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Let $A_t$ be an $\alpha$-stable symmetric process, $0<\alpha\leq 2$, on$\mathbb{R}^d$ and $D\subset \mathbb{R}^d$ be a bounded domain. This paperpresents a proof, based on the classical Brascamp-Lieb-Luttinger inequalitiesfor multiple integrals, that the distribution of the first exit time of $A_t$from $D$ increases under Steiner symmetrization. Further, it is shown that whena sequence of domains $\{D_m\}$ each contained in a ball $B$ and satisfying the$\varepsilon$-cone condition converges to a domain $D'$ with respect to theHausdorff metric, the sequence of distributions of first exit times forBrownian motion from $D_m$ converges to the distribution of the first exit timeof Brownian motion from $D'$.  These results will then be used to establish inequalities involvingdistributions of first exit times of $A_t$ from triangles and quadrilaterals.The primary application of these inequalities is verifying a conjecture fromBa\~nuelos for these planar domains. This extends a classical result of P\'olyaand Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.

On Steiner Symmetrizations for First Exit Time Distributions