Let $f$ be a convex function and $a \in \mathbb R$. Is it true that $$\int_0^\infty f(x) \cos(ax) dx \ge 0$$ holds? Of course, this would be true if we replaced $\cos$ by $|\cos|$, but I guess that the convexity of $f$ may compensate for the oscillations of $\cos$. Is it true?
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The integral need not exist. (e.g. $f(x)=1$). – Kavi Rama Murthy May 29 '22 at 12:22
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@Jun if f is negative, this fails. – user376343 May 29 '22 at 12:48
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You need to add some assumptions so that the integral is convergent. For example $f(x)$ could be decreasing and with limit $0$ at $\infty$ and $a\neq 0.$ – Ryszard Szwarc May 29 '22 at 17:22
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1A similar question was solved here by Ryszard Szwarc when integration is done over a particular interval. – Mittens May 29 '22 at 19:12