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The function g is bounded - [Iran Second Round 1987]

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December 19, 2010
functionalgebra proposedalgebra

Problem Statement

Let ff be a real function defined in the interval [0,+∞)[0, +\infty ) and suppose that there exist two functions f′,f′′f', f'' in the interval [0,+∞)[0, +\infty ) such that f′′(x)=1x2+f′(x)2+1andf(0)=f′(0)=0.f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0. Let gg be a function for which g(0)=0andg(x)=f(x)x.g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}. Prove that gg is bounded.
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