Let \(\Omega\) be a simply connected domain in the complex plane and \(\phi:\Omega\to{\mathbb D}\) be the corresponding Riemann mapping. Let \(A_{-1}^2(\Omega)\) be the Hardy-Smirnov space, and for \(\alpha>-1\), let \(A_\alpha^2(\Omega)\) be the Bergman space which is the closure of holomorphic functions on \(\Omega\) in the Lebesgue space \(L_\alpha^2(\Omega)\) equipped with the norm given by \(\|f\|_\alpha^2=\int_\Omega|f(z)|^2\,dA_\alpha(z)\), where \(dA_\alpha(z)=(1+\alpha)\frac{1-|\phi(z)|^2)^\alpha}{|\phi'(z)|^\alpha}\frac{dx\,dy}{\pi}\). By \(P\) denote the orthogonal projection from \(L_\alpha^2(\Omega)\) to \(A_\alpha^2(\Omega)\). For a suitable symbol \(\psi\), the Hankel and Toeplitz operators are defined by \(H_\psi(f)=(I-P)(\psi f)\) and \(T_\psi(f)=P(\psi f)\), respectively, where \(f\in A_\alpha^2(\Omega)\). The main result of the paper says that for \(\alpha\geq -1\) and a holomorphic function \(\psi\) on \({\mathbb D}\) such that \(\psi'\in A_\alpha^2({\mathbb D})\), one has \[ \|H_{\overline{\psi}}\|_{A_\alpha^2({\mathbb D})\to A_\alpha^2({\mathbb D})}\leq (2+\alpha)^{-1/2}\|\psi'\|_{A_\alpha^2({\mathbb D})}. \] This estimate is sharp. The above result implies the following improvement of the classical inequality of C. R. Putnam [Math. Z. 116, 323–330 (1970; Zbl 0197.10102)] for commutators of Toeplitz operators with analytic symbols: if \(\psi\) is an analytic function on a simple connected domain \(\Omega\), then \[ \|T_\psi^*T_\psi-T_\psi T_\psi^*\|_{A_\alpha^2(\Omega)\to A_\alpha^2(\Omega)} \leq (2+\alpha)^{-1/2}\|\psi'\|_{A_\alpha^2(\Omega)}, \] which was conjectured by S. R. Bell et al. [Math. Proc. R. Ir. Acad. 114A, No. 2, 115–132 (2014; Zbl 1329.47026)]. As an application, this yields a new proof of the de Saint-Venant inequality, which relates the torsional rigidity of a domain with its area.