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a <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b, f_0(x) = 2x and f_1(x) =x/(x-1)

Source: Austrian Polish 1984 APMC

April 30, 2020
functionsinequalitiesalgebra

Problem Statement

The functions f0,f1:(1,∞)→(1,∞)f_0,f_1 : (1,\infty) \to (1,\infty) are given by f0(x)=2x f_0(x) = 2x andf1(x)=xx−1 f_1(x) =\frac{x}{x-1}. Show that for any real numbers a,ba, b with 1≤a<b1 \le a < b there exist a positive integer kk and indices i1,i2,...,ik∈{0,1}i_1,i_2,...,i_k \in \{0,1\} such that a<fik(fik−1(...(fij(2))...))<ba <f_{i_k}(f_{i_{k-1}}(...(f_{i_j}(2))...))< b.
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