How much randomness is needed to symmetrize a random variable?

Given a random variable $X$, an independent random variable $Y$ is called a symmetrizer of $X$ if the distribution of $X+Y$ is symmetric about the origin. The study of symmetrization resistance asks whether every such $Y$ must contain at least as much randomness as $X$. This problem was previously investigated for binary random variables in terms of variance and Shannon entropy.
In the current paper, we establish sharp symmetrization resistance results for exponential and geometric distributions. For an exponential random variable $X$, we prove that every absolutely continuous independent symmetrizer $Y$ satisfies $$ h_\alpha (Y)\ge h_\alpha(X), $$ where $h_\alpha(\cdot)$ is Rényi entropy of order $\alpha>0$, together with analogous inequalities for Tsallis entropy and variance. The proof is based on a differential inversion formula for exponential convolution, which converts the symmetrization constraint into a hazard-rate inequality. We obtain parallel results for geometric random variables using the corresponding discrete difference operator and a discrete tail comparison argument. Furthermore, we show that equality holds if and only if $Y$ is an independent copy of $-X$. Our results provide new examples of entropic and variance symmetrization resistant distributions.