Let $\varphi(x)$ be a convex downwards function, $A>0$ a selfadjoint operator in a Hilbert space $H$, $P$ an orthogonal projector in $H$; suppose $D_A\cap PH$ is dense in $PH$, and let $A_p$ be the Friedrichs extension of the operator $PAP$ defined on $D_A\cap PH$. The inequality $\mathrm{Sp}\varphi(A_p)\leqslant\mathrm{Sp}\varphi(PAP)$ is proved. An estimate for the Jacobi $\theta$-function and a distant generalization of the Szasz inequality are obtained as corollaries. Bibliography: 3 titles.