Upper bound: Given $L$-smooth convex $f$; $( y- x)^T \left( \nabla f(z)-\nabla f(x)\right)\leq(L/2) ( \| x-z\|^2+\| x-y\|^2+\| z-y\|^2)$?
What about this simple estimate: $$ \left( y- x\right)^T \left( \nabla f(z)-\nabla f(x)\right) \le \|y-x\| \cdot \|\nabla f(z)-\nabla f(x)\| \le L \|y-x\| \cdot \|z-x\| \le \frac L2(\|y-x\|^2 + \|z-x\|^2) $$ ??