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Interesting inequality on functions - [ILL 1977]

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January 11, 2011
inequalitiesfunctionalgebra unsolvedalgebra

Problem Statement

Let ff be a strictly increasing function defined on the set of real numbers. For xx real and tt positive, setg(x,t)=f(x+t)−f(x)f(x)−f(x−t).g(x,t)=\frac{f(x+t)-f(x)}{f(x) - f(x - t)}. Assume that the inequalities2−1<g(x,t)<22^{-1} < g(x, t) < 2 hold for all positive t if x=0x = 0, and for all t≤∣x∣t \leq |x| otherwise. Show that14−1<g(x,t)<14 14^{-1} < g(x, t) < 14 for all real xx and positive t.t.
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