This brief note shows that if a production function, f, is quasiconcave, increasing and homogeneous, then f is concave if it displays nonincreasing returns to scale, and f is logconcave if it displays increasing returns to scale.
Reference:
Hjertstrand, Per (2020), "Homogeneity, Returns to Scale and (Log)Concavity". IFN Working Paper No. 1328. Stockholm: Research Institute of Industrial Economics.