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Convex functions with respect to logarithmic mean and sandwich theorem. (English) Zbl 0976.26003

Let \(L\) denote the logarithmic mean. A function \(f: (0,\infty)\to (0,\infty)\) is called \(L\)-convex if \[ f(L(x,y))\leq L(f(x), f(y)) \] for all \(x,y> 0\), \(L\)-concave if the reversed inequality holds true, and \(L\)-affine if the equality holds true. The authors prove the existence of a continuous \(L\)-convex function \(f\) and a continuous \(L\)-concave function \(g,g\leq f\), for which there is no \(L\)-affine function \(h\) such that \(g\leq h\leq f\).
Moreover, they prove the existence of some functions \(f\), \(g\) satisfying the inequality \[ g(L(x,y))\leq L(f(x), f(y)),\quad (x,y)> 0, \] for which there is no \(L\)-convex function \(h\) such that \(g\leq h\leq f\). This shows that the “sandwich” type theorem for convex functions [K. Baron, J. Matkowski and K. Nikodem, Math. Pannonica 5, No. 1, 139-144 (1994; Zbl 0803.39011)] cannot be extended to \(L\)-convex functions.

MSC:

26A51 Convexity of real functions in one variable, generalizations
26E60 Means

Citations:

Zbl 0803.39011
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