In this paper, we define positive functionals by using the Jensen's inequality, converse of Jensen's inequality, and Jensen-Mercer's inequality on time scales for superquadratic functions. We give mean-value theorems and introduce related Cauchy-type means by using the functionals mentioned above and show the monotonicity of these means. We also show that these functionals are exponentially convex and give some applications of them by using the log-convexity and exponential convexity.