On a sharp estimate for hankel operators and Putnam's inequality

We obtain a sharp norm estimate for Hankel operators with anti-analytic symbol for weighted Bergman spaces. For the classical Bergman space, the estimate improves the corresponding classical Putnam inequality for commutators of Toeplitz operators with analytic symbol by a factor of 1/2, answering a recent conjecture by Bell, Ferguson and Lundberg. 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.