On new generalizations of certain mean values by H. Haruki and Th. M. Rassias (Q2747437)

Some characterizations of mean values in the spirit of \textit{H. Haruki} and \textit{Th. M. Rassias} [J. Math. Anal. Appl. 202, No. 1, 333-348 (1996; Zbl 0878.39005)] are presented.NEWLINENEWLINENEWLINEThe author considers some mean values generalizing ordinary arithmetic, geometric and harmonic means of positive numbers \(a,b\). He considers also the power mean NEWLINE\[NEWLINEP_q(a,b)= \left( {a^q+b^q \over 2}\right)^{1\over q},NEWLINE\]NEWLINE (for \(q=0\), \(P_0(a,b)\) denotes the geometric mean). Some functional equations involving mean values and the properties of their solutions are discussed. It is worth to note that some relations of mean values and the Bessel function of the first kind are found.