Let \(A,B\geq 0\) (i.e., \(A\) and \(B\) are nonnegative definite matrices of the same dimension) satisfy \(AB+BA\geq 0\), and let \(f\) be an operator monotone function on \([0,\infty)\). Denote \[ A\,\sigma_f\,B=A^\frac{1}{2}f(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^\frac{1}{2} \] if \(A,B>0\), and \[ A\,\sigma_f\,B=\lim_{\varepsilon\to 0}(A+\varepsilon I)\sigma_f(B+\varepsilon I) \] otherwise. The authors prove that if \(f\) is positive and \(f(1)=1\), then \[ A+B-|A-B|\leq 2\,A\,\sigma_f\,B. \] The reverse Cauchy inequality for positive definite matrices \[ \frac{A+B}{2}\leq A^\frac{1}{2}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^\frac{1}{2}A^\frac{1}{2}+ \frac{|A-B|}{2} \] is so extended. Define \(g(t)=t/f(t)\) for \(t>0\), and \(g(0)=0\). The authors also prove that if \(f(0)=0\) and \(f(t)>0\) for all \(t>0\), then \[ \|A+B-|A-B|\|\leq 2\,\|f(A)^\frac{1}{2}g(B)f(A)^\frac{1}{2}\| \] for any unitarily invariant norm \(\|\cdot\|\). The generalized Powers-Størmer inequality \[ \mathrm{tr}(A+B-|A-B|)\leq 2\,\mathrm{tr}\,(f(A)^\frac{1}{2}g(B)f(A)^\frac{1}{2}), \] proved by the first author et al. [Linear Algebra Appl. 438, 242–249 (2013; Zbl 1270.46057)], is so extended.