#### Abstract

The present paper discusses in the metric S. N. Bernstein type inequalities of the most general kind on very general accessible classes of curves in a complex plane. The obtained estimations, generally speaking, are not improvable.

#### 1. Introduction

The estimations connecting the norms of derivatives of polynomials with the norm of the polynomial itself are usually called the Markov-Bernstein type estimations. Therewith, the similar global estimation in the metric was obtained by Markov. Bernstein considered the similar estimation in the metric for trigonometric polynomials and also local estimation in the metric . Bernstein type local estimation, which is precise in the sense of order in the metric , was obtained by Dzjadyk. Further, Dzjadyk considered this estimation in a complex plane. Earlier, such a global estimation in a complex plane in the metric was obtained by Mergelyan [1]. Validity of such estimations on arbitrary compacts in a complex plane in the metric was shown in the papers of Lebedev and Tamrazov [2]. Similar problems in the mean, namely, in the metric have their own specification that does not allow to consider such estimations on wide classes of sets in a complex plane. For a long time, the validity of such estimations was known on very narrow classes of curves of a complex plane. These results, in particular, are given in [3].

One of the theorems with appropriate Bernstein inequality is announced in [4]. Some auxiliary statements, by means of which such inequalities are proved, are in [4].

In the sequel, we will need the following facts.

#### 2. Preliminary Notes

Let be a closed curve in a complex plane with parametric representation (, is the length of ) of diameter the function maps the exterior of a unit circle onto the exterior of , and maps the interior of on the interior of ; the functions and are inverse to the functions and , respectively; is a level line of the curve corresponding to the equation .

Let be some fixed point on , let be the distance from the point to the curve ,, and.

Let us consider a class of curves , for which for . We denote this class of curves by . It is easy to show that the class coincides with the class introduced by Salayev [5].

Recall that the curve belongs to the class , (Salayev’s class) if there exists a constant , such that , where , (Lebesgue measure), and .

So, the following statement [4] is true.

Statement 2.1 (). By we denote a class of Jordan rectifiable curves , for which the following relation [4] is valid ( is a distance from the point to the curve ) for the given and all .

The following statement [4] is also valid.

Statement 2.2 (If then ). We will say that the function is almost increasing in uniformly in , if there exists a constant not depending on such that for any the following inequality is fulfilled.

Note that many known classes of rectifiable curves, in particular the curves of the class (Salayev's class), satisfy the condition that is almost decreasing.

By we denote a subclass of the class of curves , for which almost decreases. For the classes of curves and , the following statement is valid [4].

Statement 2.3. There hold the embeddings

Now, let us consider the quantity

#### 3. Main Results

In particular, using the previously mentioned statement, we can prove the following theorems.

Theorem 3.1. Let be an arbitrary rectifiable Jordan curve on which for any and any natural the following estimation is valid (signs and define an ordinal relation. Namely, means . And means ): where
Then where is an algebraic polynomial of degree , .

We can also prove a theorem of independent character used in the proof of Theorem 3.1.

Theorem 3.2. Under the conditions of Theorem 3.1 on the curve , whatever was the natural number and for the th-order derivative of the polynomial of degree , for the following inequality is valid.

A special case of these theorems is similar theorems for concrete classes of curves, namely, for the following classes.

() -quasiconformal mapping. The curve , being an image of the circle under some -quasiconformal mapping of the plane onto itself, is said to be -quasiconformal curve. The class of curves will be denoted by .

() We will say that the set with rectifiable Jordan curve belongs to the class [3] for some (or ), if and satisfies the following conditions:

()

()

As Dzjadyk shows [3, page 393], the validity of the condition

that is equivalent to the following geometric property of domain [6] follows from conditions () and ) of the class .

() We can connect any points of by the arc whose length satisfies the inequality

Furthermore, [6, Lemmas and ], the following conditions are valid for the set of the class :

() if , then

() if , , then

Note that the -quasiconformal curves [7] satisfy conditions ()–() and relation (3.5).

Consider some more general classes.

() We will say that (or ), if conditions () and () are fulfilled.

() We will say that with a rectifiable boundary belongs to (or ), if , and conditions () or its equivalent relation (3.5) is fulfilled for it.

Obviously, the class of the sets , possessing pure geometric description, contains the classes of the sets .

So, the following theorems are true.

Theorem 3.3. Let be an arbitrary rectifiable -quasiconformal curve. Then, whatever was the natural number and the number for the th order derivative of the polynomial of power for , the following inequality is valid:

Theorem 3.4. Let for some natural belong to the class . Then, whatever was the natural number and (under some additional condition on the curve , Theorem 3.4 remains valid for any (see Remark 5.1).) for the th-order derivative of the polynomial of power for , the following inequality is valid:

The special case of these theorems is announced in [8] and is cited in [9, 10] with incomplete proof.

Remark 3.5. The special case of Theorems 3.3 and 3.4 was also proved in [11] for curves consisting of infinitely many smooth arcs; each of these arcs has continuous curvature, and at the joint points they form between themselves external angles such that , that is, on the curves of the class .

In this paper, we give a complete proof of Theorems 3.3 and 3.4. Theorems 3.1 and 3.2 are proved by the same method Theorems 3.3 and 3.4 with the usage of Statements 2.12.3.

#### 4. Auxiliary Lemmas

When proving Theorems 3.3 and 3.4 we’ll need the following.

A nonnegative function given on the plane will be said to be admissible if

If is a family of locally rectifiable curves on the plane, we put (if is not measurable on , we assume that). If is a class of admissible functions, then the quantity is said to be external length of , and its inverse quantity a modulus of is

Let be an arbitrary one-connected domain of a complex domain containing the point ; let be a complement to ; let be their common boundary; let be a function that conformally and univalently maps onto exterior of a unit circle and is normed by the following condition:

be a level line of the continuum ; let , for ; let , for .

The following statements are valid.

Lemma A [see [12, Lemma ]] Let be an arbitrary continuum with connected complement , ,

If , , then

where is a family of curves isolating the points and in (in simplest cases ) from the points and and .

Lemma B [see [13, Theorem ]] Let be an arbitrary continuum with connected complement. Then for or where is a distance from the point to .

Lemma C [see [14, Lemma ]] Let realizes -quasiconformal mapping of plane onto itself, . are, respectively, - and -complex planes; .

Then we have the following:

) the conditions and are equivalent, and consequently the conditions are also equivalent;

() if , then where .

Lemma D [see [11]] Let be a domain with a rectifiable boundary , and . If , then for any and for all the following inequality holds:

Let be some arbitrary fixed point lying outside of , and let be a distance from the point to , and .

To prove these theorems we will need the following lemmas.

Lemma 4.1. Let a rectifiable curve , then for a polynomial of power for the following inequality is valid for :

Lemma 4.2. Under conditions of Lemma 4.1 on the curve , for a polynomial of power for , and the following inequality is valid: where under one understands a distance from the point to the level line , where .

Lemma 4.3. Let . Then whatever was a natural number and , the inequality is valid.

Lemma 4.4 (see [9]). Let . Then for and all the relation (2.1) is valid; that is, the imbedding is valid.

Lemma 4.5 (see [9]). Let . Then for and all the inequality is valid.

Proof of Lemma 4.1. Let an arbitrary rectifiable curve . At first we consider the case . İntroduce some auxiliary function where
Obviously, , as and each of its branchs is holomorphic in and continuous in . Therefore . Consequently, we can apply to Lemma D where by estimation of Lemma B we will have

Now, if we consider that , for and the relations , which is valid for any , then for the proof of (4.13) it suffices to prove the validity of the relation

where .

Let Obviously . By the property of curves of the class , we have

Prove that

Obviously, it suffices to prove that

Let , be such that

Following Belyi [7], we take in the ring

a segment and an arc of a circle connecting the points and . Let . Construct a family of circles with a center at the point , intersecting . Each of these has an annular arc in , intersecting . We denote a family of such arcs by . Obviously, the family separates in the point and some point (in the simplest cases ) from and . Therefore, by Lemma A we have

Hence (4.22) and relation (4.21) together with (4.20) prove (4.19),

So, Lemma 4.1 is proved in the case .

The proof in the case is conducted by means of analytic reasoning after introducing the auxiliary function

The proof of Lemma 4.2 is conducted in the same way.

Indeed, in the case , instead of relation (4.18) from Lemma D we’ll have

Therefore, in order to prove the statement of Lemma 4.2, obviously, it suffices to see the validity of the relation

and since the estimation is obvious, we have to show that

This relation is proved exactly in the same way as relation (4.19) in Lemma 4.1.

The case is proved similarly.

Proof of Lemma 4.3. Let . Consider two possible cases.
() We have . The case follows from Lemma 4.4.
Let , where , and . Then . By the property of the class , we will have and (see [3, page 393]) Now, by (4.30) and (4.31), we will get Hence we will get Now, if we take into account and then by Lemma 4.4, for we will get
() We have . By the property of the class of curves , we will have hence Hence, using Lemma 4.4 So, Lemma 4.3 is proved.

#### 5. Proofs of Theorems

Proof of Theorem 3.3. Consider the case . Let be an arbitrary rectifiable -quasiconformal curve. By the Cauchy formula, we will have where denotes a closed curve containing the point interior to itself, and that is defined in the following way.
Let the point under the mapping go over to the point (Figure 1).Draw a circle with a center at the point of radius . Denote preimage of this circle under the mapping by .
With such a construction of it is easy to see that by Lemma C, for all the relation will be valid.
Really, since the relation is valid for all , then by Lemma C we will have And since (see [7]), then .
Therefore, by Lemma C from relation (5.1) we find and under we understand a distance from the point to the level line , where . Therewith, by Lemma C, we take into account that this distance has the same order of , that is, Really, is obvious.
Hence, by Lemma C it follows that And since (see [7]), then It remains to show that And since the relation is obvious, it suffices to show that
Let be a point for which . Obviously, . Hence, by Lemma C, it follows the estimation that proves relation (5.11) and (5.13); hence the relation (5.6) that we need follows.
Now, in order to estimate the right-hand side of relation (5.5), we divide the circle into the arc , situated interior to the circle with the ends at the points and (see Figure 2) and the arc . In its turn, we divide the arc into and , where part of the arc , are situated from the left of the ray , connecting the origin of coordinates with the point and from the right of this ray.
Obviously, we will have Estimate the quantity that will be represented in the form Obviously, for the estimation of , it suffies to estimate the quantity , since the obtained estimation remains valid for the quantity as well, because of symmetric arrangement of arcs and with respect to the arc .
Let . Then obviously, it will lie on some circle with center in and radius equal , where .
Since , then (see Figure 2), where is an angle between the ray and a real axis. Obviously, , where is an angle between the radii and (see Figure 2) that may be determined by the cosines theorem from the triangle Hence, we directly have Estimating the quantity , we’ll get
Now, making substitution and considering that (we can determire this from the triangle where the sides and are constant by the sines theorem) we will have Hence, by Lemma 4.2, we will find
As it was said above, this estimation remains valid for the quantity , as well.
The same estimation is similarly proved for the quantity , as well that allows us to see validity of the relation and hence, considering (5.1), the statement of Theorem 3.3 follows for when is an arbitrary restifiable -quasiconformal curve. The case is proved similarly. Really, by Lemmas B and C, 4.2, relation (5.6) and relation (5.5), and the Holder inequality, we get
Later on, by Lemmas B and C and relation (5.6) it is easy to see the validity of the relation where is a circle with a center at the point and of radius equal .
Hence, we directly get
Further, the proof is completed in the same way as in the case .

So, Theorem 3.3 is proved for the case when . The same reasoning allow us to affirm that Theorem 3.3 will be valid in the case , as well.

Finally, we give the proof of Theorem 3.4.

Proof of Theorem 3.4. Let and . Consider the case .
Apply the Holder inequality to inner integral of the right-hand side of the relation where .
By Lemma 4.5
Hence, changing the integration order and applying the statements of Lemmas 4.3 and 4.1, we get the required inequality (3.10) in the case .
In order to see validity of Theorem 3.4 in the case , in the right-hand side of the obvious relation it siffies to change the integration order and apply the statements of Lemmas 4.3 and 4.1.
Remark 5.1. It is easy to show that Theorem 3.4 is valid for any , if is fulfiled as the condition (obviously, this condition is always fulfilled if is a boundary of an arbitrary convex domain) for all .
Really, let . Choose such that the condition is fulfilled. Then repeating the reasoning mentioned above in the case , we get
Now, expand the function in Taylor’s series in the vicinity of the point
Further, divide both parts of this equality into , and consider that (see (5.6)) raise to the th power, integrate with respect to and take the th power root. We will have
Now considering Lemmas B and C, 4.1, and Theorem 3.3 and making substitution , we get (here in our reasoning we assume, for all ): All remaining integrals on the right-hand side of relation (5.31) are similarly estimated except for the last one, for which following the proof of Theorem 3.3 we find
Reasoning in the same way as in obtaining estimation (5.5), we’ll have
Hence by (5.31) the statement of Theorem 3.4 will follow in the case .
So, Theorem 3.4 is proved.

Remark 5.2. Note that by Lemma 4.4 and the inverse to it of result proved in the paper [15], we will have . Obviously, this result will allow us to derive from Theorems 3.1 and 3.2 the validity of these theorems on arbitrary curves as a corollary.

#### Acknowledgment

The authors would like to thank the reviewers for carefully checking the manuscript and for his/her valuable comments and suggestions.