Let $w:\mathbb{R}^m\rightarrow\mathbb{R}^m$ a strongly monotone map, that is, there exists a $\gamma>0$ such that
$$ [w(x+h)-w(x)]^\top h \geq \gamma\cdot \vert\vert h\vert\vert ^2. $$
I have a couple of questions then. First, let $K=K^\top\in\mathbb{R}^{m\times m}$ be positive definite. Is the following true?
$$ [w(x+Kh)-w(x)]^\top h \geq \gamma\cdot h^\top Kh. $$
I would say yes, but which is the (simple) argument to use in order to prove it?
A further question then. If $w$ is sufficiently smooth, is the following true?
$$ [w(x+h)-w(x)]^\top y = h^\top \frac{\partial w(x)}{\partial x} y,\qquad \forall x,y,h\in\mathbb{R}^m. $$
All in all, this would help me in establishing that: $$ h^\top K\frac{\partial w(x)}{\partial x} h\geq\gamma \cdot h^\top Kh,\qquad\forall h\in\mathbb{R}^m, $$
but I would like also to know if all assertions are true.