The author proves certain integral inequalities related to the Jensen-Hadamard inequalities for convex functions. We quote only the following result: Let \(f:[a, b]\to \mathbb{R}\) be a convex function, and put \[ F(x,y)(t)= \textstyle{{1\over 2}}[f(tx+ (1- t)y)+ f((1- t) x+ ty)], \] where \(x,y\in [a,b]\), and \(t\in [0,1]\). Let \[ G(t)= {1\over b-a} \int^b_a F\Biggl(x,{a+ b\over 2}\Biggr)(t)\,dx. \] Then \(G\) is convex on \([0,1]\), and if \(f\) is differentiable, then \(G(t)\leq M(f,a,b)\) – the integral mean of \(f\) on \([a,b]\). Results of this type have been obtained also by S. S. Dragomir, D. M. Milošević and the reviewer [Publ. Elektroteh. Fak., Univ. Beogr., Ser. Mat. 4, 3–10 (1993; Zbl 0813.26005)].