In this work, we introduce and study the class of k-convex functions, that is, the class of functions \(u\in C^2(I)\) satisfying the second-order differential inequality \( u''(t)+ku'(t)\ge 0,\quad t\in I, \) where I is an interval of \(\mathbb {R}\) and \(k\ne 0\) is a constant. Among many other results, a Fejér-type inequality for k-convex functions is established. Making use of the obtained inequality, a general Lyapunov-type inequality is obtained for the eigenvalue problem \( -\left( u''(t)+ku'(t)\right) =\lambda w(t)u(t),\quad a<t<b, \) where \(k>0\) , \(\lambda >0\) , and \(w\in C([a,b])\) is a positive function. Next, different boundary conditions are investigated. To the best of our knowledge, this work is the first one showing a connection between Fejér-type inequalities and Lyapunov-type inequalities.
