Asymptote of the eigenvalues of a completely continuous operator

It is proved that if $\varphi(x)$ is the majorant of the $s$-numbers of a completely continuous operator $A$ (i.e., $\varphi'(x)\le0$, $s_n(A)\le\varphi(n)$) and if there are found numbers $\rho\in[0,1]$ and $r_0>0$ such that $r^\rho\varphi'(r)/\varphi(r)$ will be monotonic in $(r_0,\infty)$, then for some $\alpha>0$, $\varphi(\alpha x)$ will be a majorant of the eigenvalues of $A$.