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

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.


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
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