On quasi-gamma functions (Q2922474)

A function \(f:(0,\infty)\to (0,\infty)\) is called quasi-gamma function, if it satisfies the following conditions: \(f(1)= 1\); \(f(x+1)= xf(x)\) \((x>0)\); and \(f\) is quasi-convex. The authors study properties of such functions, by proving e.g. that they are continuous, are \(\leq 1\) on \([1,2]\), are decreasing on \((0,1]\) and increasing on \([2,\infty)\). Further, they prove that the set of all quasi-gamma functions is not a convex set, and is topologically bounded in \(C(0,\infty)\). Some subclasses are also introduced and their study initiated.