Let \(p:\mathbb{R}^{+}\rightarrow \mathbb{R}\) be a strictly monotonic function, \(H\) denote the harmonic mean, \(r\) a real number in \([0,1],\) and \[ t_{n}(\theta)=\left( a^{2n}\cos ^{2}\theta +b^{2n}\sin ^{2}\theta \right) ^{1/n} \quad (n\neq 0). \] The authors study mean values defined by the expressions: \[ M(a,b;p,t_{n})=\frac{1}{H(a,b)}p^{-1}\left( \frac{1}{2\pi }\int_{0}^{2\pi }p(t_{n}(\theta)) d\theta \right) \] or \[ M_{r}(a,b;p,t_{n})=\frac{ra+(1-r)b}{ab}p^{-1}\left( \frac{1}{2\pi } \int_{0}^{2\pi }p(t_{n}(\theta)) d\theta \right) . \] For \(n=\pm 1\) , the first expression was studied by the second author [\textit{Y.-H. Kim}, J. Math. Anal. Appl. 235, No. 2, 598-607 (1999; Zbl 0954.26004)].