On continuous convex or concave functions with respect to the logarithmic mean. (English) Zbl 1098.26008

Zgraja, Tomasz

Acta Univ. Carol., Math. Phys. 46, No. 1, 3-10 (2005).

A function \(f\) is called \(M\)-convex, if \(f(M(x,y))\leq M(f(x),f(y))\) for all \(x,y\), where \(M\) is a mean. In this very interesting paper, the authors study certain properties of \(L\)-convex functions, where \(L\) is the famous logarithmic mean. For example, they prove that if \(f:(a,b)\to(0,+\infty)\) is a monotonic \(L\)-convex function, then it is continuous. Every continuous \(L\)-convex function is quasi-convex. If \(f:(a,b)\to(0,+\infty)\) is \(L\)-convex and \(f(x)/x\) increases with \(x\), then \(f\) is convex.

Reviewer: József Sándor (Cluj-Napoca)

MSC:

26A51	Convexity of real functions in one variable, generalizations
26D15	Inequalities for sums, series and integrals
39B72	Systems of functional equations and inequalities
26E60	Means

Keywords:

means and their inequalities