Upper bound of infinity norm of stochastic matrix

Question

If $W$ is a doubly stochastic, symmetric positive definite matrix, each entry of $W $ is nonnegative, then we know $\mathbf{1}$ is an eigenvector of $W$ corresponding to eigenvalue 1. If a vector $y$ satisfies $y\bot\mathbf{1}$, how do we prove that $y^TW^TWy\leq \lambda_2(W^TW)\left\Vert y\right \Vert^2$, where $\lambda_2(W^TW)$ is the second largest eigenvalue of $W^TW$. Further, can we obtain a bound for $\left\Vert Wy\right\Vert_\infty$ where the bound is not $\left\Vert W\right\Vert_\infty\left\Vert y \right\Vert_\infty$?