Homogeneous symmetric means of two variables (Q2474087)

Let \(I\) be an open interval containing \(1\), \(f, g: \to \mathbb R\) be given continuous functions such that \(g(x) \neq 0\) for \(x \in I\) and \(h(x) := \frac{f(x)}{g(x)}\) \((x \in I)\) be strictly monotonic on \(I\). Let \(\mu\) be an increasing nonconstant function on \([0, 1]\) and \[ M_{f,g,\mu}(x,y) := h^{-1}\left(\frac{\int_{0}^{1}f(tx + (1-t)y)\,d{\mu(t)}}{\int_{0}^{1}g(tx + (1-t)y)\,d{\mu(t)}}\right). \] In the above formula the integrals are Riemann-Stieltjes ones. \(M_{f,g, \mu}\) is a mean i.e. \(\min \{x, y \} \leq M_{f,g, \mu}(x,y) \leq \max\{x, y \}\) \((x \in I)\). Suppose that \(f\), \(g\) are six times continuously differentiable and \(h'(x) \neq 0\) \((x \in I)\). If \({\mu}_{1}(0) = 0\), \({\mu}_{1}(1) = 1\) and \({\mu}_{1}(t) = \frac{1}{2}\) for \(0 < t < 1\) and \(M_{f,g,\mu_{1}}\) is homogeneous i.e. \(M_{f,g,\mu_{1}}(tx,ty) = tM_{f,g,\mu_{1}}(x,y)\) for \(t \in I_{x} \cap I_{y}\), \(x, y \in I\), where \(I_{x} = \{t \in \mathbb R: tx \in I \}\), then \(h'(x)g(x)^{2} = \beta x^{\alpha - 1}\) \((x \in I)\) and \[ S(x) = \frac{1 - \alpha^{2}}{4} x^{-2} + (k - \frac{1 - \alpha^{2}}{4})x^{2(\alpha - 1)}\quad\text{or}\quad S(x) = lx^{-2} \tag{*} \] for \((x \in I)\) and for some \(\alpha, \beta, k, l \in \mathbb R\), where \(H(x) := \frac{h''(x)}{h'(x)}\) and \(S(x) := \frac{1}{2}(H'(x) - \frac{H^{2}(x)}{2})\) \((x \in I)\). If \(\mu_{2}(t) = t\) for \(t \in [0, 1]\) and \(M_{ f,g,\mu_{2}}\) is homogeneous, then \(h'(x)g(x)^{2} = \beta^{3(\alpha - 1)}\) \((x \in I)\) and (*) holds. Let \(W_{f,g}(x,y) := M_{f,g, \mu_{1}}(x,y)\) and \(D_{f,g}(x,y) = M_{f',g',\mu_{2}}(x,y)\). The author finds all pairs of functions \(f,g\) for which \(W_{f,g}\) and \(D_{f,g}\) are homogeneous.