A generalization of Ky Fan's inequality
Summary: Let \(P_{n,r}(\mathbf {x})\) be the generalized weighted means. Let \(F(x)\) be a \(C^1\) function, \(y = y(x)\) an implicit decreasing function defined by \(f(x,y) = 0\) and \(0 < m < M \leq m'\), \(n \geq 2\), \(x_i \in [m,M]\), \(y_i \in [m',M']\). Then for \(-1 \leq r \leq 1\), if \(f'_{x}/f'_{y} \leq 1\), \[ |(F(P_{n,1}(\mathbf{y})) - F(P_{n,r} (\mathbf{y}))) /(F(P_{n,1}(\mathbf{x})) - F(P_{n,r} (\mathbf{x})))| \] \[ < \Bigl({\max_{m' \leq \xi \leq M'}|F'(\xi)|}\Bigr)\left/ \Bigl({\min_{m \leq \eta \leq M}|F'(\eta)|}\Bigr) \cdot {M}/{m'}.\right. \] A similar result exists for \(f'_{x}/f'_{y} \geq 1\). By specifying \(f(x,y)\) and \(F(x)\), we get various generalizations of Ky Fan's inequality. We also present some results on the comparison of \(P_{n,s}^{\alpha}(\mathbf{y}) - P_{n,r}^{\alpha} (\mathbf{y})\) and \(P_{n,s}^{\alpha}(\mathbf{x}) - P_{n,r}^{\alpha} (\mathbf{x})\) for \(s \geq r\), \(\alpha \in \mathbb{R}\).
scientific article; zbMATH DE number 1780280