Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\leq y\leq x^{3}$,then$(f(x))^2\leq f(y)\leq(f(x))^3.$ Please provide the smallest value of c when assuming $f(x)=x^c$ and c is a specific constant value such that the function satisfies the given property.

Let's think step by step and output the final answer within \boxed{}.
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