A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L [ f ] , we prove that there exists a unique two variable mean M [ f ] such that f ( x ) − f ( y ) x − y = M [ f ] ( f ′ ( x ) , f ′ ( y ) ) for all x , y ∈ I , x ≠ y . The mean M [ f ] is closely related L [ f ] . Necessary and sufficient condition for the equality M [ f ] = M [ g ] is given. A family of means { M [ t ] : t ∈ R } relevant to the logarithmic means is introduced. The invariance of geometric mean with respect to mean-type mappings of this type is considered. A result on convergence of the sequences of iterates of some mean-type mappings and its application in solving some functional equations is given. A counterpart of the Cauchy mean-value theorem is presented. Some relations between Stolarsky means and M [ t ] means are discussed.