In introductory books on C*-algebra, functionnal calculus is quicly presented as it is a powerful tool in the field. It is for example prooved that for an operator $a$ and a function $f$ continous on Sp$(a)$, $f(a)$ is well defined and that $||f(a)|| \leq \sup\{|f(x)|, x \in \text{ Sp}(a) \}$.
However for the moment I never find some reference that deal with functionnal inequality with two operators. For exemple it is known that for two operators a,b we have that Sp$(a+b) \subset$ Sp$(a) +$ Sp$(b)$ so personnaly for a function $f$ continuous on Sp$(a) +$ Sp$(b)$, I would expect to have an explicit inequality that bound $f(a+b)-f(a)$ and decay when $b$ decrease toward 0.
For the moment I only find one proof (see Continuity of functional calculus) that show that the it go to zero when b go to zero. But it's only a proof with abstract epsilon cutting that do not give any uniform bound. So I would like to know if there is any litterature that go into a more uniform majoration of this quantity.
Bounding differences $\|f(a+b)-f(a)\|$ in terms of $\|b\|$ is much harder than the norm bound from the first paragraph, so I'll just give you some pointers to the relevant literature.
As a first example, it is known [1] that there does not exist a constant $C>0$ such that $$ \|\,|a+b|-|a|\,\|\leq C\|b\| $$ for all bounded self-adjoint operators $a,b$.
Continuous functions $f\colon \mathbb{R}\to\mathbb{R}$ with the property that there exists a constant $C>0$ such that $\|f(a+b)-f(b)\|\leq C\|b\|$ for all bounded self-adjoint operators are called operator Lipschitz functions [2]. Similarly one can define operator Hölder functions, and this set is again a proper subset of Hölder functions.
To get bounds on the difference $\|f(a+b)-f(b)\|$, one typically uses the theory of double operator integrals [3,4]. If $a,b$ are bounded self-adjoint operators on $H$ with spectral measures $e,f$, then $$ f(a+b)-f(b)=\int_{\mathbb{R}}\int_{\mathbb{R}}\phi_f(s,t)\,de(s)\,b\,df(t) $$ with $$ \phi_f(s,t)=\begin{cases}\frac{f(s)-f(t)}{s-t}&\text{if }s\neq t,\\ 0&\text{if }s=t.\end{cases} $$ Then $\|f(a+b)-f(b)\|$ can be bounded by the norm of $\phi_f$ in the extended Haagerup tensor product $L^\infty(\mathbb{R},e)\otimes_{eh}L^\infty(\mathbb{R},f)$ times the norm of $b$.
[1] T. Kato: Continuity of the map $S \mapsto |S|$ for linear operators. Proc. Jpn. Acad. 49, 157–160 (1973)
[2] A. B. Aleksandrov and V. V. Peller. Operator Lipschitz functions. Russ. Math. Surv. 71 605 (2016)
[3] M. Sh. Birman and M. Solomyak. Double operator integrals in a Hilbert space. Integral Equations Operator Theory, 47(2):131–168 (2003)
[4] B. de Pagter, H. Witvliet, and F.A. Sukochev. Double Operator Integrals, Journal of Functional Analysis, 192(1):52-111 (2002)