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Let $A$ and $B$ be not necessarily commuting self-adjoint operators. We define functions $f(A,B)$ for a certain class of functions $f$ with the help of double operator integrals. We are interested in the problem when we have Lipschitz-type estimates in different norms. We obtain the following Lipschitz norm estimate in the trace norm: $ \|f(A_1,B_1)-f(A_2,B_2)\|_{S_1}\leq{\rm const}\max\{\|A_1-A_2\|_{S_1},\|B_1-B_2\|_{S_1}\} $ for functions $f$ on the plane that belong to the Besov space $B_{\infty,1}^1$. On the other hand, unlike in the case of commuting operators, for functions in the Besov space $B_{\infty,1}^1$ there is no Lipschitz type estimates in the operator norm. The main tool is triple operator integrals.