Steiner symmetrization and the initial coefficients of univalent functions

We establish the inequality $|a_1|^2-\operatorname{Re}a_1a_{-1}\ge |a_1^*|^2-\operatorname{Re}a_1^*a_{-1}^*$ for the initial coefficients of any function $f(z)=a_1z+a_0+{a_{-1}}/z+\dotsb$ meromorphic and univalent in the domain $D=\{z\colon |z|>1\}$, where $a_1^*$ and $a_{-1}^*$ are the corresponding coefficients in the expansion of the function $f^*(z)$ that maps the domain $D$ conformally and univalently onto the exterior of the result of the Steiner symmetrization with respect to the real axis of the complement of the set $f(D)$. The Pólya–Szegő inequality $|a_1|\ge |a_1^*|$ is already known. We describe some applications of our inequality to functions of class $\Sigma$.