I saw the following version of the Hille-Yosida theorem in a book:
However, from the proof that I have seen, they only proof the bound
$$\left|\left|\frac{\partial Y}{\partial s}(s)\right|\right|=\left|\left|AY(s)\right|\right| \le \left|\left|AY_0\right|\right|$$
I know that the proof of the theorem follows from studying the Yosida approximations, but I am unable to prove $\left|\left|AY(s)\right|\right| \le \frac 1s\left|\left|Y_0\right|\right|$ from that.