Let \(f: (0,\infty)\to\mathbb{R}\) be a continuous, strictly monotonic function, and let \(\mu\) be a probability measure on \([0,1]\). For \(x,y>0\), let \[ {\mathcal M}_f(x,y;\mu)= f^{-1} \Biggl[\int^1_0 f(\lambda x+ (1-\lambda)y)d\mu(\lambda)\Biggr], \] called the functional mean of \(x\), \(y\) with respect to the measure \(\mu\). For a particular function \(f\), or special measures \(\mu\) one can reobtain various means of two arguments, including the arithmetic, geometric, logarithmic, identric and power means. The author proves certain interesting theorems on the characterization of equality, homogeneity or monotonicity of such means, related to the function \(f\). We quote the following result: Let \(f\), \(g\) be continuous and strictly increasing on \((0,\infty)\). Then a necessary and sufficient condition in order that \({\mathcal M}_f(x,y;\mu)\leq{\mathcal M}_g(x,y; \mu)\) for all \(x\), \(y\) and \(\mu\), is that \(g\circ f^{-1}\) is convex. Functional and harmonic means of \(n\) arguments are introduced, too; and the corresponding properties are stated without proof.