I have a function $f(x_1,x_2) \colon \mathbb{R}^2_{+} \to \mathbb{R}_{+}$ positive homogenous: $ f(\lambda x_1, \lambda x_2) = \lambda f(x_1,x_2), \; \forall \lambda > 0 $ and such that $f(x_1,x_2)$ permits decomposition $ f(x_1,x_2) = h(g(x_1)+g(x_2)) $ where $h,g$ are some continuous functions. One of appropriate functions is $ f_{0}(x_1,x_2) = C(x_1^{\gamma}+x_{2}^{\gamma})^{\frac{1}{\gamma}}, \; \gamma > 0, \; C \geq 0 $ Are there some other functions that satisfy specified conditions or $f_{0}(x_1,x_2)$ is unique?
Update If $h = g^{-1}$ then $f_{0}$ is a unique family of solutions (Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952) Inequalities. 2nd ed, page 68).