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Function and inequality

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October 5, 2010
functioninequalitieslogarithmsalgebra unsolvedalgebra

Problem Statement

Let a,b,ca, b, c be positive real numbers and let [x][x] denote the greatest integer that does not exceed the real number xx. Suppose that ff is a function defined on the set of non-negative integers nn and taking real values such that f(0)=0f(0) = 0 and f(n)≤an+f([bn])+f([cn]), for all n≥1.f(n) \leq an + f([bn]) + f([cn]), \qquad \text{ for all } n \geq 1. Prove that if b+c<1b + c < 1, there is a real number kk such that f(n)≤kn for all n(1)f(n) \leq kn \qquad \text{ for all } n \qquad (1) while if b+c=1b + c = 1, there is a real number KK such that f(n)≤Knlog⁡2nf(n) \leq K n \log_2 n for all n≥2n \geq 2. Show that if b+c=1b + c = 1, there may not be a real number kk that satisfies (1).(1).
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