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Approximate convexity of Takagi type functions. (English) Zbl 1201.26003

The authors provide sufficient conditions on a bounded function \(a: \mathbb{R}\to \mathbb{R}\) in order that the Takagi type function \[ T(x)= \sum^{+\infty}_{n= 0} {a(2^n\cdot x)\over 2^n} \] to be approximately Jensen convex, to the effect that \[ T\Biggl({x+ y\over 2}\Biggr)\leq {T(x)+ T(y)\over 2}+ a\Biggl({x-y\over 2}\Biggr)- a(0) \] for all \(x,y\in\mathbb{R}\). Applications to the theory of approximately convex functins are also pointed out.

MSC:

26B25 Convexity of real functions of several variables, generalizations
26A51 Convexity of real functions in one variable, generalizations
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