The problem discussed is a classical problem of calculus of variations: \[ \begin{cases} J(w)= \int_G \{a(x, Dw)+ b(x, w)\}dx\to \min\\ \text{with } w(x)= g(x)\text{ on }\partial G,\end{cases}\tag{1.1} \] (\(Dw\) is the gradient). The authors consider the class of such problems where rates of growth of the functions \(a\) and \(b\) are given and the bounds for \(g(x)\) and measure of \(G\) are fixed and the subsets of such problems whose minimizers are spherically symmetric and upper bounds are spherically symmetric rearrangements of other problems. Symmetrization has been considered as an important tool in the past. For example, \textit{H. Weinberger} applied it to uniformly elliptic problems in 1962 [Stud. Math. Anal. related Topics, Essays in Honor of G. Pólya 424-428 (1962; Zbl 0123.072)], \textit{C. Bandle} to parabolic problems in 1976 [J. Anal. Math. 30, 98-112 (1976; Zbl 0331.35036)]. First, the authors prove that any minimizer of the problem (1.1) can be estimated via rearrangement in the style of \textit{G. H. Hardy}, \textit{J. E. Littlewood} and \textit{G. Pólya} [``Inequalities'' (1988; Zbl 0634.26008)]. Then a comparison of Orlicz norms and lengthy strings of inequalities produce the required estimate.