y)\) holds, where \(h\:J\rightarrow \mathbb R\), \((0,1)\subseteq J\), is a nonnegative function. If \(f\in L_1(a,b)\), \([a,b]\subseteq I\) and \(g\:[a,b]\rightarrow \mathbb R\) is a nonnegative, integrable and symmetric function with respect to \({a+b \over 2}\), then the following inequality is proved: \[ \int \limits _a^b f(x)g(x)\,dx\leq {f(a)+f(b) \over 2}\int \limits _a^b \Biggl (h\left (\frac {b-x}{b-a}\right)+h\left (\frac {x-a}{b-a}\right)\Biggr) g(x)\,dx. \] Also, some other inequalities of similar type are given.