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Given 3 different investments F, G and H, all of them are part of the FSD set (neither of them is superior FSD on the other). Is there any investment H so no investor with utility function U'>0 would prefer it?

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There are plenty. Consider $M(x) = \max \{ F(x), G(x), H(x) \}$. You can show that $M(x)$ is also a CDF and that $M(x)$ is weakly FSD-dominated by any of $F, G, H$. To play safe, shift $M(x)$ leftwards by an amount $a > 0$ and consider $M_a(x) = M(x+a)$. Now $M_a(x)$ is strictly FSD-dominated by any of $F, G, H$ for any $a>0$.

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    I don't understand how this addresses the question I want to show that there is (or not) investment that is **part of the FSD set** but no investor would prefer it, your investment is dominated by all the given set so clearly no investor would prefer it, moreover I asked about 3 **given** investments and you created another one. – 2017-03-19
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    Perhaps I do not understand your question. It said "is there any **other** investment". And "no investor would prefer it" to me means that it is FSD-dominated. – 2017-03-19
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    No, the **"other"** is not part of the question. also the investment H is part of the three – 2017-03-26
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    So you want to know whether is it possible that there is no investor with increasing utility function who would pick an investment $H$ in the FSD-set? – 2017-03-26
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    Basically yes, appreciate your dedication. – 2017-03-27
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    The answer is no. But it's a bit of a pain to write a full proof. – 2017-03-27