Let $x, y \in \mathbb{R}^{d}$ and let $d \times d$ matrix $A$ be symmetric and positive definite. We know that
$$x^{T}Ax \leq \lambda_{\max}(A) x^{T}x$$
where $\lambda_{\max} (A)$ is the maximum eigenvalue of $A$. Similarly, can we say that $x^{T}Ay \leq \lambda_{\max}(A)x^{T}y$?
Edit: there was counter example. Now is there any constant $c$ such that $x^{T}Ay \leq c x^{T}y$?