A sharp Remez type inequalities for the functions with asymmetric restrictions on the oldest derivative

. For odd r ∈ N ; α, β > 0 ; p ∈ [1 , ∞ ] ; δ ∈ (0 , 2 π ) , any 2 π -periodic function x ∈ L r ∞ ( I 2 π ) , I 2 π := [0 , 2 π ] , and arbitrary measurable set B ⊂ I 2 π , µB (cid:54) δ/λ, where λ = (cid:16)(cid:13)(cid:13) ϕ α,βr (cid:13)(cid:13) ∞ (cid:13)(cid:13)(cid:13) α − 1 x ( r )+ + β − 1 x ( r ) − (cid:13)(cid:13)(cid:13) ∞ E − 1 0 ( x ) ∞ (cid:17) 1 /r , we obtain sharp Remez type inequality E 0 ( x ) ∞ (cid:54) (cid:107) ϕ α,βr (cid:107) ∞ E

For r ∈ N, G = R or G = I d , via L r ∞ (G) denote the space of all functions x ∈ L ∞ (G) having locally absolutely continuous derivatives up to (r − 1)-th order and satisfying the condition x (r) ∈ L ∞ (G).
In approximation theory, Remez-type inequalities play an important role in class T n (trigonomotric polynomials T of order no higher n), where B is arbitrary Lebesgue measurable set, B ⊂ I 2π , µB ≤ β.
This topic originates from Remez's article [1], in which he found the sharp constant C(n, β) in the inequality of the form (1.1) for algebraic polynomials.
In the inequality of the form (1.1) for trigonometric polynomials, in a number of works two-sided estimates for exact constant C(n, β) have obtained. Moreover, the asymptotic behavior of the constants C(n, β) is known for β → 2π [2] and for β → 0 [3]. A bibliography of papers on this subject can be found in [2] [5]. In the paper [3] proved the inequality for an arbitrary polynomial T ∈ T n with minimal period 2π/m and an arbitrary Lebesgue measurable set B ⊂ I 2π , µB ≤ β, where β ∈ (0, 2πm/n). Equality in (1.2) is achieved for the polynomial T (t) = cos nx + 1 2 (1 − cos β/2).

RI
In particular, such inequalities have been obtained for dierentiable periodic functions, trigonometric polynomials, and splines. In addition, in paper [15] investigates the relationship between sharp constants in Kolmogorov-type inequalities and in the corresponding Kolmogorov-Remez-type inequalities.
The relationship between sharp constants in Kolmogorov-type inequalities for periodic functions and the same inequalities for non-periodic functions on the real axis is studied in [18].
In this paper, we obtain the sharp Remez type inequality of various metrics on classes of functions with nonsymmetric restrictions on the highest derivative.

RP
Proof. It suces to prove the equality of the lemma for p < ∞.
Since the function ϕ is odd, it suces to take the minimum in the assertion of the lemma over c > 0.
Consider the function f (c) := To do this, consider two cases: Since there are equalities Since the function ϕ is upwardly convex on [0, ω], then for the points there is an inequality Therefore, taking into account the equality |ϕ (m + u 2 ) + c| = |ϕ (M − v 2 ) + c| , we conclude that Similarly, it is proved that In this way f (c) ≤ 0 for c ∈ 0, 1 2 ( ϕ ∞ − ϕ (M + δ 1 )) and inequality (2.4) is proved in this case. Let now c > ϕ ∞ . Then ϕ(t)+c > 0 for all t ∈ R where the function g t (c) := ϕ(t) + c strictly increasing in the variable c for every x t. Therefore, the function f (c) also strictly increases and cannot reach a minimum at c > ϕ ∞ . Thus (2.4) is completely proved. So, the function f (c) reaches its minimum in the interval M δ . In this case where δ 1 and δ 2 dened by the relations (2.3), and It is clear that as c ∈ M δ increases, the quantity z 1 = z 1 (c) is decreasing, while z 2 = z 2 (c) is increasing. In this case, the modules of the rst and fourth integrals in (2.5) decrease, while the modules of the second and third ones increase. Moreover, it is obvious that f ( ϕ ∞ ) > 0 and previously proved the inequality f 1 2 ( ϕ ∞ − ϕ (M + δ 1 )) ≤ 0.
RS Therefore, the minimum of the function f (c) is attained at the point c ∈ M δ satisfying f (c) = 0, which, in view of (2.5), can be rewritten in the form I 2ω \B δ |ϕ(t) + c| p−1 sgn(ϕ(t) + c)dt = 0. (2.6) From the last equality, by virtue of the criterion of the element of the best approximation in the space L p follows the assertion of Lemma 1.
If the function x ∈ L r ∞ (I 2π ) satises the condition where x ± (t) := max{x ± (t), 0}, and λ is chosen so that then for an arbitrary measurable set B ⊂ I 2π , µB ≤ δ/λ, the following inequality holds: where B δ := [M − δ 2 , M + δ 1 ], M is the local maximum point of the function ϕ α,β r from the interval I 2π , and δ 1 , δ 2 > 0 are such that Proof. Fix a function x ∈ L r ∞ (I 2π ) satisfying the conditions of the lemma. We denote by m the point of local minimum of the function ϕ α,β r from the interval I 2π nearest to the left of the point M . Then the points m/λ and M/λ are local minimum (maximum) points of the function ϕ α,β λ,r from the interval I 2π/λ . As in Lemma 1, for brevity we set ϕ := ϕ α,β λ,r .
It immediately follows from this that r(x, t) ≥ r(ϕ(·) + c, t), t ≥ 0. (3.6) Note that for any measurable set B ⊂ I 2π , µB ≤ δ/λ, the inequality holds and since the rearragment preserves the L p -norm of the function, then Hence, taking into account inequality (3.6) and the relation 2π ≥ 2ω, we obtain is dened by the equalities (2.1) and (2.2). Now, applying Lemma 1, we conclude that for any measurable sets B, µB ≤ δ/λ, we have the inequality Lemma 2 is proved.
Inequality (3.7) is sharp and becomes the equality for function x(t) = ϕ α,β r (t) − c p (ϕ α,β r , B δ ) and set B = B δ , where c p (ϕ α,β r , B δ ) is the constant of the best approximations of the function ϕ α,β r in the space L p (I 2π \ B δ ).
Proof. We x the function x ∈ L r ∞ (R). In view of the homogeneity of inequality (3.7), we can assume tha and, taking into account the denition of γ, we get .
Applying the Hörmander inequality [23] x (k) and then estimating E 0 (x) ∞ using inequality (3.7), we obtain the following inequality of the Kolmogorov-Remez type.
Theorem 2. . Under the conditions of Theorem 1, for any k ∈ N, k < r, has place inequality (3.10) where γ = r−k r+1/p , B δ := [M − δ 2 , M + δ 1 ], M is local maximum point of the function ϕ α,β r from the interval I 2π , and δ 1 , δ 2 > 0 are such that Inequality (3.10) is sharp and becomes the equality for those the same function and set as inequality (3.7).