Best constants in reversed Hardy's inequalities for quasimonotone functions

Let f be a nonnegative measurable function on (0,infty) and define the operators (Hsb1f)(x)=x^{-1}intsb0sp xf and (Hsb2f)(x)=x^{-1}intsb xsp infty f. Then consider the inequalities (1) |xspalpha(Hsb if)|sb{Lsp p}le Csb1 |xspalpha f|sb{Lsp p} and (2) |xspalpha f|sb{Lsp p}ge Csb2 |xspalpha(Hsb if)|sb{Lsp p}, where Csb1, Csb2>0 are constants independent of f, 0<pleinfty, i=1,2, alphainbold R. By a classical result of G.~H.~Hardy it is known that, under suitable assumptions on alpha, inequality (1) holds when pge1, while inequality (2) holds when 0<p<1, on the class of all nonnegative functions f. Moreover, in this case the best possible constants Csb1, Csb2 are known. par The authors present several proofs of "reversed Hardy inequalities", i.e. inequalities of the type (1) with 0<p<1 or of the type (2) with pge1, on the classes of (quasi-) monotone functions and find the best possible constants in all cases. They also give some applications. par The results of the paper may be regarded as a unification and generalization of some results recently obtained by Bergh [Math. Z. {bf 202} (1989), no.~1, 147--149; [msn] MR1007745 (90h:26028) [/msn]], Burenkov [Trudy Mat. Inst. Steklov. {bf 194} (1992), Issled. po Teor. Differ. Funktsiĭ Mnogikh Peremen. i ee Prilozh. 14, 58--62] and Persson [in {it Nonlinear analysis, function spaces and applications, Vol. 4 (Roudnice nad Labem, 1990)}, 127--148, Teubner, Leipzig, 1990; [msn] MR1151434 (93b:46067) [/msn]].