The quasianalyticity problem of the class $C_{I}(M_n)$ for interval $I$ is known to be solved by the Denjoy-Carleman theorem. It follows from well-known Men'shov example that not only this theorem but the very statement of the quasianalyticity problem of the class $C_{K}(M_n)$ doesn't expand on the case of arbitrary continuum $K$ of the complex plain. The quasianalyticity problem was studied for Jordan domains and rectifiable arcs including quasismooth arcs by a number of authors. We discuss in this article theorems of Denjoy-Carleman type in the convex domains of the complex plane, more precisely, connection between R. S. Yulmukhametov criterion of quasianalyticity of the Carleman class $H(D,M_n)$ for arbitrary convex domain $D$ and R. Salinas criterion for the class $H(\Delta_{\alpha},M_n)$ with angle $\Delta_{\alpha}=\{z: |\arg z|\leq\frac{\pi}{2}\alpha,\ \ 0<\alpha\leq1\}$. The problem of quasianalyticity of the class $H(D,M_n)$ is to find necessary and sufficient conditions for sequence $M_n$ and point $z_0\in\partial D$ for quasianalyticity of the class $H(D,M_n)$ at this point. The answer to question of simultaneous quasianalyticity or nonquasianalyticity these Carleman classes at a point $z=0$ has been obtained in therms of special integral condition which characterizes the degree of proximity of the domain boundaries $D$ and the angle $\Delta_{\alpha}$ in the neighbourhood of origin. Geometric interpretation of this integral condition and explicit examples illustrating essentiality of this condition are given.