On One Property of the Mean-Arithmetic Oscillations of a Monotone Sequence

For a number sequence Y = y i i = P + 1 Q ( the numbers P,Q ∈ Z are fixed and P < Q ) , we consider the mean-arithmetic oscillations Ω Y p q = 1 q − p ∑ i = p + 1 q y i − σ Y p q , where σ Y p q = 1 q − p ∑ i = p + 1 q y i is the arithmetic mean of the sequence Y on the segment [ p, q ] and P ≤ p < q ≤ Q are arbitrary numbers. These oscillations coincide with the mean-integral oscillations of the function f Y = ∑ i = P + 1 Q y i χ i − 1 i (χ E is the characteristic function of the set E ) : Ω f Y p q = 1 q − p ∫ p q f Y x − σ f Y p q dx , σ f Y p q = 1 q − p ∫ p q f Y x dx , on segments with integer boundaries. The main result of the paper is the following equality: max p q : P ≤ p < q ≤ Q Ω Y p q = max r ∈ Z : P ≤ r ≤ Q max Ω Y P r Ω Y r Q , which is true for any monotone sequence Y. In this case, the main point is the fact that the maximum on the right-hand side is taken only over all integer numbers r. This equality turns into a well-known equality if we take a function f Y instead of the sequence Y, replace the mean-arithmetic oscillations by the mean-integral oscillations and, in addition, assume that the number r on the right-hand side is not necessarily integer.