A functional inequality (Q5917811)

Given extended real numbers \(a,b\) with \(a<b\) put \(C=\{(x,y)\in\mathbb{R}^2\): \(x>0\), \(a<y/x<b\}\). It is observed (among others) that a function \(f:C\to\mathbb{R}\) is subadditive and positively homogeneous iff \(f(1,\cdot)\) is convex and \(f(x,y)=xf(1,\frac yx)\) for \((x,y)\in C\). A generalization and examples of applications of this observation are also presented. Reviewer's remark: The above quoted observation for \(a=0\) and \(b=+\infty\) as well as its applications may also be found in some papers by \textit{J. Matkowski} [cf. e.g. the last part of Aequationes Math. 40, No. 2/3, 168-180 (1990; Zbl 0715.39013), part 4 of Proc. Am. Math. Soc. 117, No. 4, 991-1001 (1993; Zbl 0777.26013), and part 1 of Aequationes Math. 46, No. 3, 220-228 (1993; Zbl 0790.26016) (with coauthor \textit{J. Rätz})].