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First of all we need to see whether the possibilities are there for positive x and if they do then in what range. So below is list of possible scenarions discussed in question and the possible values for x in given inequality.
x^2 > 2x => x^2 - 2x > 0 => x(x-2)>0 => x > 2
x^2 < 2x => x^2 - 2x < 0 => x(x-2)<0 => x < 2
x^2 < 1/x => x^3 < 1 => x < 1
1/x < 2x => x^2 > 1/2 => x > 1/Sqrt(2)
1/x > 2x => x^2 < 1/2 => x < 1/Sqrt(2)
1) 1/x > 2x > x^2 (possible for all x less than 1) 2) 2x > 1/x > x^2 (possible for all 1 > x > 1/Sqrt(2)) 3) 1/x > x^2 > 2x (Not possible because x^2 to be greater than 2x x needs to be greater than 2 and 1/x to be greater than x^2, x should be less than 1 so contradicting values)