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In this paper we shall present discrete Kolmogorov type inequalities for multiply monotone sequences defined on non-positive integers. Moreover, we will provide a more delicate information by obtaining the description of the following modulus of continuity ω j,r p,q (δ ,e) = sup{‖∆ x‖q : x,∆x, . . . ,∆ j x ! 0, ‖x‖p = δ , ‖∆ x‖∞ = e} for δ ! e > 0 and values of j = r−2 or j = r−1 depending on values of other parameters. 1. Notation, definitions, and history Let M := Z− ∪ {0} = {· · · ,−2,−1,0} . We shall define the (forward) difference operator as follows: for the sequence x = {xm}m∈M set ∆x := {xm − xm−1}m∈M, ∆x := x, ∆x := ∆x, and, recursively, ∆ x := ∆(∆ )x for j = 2,3, . . . . Let lp = lp(M) , p ∈ [1,∞] , be the space of all real-valued sequences defined on M such that the norm ‖x‖p :=    ( ∑ m∈M |xm| p )1/p , 1 " p < ∞, sup m∈M |xm|, p = ∞, is finite. Note that in contrast with the derivative operator D = d dt , the difference operator ∆ is a bounded linear operator defined on all lp(M) for any 1 " p " ∞ . In this paper we shall consider sharp discrete inequalities of Kolmogorov type, i.e. inequalities of the form ‖∆x‖q "C‖x‖ α p‖∆ x‖ s , x ∈ lp(U), ∆ x ∈ ls(U) (1) where U = M or U = Z and α ∈ [0,1] is defined by relation (9) below. The exact constants C = C(k,r, p,q,s) in inequalities of this form have been found in the following cases: Mathematics subject classification (2010): 26D10, 47B39, 39A70, 65L12.

Inequalities for the norms of finite difference operators of multiply monotone sequences