Let $G\subset \mathbb{R}^2$ be some bounded domain such that $\overline{G}\subset (0,\infty)\times \mathbb{R}$. I want to estimate the integral $\int_G x^2 \, d \lambda$ from below in terms of $\int_G x\, d\lambda$. Using Cauchy-Schwarz inequality (as was suggested by Daniel Schepler below), we have:
$$ \int_G x^2 \, d\lambda\geq \frac{ \left(\int_G x \, d\lambda\right)^2}{\int_{G}1 \,d\lambda}. $$
My question is: Is there a better estimate?
Best wishes
There certainly isn't a uniform bound for all domains: If $G=(0,1/n)\times[0,1]$ then the ratio between the integral of the square and the integral without the square tends to zero as $n\to\infty$. $\int_Gx^2\,d\lambda$ is much smaller than $\int_G x\,d\lambda$. In fact, the first integral is $n^{-3}/3$ and the second one is $n^{-2}/2$.