This exercise is a step to proving inequalities involving non-commuting elements of a C*-algebra. (In particular in the subsequent exercise 12). Unfortunately I do not see, how to prove part (a):
For $α > 0, t ≥ 0$ define $$f_α(t) := \frac{t}{1 + αt} = \frac{1}{α}(1-\frac{1}{1+αt}),$$ then for any hermitian elements $a, b$ in a C*-algebra $\mathscr{A}$ with $0 ≤ a ≤ b$ we have that $$f_α(a) ≤ f_α(b).$$
For $a$ and $b$ commuting it is trivial, but how can it be done in the general case? Is there maybe a clever way to write $f_α(b) - f_α(a)$? The biggest obstacle is, that a product of positive elements does not have to be positive itself.
I hope that the argument can also be used for unbounded operators.
Note that all of the work is in Exercise 8, which says that if $0\leq x\leq y$ in a C*-algebra and $x$ is invertible, then $y^{-1}\leq x^{-1}$. You will apply this with $x=1+\alpha a$ and $y=1+\alpha b$, and the rest follows easily.
For a solution to Exercise 8, see Does inversion reverse order for positive elements in a unital C* algebra? or For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$? or If $0\leq A\leq B$ on Hilbert space and $A^{-1}$ exists, show that $A^{-1}\geq B^{-1}$ (soon to have better titles).