This paper investigates the bounds on channel capacity and constellation shaping under memoryless mixed noise, which is composed of impulsive noise (IN) and white Gaussian noise (WGN). The capacity bounds are derived using the entropy power inequality and the dual expression of capacity. It is then shown that the proposed lower and upper bounds asymptotically converge to the true channel capacity, and the analytic asymptotic capacity expression is obtained. Leveraging this property, we design a low-complexity constellation shaping method that operates without iterative procedures. Simulation results demonstrate that the derived bounds are remarkably tight, and the shaped constellation achieves the highest mutual information among all considered baseline schemes.