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Let $X, Y$ be two independent random variables, both uniformly distributed on $[0, 1]$ and let $g:\mathbb{R}\rightarrow \mathbb{R} $ be continuous function. Does following inequality always hold?

$E\left[\,{\left\vert\,{g(X)+g(Y)}\,\right\vert}\,\rule{0pt}{4mm}\right] \geq E[|g(Y)|]$

I think this is more calculus problem, as this inequality can be rewritten in the following way:

$\int_0^1\int_0^1|g(x)+g(y)|dxdy \geq \int_0^1|g(y)|dy$

It is easy to show that

$\int_0^1\int_0^1(g(x)+g(y))^2dxdy\geq \int_0^1g(y)^2dy$

but I don't know how to derive initial inequality.

River Li
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innerproduct
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  • Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Apr 22 '24 at 20:56
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    Please, use descriptive titles. – jjagmath Apr 23 '24 at 00:16
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    The inequality is true. It is an old problem. – River Li Apr 23 '24 at 06:51
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    See: https://math.stackexchange.com/questions/4016141/the-weighted-average-absolute-value-of-pairwise-sum-of-any-finite-sequence-is-la/4016986#4016986, and https://math.stackexchange.com/questions/73945/stuck-trying-to-prove-an-inequality, and https://math.stackexchange.com/questions/507930/how-prove-this-inequality-sum-i-j-1nx-ix-j-ge-n-sum-i-1nx – River Li Apr 23 '24 at 06:57

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