Sharp Nagy type inequalities for the classes of functions with given quotient of the uniform norms of positive and negative parts of a function

For any $p\in (0, \infty],$ $\omega > 0,$ $d \ge 2 \omega,$ we obtain the sharp inequality of Nagy type$$\|x_{\pm}\|_\infty \le\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega})}} \left\|x \right\|_{L_{p} \left(I_d  \right)}$$on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodiccomparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ is such that$$ \|x_{+}\|_\infty \cdot\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot\|(\varphi+c)_{-}\|^{-1}_\infty .$$In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms $\|x_{+}\|_\infty / \|x_-\|_\infty$.

Ключовi слова: Нерiвнiсть типу Надя, клас функцiй iз заданою функцiєю порiвняння, соболєвський клас, полiном, сплайн. For any p ∈ (0, ∞], ω > 0, d ≥ 2ω, we obtain the sharp inequality of Nagy type on the set S ϕ (ω) of d-periodic functions x having zeros with given the sine-shaped 2ω- In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and polynomial splines with given quotient of the norms x + ∞ / x − ∞ .
Key words: Nagy type inequality, a class of functions with given comparison function, Sobolev class of functions, polynomial, spline.
MSC2010: Pri 41A17, Sec 41A44, 42A05, 41A15 1. Introduction. Let G ⊂ R. We will consider the spaces L p (G), 0 < p ≤ ∞, of all measurable functions x : Let d > 0 and I d denote the circle which is realized as the interval [0, d] with coincident endpoints. For r ∈ N, G = R or G = I d , denote by L r ∞ (G) the space of all functions x ∈ L ∞ (G) for which x (r−1) is locally absolutely continuous and x (r) ∈ L ∞ (G).
For 2ω-periodic S-function ϕ denote by S ϕ (ω) the class of functions x ∈ L 1 ∞ (R) for which ϕ is the comparison function. Note that the classes S ϕ (ω) were considered in [1], [2]. Examples of such classes S ϕ (ω) are the Sobolev classes {x ∈ L r ∞ (I d ) : x (r) ∞ ≤ 1}, the bounded subsets of the space T n of all trigonometric polynomials of order at most n, and the same subsets of the space S n,r of polynomial splines of order r having defect 1 with knots at the points kπ/n, k ∈ Z.
It is shown in [3] that for p ∈ [1, ∞] and x ∈ L r ∞ (I 2π ) there holds the following sharp inequality of Nagy type where α = r r+1/p , ϕ r is the perfect Euler spline of order r and E 0 (x) ∞ is the best uniform approximation of the function x by constants.
In this paper we generalize the inequality (1.1) on the classes S ϕ (ω) of a function with given quotient positive and negative parts of a function (Theorem 1). In particular, we obtain such type inequalities for a function x ∈ L r ∞ (I 2π ) (Theorem 2) and for functions in spaces T n and S n,r (Theorem 3 and Theorem 4) with given quotient 2. The inequalities of various metrics on the classes of the functions with given comparison function. For a function f ∈ L 1 (I d ) denote by m(f, y), y > 0, the distribution function defined below and let r(f, t) be decreasing rearrangement (see, for example, Theorem 1. Let p ∈ (0, ∞] and ϕ is 2ω-periodic S-function. For any d-periodic function x ∈ S ϕ (ω) having zeros the inequality The inequality (2.2) is sharp on the classes of functions x ∈ S ϕ (ω) having zeros with given quotient x + ∞ / x − ∞ and becomes equality for the function x(t) = ϕ(t) + c.
Proof. Fix any d-periodic function x ∈ S ϕ (ω) having zeros. Since ϕ is comparison function for x, then there exists a constant c ∈ R satisfying By definition, the function ϕ is strictly increasing on [− ω 2 , ω 2 ]. For τ ∈ R set x τ (t) := x(τ + t), t ∈ R. Choose τ 1 , τ 2 ∈ R such that Since ϕ is comparison function for x, then
Denote by E 0 (f ) Lp(G) the best approximation of the function f by constants in the space L p (G) and let be the best one-sided approximations by constants of the function f in that space.

Corollary 1. Under the assumptions of Theorem 1 for any
and Besides, for a function x ∈ S ϕ (ω) having zeros the inequality x Lp(I d ) holds true.

1)
where α = r r+1/p and c ∈ [−K r , K r ] is such that The inequality (3.1) is sharp on the class of functions x ∈ L r ∞ (I 2π ) having zeros with given quotient x + ∞ / x − ∞ and becomes equality for the function x(t) = ϕ r (t) + c.
Proof. Fix a function x ∈ L r ∞ (R) having zeros. In view of homogeneity of the inequality (3.1) we can assume that Then by the Kolmogorov comparison theorem [5] the spline ϕ := ϕ λ,r is the comparison function for the function x. Consequently x ∈ S ϕ ( π λ ), and by Theorem 1 we have Besides, it follows from (3.3) in view of condition of Theorem 2 for constant c that Combining (3.4) and (3.5) we get Applying (3.5), (3.6) and taking into account the definition of α we obtain . It follows (3.1) in view of (3.2). Theorem 2 is proved.

Corollary 2. Under the assumptions of Theorem 1 for any
Besides, for a function x ∈ L r ∞ (I 2π ) having zeros the inequality holds true.
Remark 2. The first inequality is proved in [3] and the rest ones are proved in [6]. 4. Nikolskii type inequalities for trigonometric polynomials. Let us recall that T n is the space of all trigonometric polynomials of degree at most n.
Theorem 3. Let p ∈ (0, ∞], n, m ∈ N, m ≤ n. For any trigonometric polynomial T ∈ T n with minimal period 2π/m having zeros the inequality holds true, where c ∈ [−1, 1] is the constant satisfying The inequality (4.1) is sharp for m = 1 in the sense sup n∈N sup T ∈Tn(c) where T n (c) is the set of all trigonometric polynomials T ∈ T n having zeros with given quotient T + ∞ / T − ∞ satisfying (4.2).
Proof. Fix a polynomial T ∈ T n with minimal period 2π m having zeros. In view of homogeneity of the inequality (4.1) we can assume that Then the polynomial ϕ(t) := sin nt is comparison function for the polynomial T (t) (see, for example, the proof of Theorem 8.1.1 [7]). It is clear that ϕ is 2π n -periodic S-function. Hence T ∈ S ϕ ( π n ). Then by Theorem 1 (sin n(·) + c) ± ∞ = (sin(·) + c) ± ∞ .
Theorem 3 is proved.
Under the assumptions of Theorem 3 for a polynomial T ∈ T n with minimal period 2π/m we have and .
Besides, for a polynomial T ∈ T n with minimal period 2π/m having zeros T Lp(I 2π ) .

Remark 3.
For m = 1 the first inequality is proved in [3] and the rest ones are proved in [6].
5. Nikolskii type inequalities for periodic polynomial splines. Let r, n ∈ N. Recall that S n,r stands for the space of polynomial splines of order r having defect 1 with knots at the points kπ/n, k ∈ Z. It is clear that S n,r ⊂ L r ∞ (R). Theorem 4. Let p ∈ (0, ∞]; n, m ∈ N, m ≤ n. For a spline s ∈ S n,r with minimal period 2π/m having zeros the inequality The inequality (5.1) is sharp for m = 1 in the sense sup n∈N sup s∈Sn,r(c) where S n,r (c) is the set of all splines s ∈ S n,r having zeros with given quotient s + ∞ / s − ∞ satisfying (5.2).
Proof. Fix a spline s ∈ S n,r with minimal period 2π/m having zeros. In view of homogeneity of the inequality (5.1) we can assume that Then by the Tikhomirov inequality [8] s (r) ∞ ≤ E 0 (s) ∞ ϕ n,r ∞ = 1.
Hence all conditions of Kolmogorov comparison theorem [5] are fulfilled. By this Theorem the function ϕ(t) := ϕ n,r (t) is comparison function for the spline s. It is clear that ϕ is S-function with period 2π/n. So s ∈ S ϕ ( π n ). Then by Theorem 1 s ± ∞ ≤ (ϕ n,r + n −r c) ± ∞ ϕ n,r + n −r c Lp(I 2π (ϕ n,r + n −r c) ± ∞ = n −r (ϕ r + c) ± ∞ .
Besides, for a spline s ∈ S n,r with minimal period 2π/m having zeros s Lp(I 2π ) .

Remark 4.
For m = 1 the first inequality is proved in [3] and the rest ones are proved in [6].