Article Open Access

Inequalities between height and deviation of polynomials

  • EMAIL logo
Published/Copyright: July 5, 2021
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this paper, for polynomials with real coefficients P , Q satisfying ∣ P ( x ) ∣ ≤ ∣ Q ( x ) ∣ for each x in a real interval I , we prove the bound L ( P ) ≤ c L ( Q ) between the lengths of P and Q with a constant c , which is exponential in the degree d of P . An example showing that the constant c in this bound should be at least exponential in d is also given. Similar inequalities are obtained for the heights of P and Q when the interval I is infinite and P , Q are both of degree d . In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials.

Keywords: height of a polynomial; Chebyshev polynomials
MSC 2010: 11C08; 12D10; 26C10; 41A50

1 Introduction

Throughout, for a polynomial

f ( x ) = a 0 + a 1 x + ⋯ + a d x d ∈ C [ x ] , a d ≠ 0 ,

of degree d by H ( f ) ≔ max 0 ≤ k ≤ d ∣ a k ∣ and L ( f ) ≔ ∑ k = 0 d ∣ a k ∣ we denote its height and its length, respectively. It is clear that

(1) H ( f ) ≤ ∑ k = 0 d ∣ a k ∣ 2 1 / 2 ≤ max ∣ z ∣ = 1 ∣ f ( z ) ∣ ≤ L ( f ) ≤ ( d + 1 ) H ( f )

(see, e.g., [1]) and

(2) ∣ f ( x ) ∣ ≤ L ( f ) max { 1 , ∣ x ∣ } d

for any x ∈ C . In particular, by (1), for polynomials f , g ∈ C [ x ] the inequality for their maximums on the unit circle

(3) max ∣ z ∣ = 1 ∣ f ( z ) ∣ ≤ max ∣ z ∣ = 1 ∣ g ( z ) ∣

implies the inequality for their heights

H ( f ) ≤ ( deg g + 1 ) H ( g ) .

See also [2–11] for some other inequalities between various heights of polynomials and their factors, not only H ( f ) , L ( f ) and max ∣ z ∣ = 1 ∣ f ( z ) ∣ but also their house ∣ α ∣ ¯ ≔ max α : f ( α ) = 0 ∣ α ∣ , Mahler’s measure M ( f ) ≔ ∣ a d ∣ ∏ α : f ( α ) = 0 max { 1 , ∣ α ∣ } , etc.

In this paper, we will consider the case when in (3) the unit circle is replaced by a real (finite or infinite) interval. The following questions related to his research in the study of solutions of singular parabolic differential equations were recently asked by my colleague Prof. V. Mackevičius.

Find an explicit constant ξ ( d ) for which the inequality

(4) H ( P ) ≤ ξ ( d ) H ( Q )

holds for any P , Q ∈ R [ x ] of the degree d satisfying

(5) ∣ P ( x ) ∣ ≤ ∣ Q ( x ) ∣ for x ≥ 0 .

He is also interested in an explicit constant η ( d ) for which the inequality

(6) H ( P ) ≤ η ( d ) H ( Q )

holds for any P , Q ∈ R [ x ] of the degree d satisfying

(7) ∣ P ( x ) ∣ ≤ ∣ Q ( ∣ x ∣ ) ∣ for x ∈ R .

Note that for d = 1 we have P ( x ) = a 0 + a 1 x and Q ( x ) = b 0 + b 1 x . Inserting x = 0 into (5) or (7), we get ∣ a 0 ∣ ≤ ∣ b 0 ∣ . Similarly, inserting x = r and letting r → ∞ , in both cases, (5) and (7), we obtain ∣ a 1 ∣ ≤ ∣ b 1 ∣ . Hence,

(8) ξ ( 1 ) = η ( 1 ) = 1

are the best possible constants in (4) and (6) for d = 1 .

Likewise, for d = 2 , assuming that P ( x ) = a 0 + a 1 x + a 2 x 2 and Q ( x ) = b 0 + b 1 x + b 2 x 2 , we find that ∣ a 0 ∣ ≤ ∣ b 0 ∣ and ∣ a 2 ∣ ≤ ∣ b 2 ∣ . Thus, ∣ a 0 ∣ , ∣ a 2 ∣ ≤ H ( Q ) . Inserting x = 1 into (5) we deduce ∣ a 0 + a 1 + a 2 ∣ ≤ 3 H ( Q ) . Thus, ∣ a 1 ∣ ≤ 3 H ( Q ) + ∣ a 0 ∣ + ∣ a 2 ∣ ≤ 5 H ( Q ) . Consequently, H ( P ) ≤ 5 H ( Q ) for any quadratic polynomials P , Q satisfying (5). This, combined with the example P ( x ) = 1 − 5 x + x 2 , Q ( x ) = 1 + x + x 2 satisfying (5) and H ( P ) = 5 H ( Q ) , shows that

(9) ξ ( 2 ) = 5

is the best possible constant in (4) for d = 2 .

A simple example of polynomials P ( x ) = ( x − 1 ) d and Q ( x ) = 1 + x d satisfying (5) shows that the constant ξ ( d ) should be at least exponential in d . Indeed, by Stirling’s formula (see, e.g., [12]), for each n ∈ N one has

(10) n ! = 2 π n ( n / e ) n e θ n ,

where 1 12 n + 1 < θ n < 1 12 n . Hence, H ( P ) / H ( Q ) = d ⌊ d / 2 ⌋ ∼ 2 d + 1 / 2 / π d as d → ∞ .

First, we will show the following lower bounds with better exponents:

Theorem 1

The constants ξ ( d ) and η ( d ) for which (4) and (6) hold under corresponding assumptions (5) and (7) must satisfy

ξ ( d ) ≥ 1 2 d + 2 3 + 5 2 d + 3 − 5 2 d

and

η ( d ) ≥ 1 2 d + 2 1 + 5 2 d + 1 − 5 2 d .

In the next theorem, we compare the lengths of two polynomials P , Q , which have not necessarily the same degree but the deviation of P from zero in an interval is smaller than that of Q .

Theorem 2

Let P , Q ∈ R [ x ] be polynomials satisfying

(11) max 0 ≤ x ≤ 1 ∣ P ( x ) ∣ ≤ max 0 ≤ x ≤ 1 ∣ Q ( x ) ∣ .

Then,

(12) L ( P ) ≤ 1 2 ( ( 3 + 2 2 ) d + ( 3 − 2 2 ) d ) L ( Q ) ,

where d = deg P . Furthermore, if P , Q ∈ R [ x ] are polynomials satisfying

(13) max − 1 ≤ x ≤ 1 ∣ P ( x ) ∣ ≤ max − 1 ≤ x ≤ 1 ∣ Q ( ∣ x ∣ ) ∣

and d = deg P , then

(14) L ( P ) ≤ 1 2 ( ( 1 + 2 ) d + ( 1 − 2 ) d ) L ( Q ) .

From Theorem 2 it is easy to find some explicit ξ ( d ) and η ( d ) for which (4) and (6) hold under the corresponding assumptions (5) and (7). Indeed, by (1), we have H ( P ) ≤ L ( P ) and L ( Q ) ≤ ( deg Q + 1 ) H ( Q ) . Therefore, (12) implies that

H ( P ) ≤ deg Q + 1 2 ( ( 3 + 2 2 ) d + ( 3 − 2 2 ) d ) H ( Q ) .

In particular, for deg Q = d , since (5) implies (11), we see that, under assumption (5), inequality (4) holds with the constant

ξ ( d ) = d + 1 2 ( ( 3 + 2 2 ) d + ( 3 − 2 2 ) d ) .

Likewise, in view of (14), as (7) implies (13), under assumption (7), we derive that inequality (6) holds with the constant

η ( d ) = d + 1 2 ( ( 1 + 2 ) d + ( 1 − 2 ) d ) .

In the next theorem, we will improve these bounds as follows:

Theorem 3

Under assumption (5), inequality (4) holds with the constant

ξ ( d ) = 3 3 d / 2 d ,

while under assumption (7), inequality (6) holds with the constant

η ( d ) = 3 3 d / 4 7 d .

Note that

3 3 / 2 = 5.196152 … < 3 + 2 2 = 5.828427 …

and

3 3 / 4 = 2.279507 … < 1 + 2 = 2.414213 … .

By Theorem 1, the constants 3 3 / 2 and 3 3 / 4 cannot be replaced by the constants smaller than 3 + 5 2 = 2.618033 … and 1 + 5 2 = 1.618033 … , respectively, so there is still a gap for improvement of those constants in one or the other direction.

In Section 2, we give some auxiliary lemmas and then prove Theorems 1, 2 and 3 in Section 3. The proofs are self-contained except for some basic properties of Chebyshev polynomials and Lemma 4.

2 Auxiliary results related to Chebyshev polynomials

Let T d ( x ) = cos ( d arccos x ) , where − 1 ≤ x ≤ 1 be the Chebyshev polynomial of the first kind. Below, we shall use the following standard formulas:

(15) T d ( x ) = ∑ k = 0 ⌊ d / 2 ⌋ d 2 k ( x 2 − 1 ) k x d − 2 k ,

(16) T d ( x ) = d 2 ∑ k = 0 ⌊ d / 2 ⌋ ( − 1 ) k ( d − k − 1 ) ! k ! ( d − 2 k ) ! ( 2 x ) d − 2 k

for all x ∈ R and

(17) T d ( x ) = 1 2 ( ( x + x 2 − 1 ) d + ( x − x 2 − 1 ) d )

for x ≥ 1 that can be found, for instance, in [13, Chapter 22]. See also [14] and [15].

The most useful property of Chebyshev polynomials that we will use is the inequality

(18) ∣ T d ( x ) ∣ ≤ 1 for − 1 ≤ x ≤ 1 .

Some basic properties of Chebyshev polynomials were established by Chebyshev himself, Markov and Bernstein (see [16]). We first state the following lemma, which is a combination of Theorem 2.1 (with ξ = 0 ), Corollary 2.1 and Theorem 2.2 in the paper of Osipov and Sazhin [17]. (Its partial case has been proved in [18]. See also [19].)

Lemma 4

Let P ∈ R [ x ] be a degree d polynomial satisfying ∣ P ( x ) ∣ ≤ 1 for x ∈ [ 0 , a ] . Then, for each k , 0 ≤ k ≤ d , the absolute value of the kth coefficient of P ( x ) does not exceed that of the kth coefficient of the polynomial T d ( 2 x / a − 1 ) . In particular,

(19) L ( P ) ≤ L ( T d ( 2 x / a − 1 ) ) .

Moreover, if P ∈ R [ x ] is a degree d polynomial satisfying ∣ P ( x ) ∣ ≤ 1 for x ∈ [ − 1 , 1 ] , then the length of the polynomial P ( x ) does not exceed that of T d ( x ) .

Next, we calculate the lengths of some shifted Chebyshev polynomials.

Lemma 5

For each a > 0 , we have

(20) L ( T d ( 2 x / a − 1 ) ) = 1 2 1 + 2 a + 2 a a + 1 d + 1 + 2 a − 2 a a + 1 d

and

(21) L ( T d ( x / a ) ) = 1 2 1 + a 2 + 1 a d + 1 − a 2 + 1 a d .

Proof

In view of arccos ( 2 y − 1 ) = 2 arccos ( y ) for y ∈ [ 0 , 1 ] , we obtain T d ( 2 y − 1 ) = T 2 d ( y ) . Hence, T d ( 2 x / a − 1 ) = T 2 d ( x / a ) and from (16) it follows that

(22) T d ( 2 x / a − 1 ) = d ∑ k = 0 d ( − 1 ) k 4 a d − k ( 2 d − k − 1 ) ! k ! ( 2 d − 2 k ) ! x d − k

for x ∈ [ 0 , a ] . By the uniqueness theorem for holomorphic functions (see [20, p. 122]), this formula must hold for each x ∈ C .

From (22) we see that the coefficients of the polynomial T d ( 2 x / a − 1 ) have alternating signs, so its length equals the modulus of its value at x = − 1 , namely,

L ( T d ( 2 x / a − 1 ) ) = ∣ T d ( − 2 / a − 1 ) ∣ = ∣ T d ( 2 / a + 1 ) ∣ .

Inserting x = 2 / a + 1 into (17) we find that T d ( 2 / a + 1 ) is equal to the right hand side of (20), whence the result.

Similarly, by (16), the length of T d ( x / a ) is equal to ∣ T d ( i / a ) ∣ . Since

( ( i / a ) 2 − 1 ) k = ( − 1 ) k ( 1 + 1 / a 2 ) k and ( i / a ) d − 2 k = i d ( − 1 ) k a 2 k − d ,

inserting x = i / a into (15), we find that

L ( T d ( x / a ) ) = ∣ T d ( i / a ) ∣ = a − d ∑ k = 0 ⌊ d / 2 ⌋ d 2 k ( a 2 + 1 ) k = 1 2 a d ( ( 1 + a 2 + 1 ) d + ( 1 − a 2 + 1 ) d ) ,

which completes the proof of (21).□

Combining (19) with (20) we obtain the following:

Lemma 6

Let a > 0 and let P ∈ R [ x ] be a degree d polynomial satisfying ∣ P ( x ) ∣ ≤ 1 for 0 ≤ x ≤ a . Then,

L ( P ) ≤ 1 2 1 + 2 a + 2 a a + 1 d + 1 + 2 a − 2 a a + 1 d .

Likewise, combining the last statement of Lemma 4 with (21), where a = 1 , we obtain the following:

Lemma 7

Let P ∈ R [ x ] be a degree d polynomial satisfying ∣ P ( x ) ∣ ≤ 1 for − 1 ≤ x ≤ 1 . Then,

L ( P ) ≤ 1 2 ( ( 1 + 2 ) d + ( 1 − 2 ) d ) .

In the next lemma, we give the bounds for the modulus of the polynomial P ∈ R [ x ] of degree d and the modulus of its reciprocal polynomial P ∗ ( x ) = x d P ( 1 / x ) in the interval [ 0 , 1 ] under assumptions of (5) and (7).

Lemma 8

Let P , Q ∈ R [ x ] be two degree d polynomials satisfying (5). Then, ∣ P ( x ) ∣ ≤ ( d + 1 ) H ( Q ) and ∣ P ∗ ( x ) ∣ ≤ ( d + 1 ) H ( Q ) for x ∈ [ 0 , 1 ] .

Furthermore, if P , Q ∈ R [ x ] are two degree d polynomials satisfying (7), then

(23) 1 2 ∣ P ( x ) + P ( − x ) ∣ , 1 2 ∣ P ( x ) − P ( − x ) ∣ ≤ ( d + 1 ) H ( Q )

and

(24) 1 2 ∣ P ∗ ( x ) + P ∗ ( − x ) ∣ , 1 2 ∣ P ∗ ( x ) − P ∗ ( − x ) ∣ ≤ ( d + 1 ) H ( Q )

for x ∈ [ − 1 , 1 ] .

Proof

By (1) and (2), we clearly have ∣ Q ( x ) ∣ ≤ L ( Q ) ≤ ( d + 1 ) H ( Q ) for x ∈ [ 0 , 1 ] , which implies the first inequality in view of (5). To see that the second inequality holds, observe that (5) and deg P = deg Q implies ∣ P ∗ ( x ) ∣ ≤ ∣ Q ∗ ( x ) ∣ for each x ≥ 0 . In particular, this inequality holds for x ∈ [ 0 , 1 ] . Now, the second inequality of the lemma follows by ∣ Q ∗ ( x ) ∣ ≤ ( d + 1 ) H ( Q ∗ ) for x ∈ [ 0 , 1 ] and H ( Q ∗ ) = H ( Q ) .

For the second part, assume that P , Q are two degree d polynomials satisfying (7). For x ∈ [ − 1 , 1 ] , we have ∣ Q ( ∣ x ∣ ) ∣ ≤ L ( Q ) ≤ ( d + 1 ) H ( Q ) , which implies ∣ P ( x ) ∣ ≤ ( d + 1 ) H ( Q ) . Also, from (7) it follows that ∣ P ∗ ( x ) ∣ ≤ ∣ Q ∗ ( ∣ x ∣ ) ∣ for ∣ x ∣ ≤ 1 , and hence

∣ P ∗ ( x ) ∣ ≤ ∣ Q ∗ ( ∣ x ∣ ) ∣ ≤ ( d + 1 ) H ( Q ∗ ) = ( d + 1 ) H ( Q ) .

Furthermore, since the pair of polynomials P ( − x ) , Q ( x ) also satisfies (7), we obtain ∣ P ( − x ) ∣ , ∣ P ∗ ( − x ) ∣ ≤ ( d + 1 ) H ( Q ) for x ∈ [ − 1 , 1 ] . This implies the bounds (23) and (24).□

We conclude this section with the following inequality:

Lemma 9

For d ∈ N and k , 0 ≤ k ≤ d , set

(25) f ( k , d ) ≔ d d + k d + k 2 k 4 k .

Then,

(26) max 0 ≤ k ≤ ⌊ d / 2 ⌋ f ( k , d ) = f ( ⌊ d / 2 ⌋ , d ) < 0.47 ⋅ 3 3 d / 2 d .

Furthermore, for d ≥ 3 odd we have

(27) f ( ( d + 1 ) / 2 , d ) = 9 d − 3 d + 1 f ( ( d − 1 ) / 2 , d − 1 ) .

Proof

Inequality (26) can be easily verified directly for d in the range 1 ≤ d ≤ 7 . Then, the maximum of the left hand side of (26) is indeed attained at k = ⌊ d / 2 ⌋ and is equal to

1 , 8 , 18 , 160 , 400 , 3584 , 9408

for d = 1 , 2 , 3 , 4 , 5 , 6 , 7 , respectively. In each case, it is smaller than 0.47 ⋅ 3 3 d / 2 / d for the corresponding value of d .

Fix any d ≥ 8 . By (25), for k in the range 0 ≤ k ≤ ⌊ d / 2 ⌋ − 1 , we have

f ( k + 1 , d ) f ( k , d ) = 4 ( d + k ) ( d − k ) ( 2 k + 1 ) ( 2 k + 2 ) > 4 ( d 2 − k 2 ) ( 2 k + 2 ) 2 = d 2 − k 2 ( k + 1 ) 2 > 1 ,

because k 2 + ( k + 1 ) 2 < 2 ( k + 1 ) 2 ≤ 2 ⌊ d / 2 ⌋ 2 < d 2 for d ≥ 2 . Thus, the maximum of f ( k , d ) in (26) is attained at k = ⌊ d / 2 ⌋ .

Assume first that d ≥ 8 is even. Set

(28) B d ≔ 2 d + 1 ( 3 d / 2 ) ! 3 ( d / 2 ) ! d ! .

Then, ⌊ d / 2 ⌋ = d / 2 , and hence

(29) max 0 ≤ k ≤ ⌊ d / 2 ⌋ f ( k , d ) = f ( d / 2 , d ) = d d + d / 2 3 d / 2 d / 2 4 d / 2 = B d .

Applying (10) to n = 3 d / 2 , d / 2 and d , by (28), we derive that for each even d ≥ 8

B d = 2 d + 1 3 π d ( 3 d / 2 e ) 3 d / 2 e θ 3 d / 2 3 π d ( d / 2 e ) d / 2 e θ d / 2 2 π d ( d / e ) d e θ d = 2 3 π d ⋅ 3 3 d / 2 ⋅ e θ 3 d / 2 − θ d / 2 − θ d < e 1 / 144 ⋅ 2 3 π ⋅ 3 3 d / 2 d < 0.47 ⋅ 3 3 d / 2 d .

So, by (29), this implies (26) for each even d ≥ 8 .

If d ≥ 9 is odd, then D ≔ d − 1 ≥ 8 is even and ⌊ d / 2 ⌋ = ( d − 1 ) / 2 . Accordingly, we have

max 0 ≤ k ≤ ⌊ d / 2 ⌋ f ( k , d ) = f ( ( d − 1 ) / 2 , d ) .

Thus, in view of

f ( ( d − 1 ) / 2 , d ) = d 2 d 3 d − 1 ⋅ ( ( 3 d − 1 ) / 2 ) ! ( d − 1 ) ! ( ( d + 1 ) / 2 ) ! = ( D + 1 ) 2 D + 1 3 D + 2 ⋅ ( 3 D / 2 + 1 ) ! D ! ( D / 2 + 1 ) ! = D + 1 D + 2 ⋅ 3 B D ,

where the last equality follows from (28), we derive that the left hand side of (26) does not exceed 3 B D . Using the upper bound for B D for even D = d − 1 ≥ 8 we derive that

3 B D < 3 ⋅ 0.47 ⋅ 3 3 D / 2 D = 1.41 ⋅ 3 − 3 / 2 ⋅ 3 3 d / 2 d − 1 < 0.29 ⋅ 3 3 d / 2 d

for each odd d ≥ 9 . This completes the proof of (26).

The verification of (27) is straightforward. The left hand side of (27) divided by the binomial coefficient 3 ( d − 1 ) / 2 ( d − 1 ) / 2 equals

f ( ( d + 1 ) / 2 , d ) 3 ( d − 1 ) / 2 ( d − 1 ) / 2 = 2 d ( 3 d / 2 + 1 / 2 ) ( 3 d / 2 − 1 / 2 ) 2 d + 1 ( 3 d + 1 ) ( d + 1 ) d = ( 3 d − 1 ) 2 d d + 1 ,

while, by (29), the right hand side divided by the same binomial coefficient equals

9 d − 3 d + 1 ⋅ f ( ( d − 1 ) / 2 , d − 1 ) 3 ( d − 1 ) / 2 d − 1 = 9 d − 3 d + 1 ⋅ 2 ⋅ 4 ( d − 1 ) / 2 3 = ( 3 d − 1 ) 2 d d + 1 ,

which is the same.□

3 Proofs of the main results

Proof of Theorem 1

Consider the pair of degree d polynomials

P ( x ) = T d ( x / 2 − 1 ) and Q ( x ) = 1 + x d .

We first show that they satisfy condition (5). Indeed, for 0 ≤ x ≤ 4 , we have ∣ T d ( x / 2 − 1 ) ∣ ≤ 1 by (18), so ∣ P ( x ) ∣ ≤ 1 ≤ 1 + x d = Q ( x ) . Assume that x ≥ 4 . We will show that then

∣ T d ( x / 2 − 1 ) ∣ = T d ( x / 2 − 1 ) ≤ x d .

Indeed, setting y ≔ x / 2 − 1 ≥ 1 and using (17) we deduce

1 2 ( ( y + y 2 − 1 ) d + ( y − y 2 − 1 ) d ) < 1 2 ( ( 2 y ) d + 1 ) < ( 2 y + 2 ) d .

Hence, these two polynomials satisfy (5) for each x ≥ 0 . This implies the first part of the theorem, because H ( Q ) = 1 and H ( P ) is greater than or equal to

L ( P ) d + 1 = L ( T d ( x / 2 − 1 ) ) d + 1 = 1 2 d + 2 3 + 5 2 d + 3 − 5 2 d

by (1) and (20) with a = 4 .

For the second part, take P ( x ) = T d ( x / 2 ) and Q ( x ) = 1 + x d . Then, for − 2 ≤ x ≤ 2 , we have ∣ T d ( x / 2 ) ∣ ≤ 1 by (18), and hence ∣ P ( x ) ∣ ≤ 1 ≤ 1 + ∣ x ∣ d = Q ( ∣ x ∣ ) . Since ∣ T d ( x ) ∣ = ∣ T d ( − x ) ∣ , in order to verify (7) it suffices to check that ∣ T d ( x / 2 ) ∣ = T d ( x / 2 ) ≤ x d for x ≥ 2 . This is indeed the case, because

T d ( x / 2 ) = 1 2 ( ( x / 2 + ( x / 2 ) 2 − 1 ) d + ( x / 2 − ( x / 2 ) 2 − 1 ) d ) < 1 2 ( x d + 1 ) < x d

by (17). Consequently, these two polynomials satisfy (7) for each x ∈ R . As above, this implies the second part of the theorem, because H ( Q ) = 1 and H ( P ) is greater than or equal to

L ( P ) d + 1 = L ( T d ( x / 2 ) ) d + 1 = 1 2 d + 2 1 + 5 2 d + 1 − 5 2 d

by (21) with a = 2 .□

Proof of Theorem 2

Suppose that P , Q ∈ R [ x ] are two polynomials satisfying (11). Then, by (2), we have ∣ P ( x ) ∣ ≤ L ( Q ) for each x ∈ [ 0 , 1 ] . Applying Lemma 6 to a = 1 and the polynomial P ( x ) / L ( Q ) of length L ( P ) / L ( Q ) we find that

L ( P ) L ( Q ) ≤ 1 2 ( ( 3 + 2 2 ) d + ( 3 − 2 2 ) d ) ,

which implies (12).

Similarly, by (2) and (13), we obtain ∣ P ( x ) ∣ ≤ L ( Q ) for each x ∈ [ − 1 , 1 ] . So, applying Lemma 7 to the polynomial P ( x ) / L ( Q ) of length L ( P ) / L ( Q ) we derive that

L ( P ) L ( Q ) ≤ 1 2 ( ( 1 + 2 ) d + ( 1 − 2 ) d ) ,

which gives (14).□

Proof of Theorem 3

Let P and Q be two degree d polynomials satisfying (5). Then, by Lemma 8, we have

(30) ∣ P ( x ) ∣ ≤ ( d + 1 ) H ( Q ) and ∣ P ∗ ( x ) ∣ ≤ ( d + 1 ) H ( Q )

for x ∈ [ 0 , 1 ] . From (22) with a = 1 , it follows that

T d ( 2 x − 1 ) = ∑ k = 0 d ( − 1 ) d − k d d + k d + k 2 k 4 k x k .

Write

P ( x ) = a 0 + a 1 x + ⋯ + a d x d and P ∗ ( x ) = a d + a d − 1 x + ⋯ + a 0 x d .

Then, using (30), by the first part of Lemma 4 with a = 1 , we deduce that for each 0 ≤ k ≤ ⌊ d / 2 ⌋

max { ∣ a k ∣ , ∣ a d − k ∣ } ( d + 1 ) H ( Q ) ≤ d d + k d + k 2 k 4 k ,

which is less than 0.47 ⋅ 3 3 d / 2 / d by Lemma 9. Hence, from

H ( P ) = max 0 ≤ k ≤ d ∣ a k ∣ = max 0 ≤ k ≤ ⌊ d / 2 ⌋ max { ∣ a k ∣ , ∣ a d − k ∣ }

we conclude that

(31) H ( P ) < ( d + 1 ) H ( Q ) 3 3 d / 2 2 d ≤ 2 ⋅ 3 3 d / 2 − 1 d H ( Q ) ,

for each d ≥ 3 . Note that (31) also holds for d = 1 and d = 2 by (8) and (9). This implies the first part of the theorem with ξ ( d ) = 2 ⋅ 3 3 d / 2 − 1 d , which is better than claimed.

Next, suppose that P ( x ) = a 0 + a 1 x + ⋯ + a d x d and Q ( x ) satisfy (7). Assume first that d is even. Then, we can rewrite (23) in the form

∣ a 0 + a 2 x 2 + ⋯ + a d x d ∣ , ∣ a 1 x + a 3 x 3 + ⋯ + a d − 1 x d − 1 ∣ ≤ ( d + 1 ) H ( Q ) .

Replacing x 2 by y ∈ [ 0 , 1 ] and using

∣ a 1 x 2 + a 3 x 4 + ⋯ + a d − 1 x d ∣ ≤ ∣ a 1 x + a 3 x 3 + ⋯ + a d − 1 x d − 1 ∣ ,

we deduce

∣ a 0 + a 2 y + ⋯ + a d y d / 2 ∣ , ∣ a 1 y + a 3 y 2 + ⋯ + a d − 1 y d / 2 ∣ ≤ ( d + 1 ) H ( Q )

for y ∈ [ 0 , 1 ] . In the same fashion, from (24) it follows that

∣ a d + a d − 2 y + ⋯ + a 0 y d / 2 ∣ , ∣ a d − 1 y + a d − 3 y 2 + ⋯ + a 1 y d / 2 ∣ ≤ ( d + 1 ) H ( Q )

for y ∈ [ 0 , 1 ] .

Now, by the first part of Lemma 4 with a = 1 and d replaced by d / 2 , we derive that max { ∣ a 0 ∣ , ∣ a d ∣ } ≤ ( d + 1 ) H ( Q ) and for each 1 ≤ k ≤ ⌊ d / 4 ⌋

max { ∣ a 2 k − 1 ∣ , ∣ a 2 k ∣ , ∣ a d − 2 k ∣ , ∣ a d − 2 k + 1 ∣ } ( d + 1 ) H ( Q ) ≤ d / 2 d / 2 + k d / 2 + k 2 k 4 k ,

which is less than 0.47 ⋅ 3 3 d / 4 / ( d / 2 ) by Lemma 9. Consequently, the moduli of all coefficients of P except possibly for a d / 2 (which happens for d even but not divisible by 4) are bounded above by

(32) ( d + 1 ) H ( Q ) 3 3 d / 4 2 d / 2 < 3 3 d / 4 2 d H ( Q ) .

In particular, this implies the bound better than claimed for d divisible by 4.

If d is of the form 4 l − 2 , l ∈ N , then the coefficient a d / 2 = a 2 l − 1 occurs for y l . So, by the first part of Lemma 4 with a = 1 and d replaced by d / 2 , we get

∣ a d / 2 ∣ ( d + 1 ) H ( Q ) ≤ d / 2 d / 2 + l d / 2 + l 2 l 4 l = f ( ( d + 2 ) / 4 , d / 2 ) .

For d = 2 , this equals 2 and gives the bound ∣ a 1 ∣ ≤ 6 H ( Q ) implying H ( P ) ≤ 6 H ( Q ) , which is better than required. For d ≥ 6 , by the identity (27) with d replaced by d / 2 , we see that the right hand side is equal to

9 d − 6 d + 2 f ( ( d − 2 ) / 4 , d / 2 − 1 ) .

In view of (26) this is less than

0.47 ⋅ 9 d − 6 d + 2 ⋅ 3 3 d / 4 − 3 / 2 d / 2 − 1 = 0.47 ⋅ 6 ⋅ d − 2 / 3 d + 2 ⋅ 3 3 d / 4 d − 2 ,

and hence

∣ a d / 2 ∣ < 0.47 ⋅ 6 ⋅ ( d − 2 / 3 ) ( d + 1 ) d + 2 ⋅ 3 3 d / 4 d − 2 H ( Q ) .

Now, by

( d + 1 ) ( d − 2 / 3 ) < ( d + 2 ) d ( d − 2 )

and 0.47 ⋅ 6 < 1.16 , we obtain ∣ a d / 2 ∣ < 1.16 ⋅ 3 3 d / 4 d for d ≥ 6 . Combining this with (32), we complete the proof of the inequality H ( P ) < 3 3 d / 4 2 d H ( Q ) for each d ≥ 2 of the form 4 l − 2 , l ∈ N .

It remains to consider the case when d is odd. For d = 1 , we have H ( P ) ≤ H ( Q ) by (8), so we can assume that d ≥ 3 . Note that, by an argument as above, (23) implies

∣ a 0 + a 2 x 2 + ⋯ + a d − 1 x d − 1 ∣ , ∣ a 1 x + a 3 x 3 + ⋯ + a d x d ∣ ≤ ( d + 1 ) H ( Q ) ,

while (24) yields

∣ a d − 1 + a d − 3 x 2 + ⋯ + a 1 x d − 1 ∣ , ∣ a d x + a d − 2 x 3 + ⋯ + a 0 x d ∣ ≤ ( d + 1 ) H ( Q ) .

Following the above argument, replacing x 2 by y ∈ [ 0 , 1 ] , we find that

∣ a 0 + a 2 y + ⋯ + a d − 1 y d − 1 2 ∣ , ∣ a 1 y + a 3 y 2 + ⋯ + a d y d + 1 2 ∣ ≤ ( d + 1 ) H ( Q )

and

∣ a d − 1 + a d − 3 y + ⋯ + a 1 y d − 1 2 ∣ , ∣ a d y + a d − 2 y 2 + ⋯ + a 0 y d + 1 2 ∣ ≤ ( d + 1 ) H ( Q )

for y ∈ [ 0 , 1 ] .

Applying the first part of Lemma 4 with a = 1 to the above 4 polynomials of degree at most ( d + 1 ) / 2 , we derive that max { ∣ a 0 ∣ , ∣ a d − 1 ∣ } ≤ ( d + 1 ) H ( Q ) , whereas for each 1 ≤ k ≤ ⌊ ( d + 1 ) / 4 ⌋

max { ∣ a 2 k − 1 ∣ , ∣ a 2 k ∣ , ∣ a d − 2 k − 1 ∣ , ∣ a d − 2 k + 2 ∣ } ( d + 1 ) H ( Q )

does not exceed

( d + 1 ) / 2 ( d + 1 ) / 2 + k ( d + 1 ) / 2 + k 2 k 4 k .

By Lemma 9, this is less than 3 3 ( d + 1 ) / 4 / 2 d + 2 . It follows that the moduli of all the coefficients of P (except possibly for a ( d + 1 ) / 2 when d = 4 l + 1 , l ∈ N ) are less than

( d + 1 ) 3 3 ( d + 1 ) / 4 2 d + 2 H ( Q ) = 3 3 / 4 d + 1 2 3 3 d / 4 H ( Q ) < 2 d ⋅ 3 3 d / 4 H ( Q )

for d ≥ 3 . This completes the proof of the second part of the theorem for all odd d except for those of the form d = 4 l + 1 , l ∈ N .

We now turn to the case when d = 4 l + 1 . Then, the only coefficient a ( d + 1 ) / 2 = a 2 l + 1 that remains to be dealt with occurs for y l + 1 . In that case, by the first part of Lemma 4 with a = 1 and d replaced by ( d + 1 ) / 2 , and by identity (27) with d replaced by ( d + 1 ) / 2 , we find that

∣ a ( d + 1 ) / 2 ∣ ( d + 1 ) H ( Q ) ≤ ( d + 1 ) / 2 ( d + 1 ) / 2 + l + 1 ( d + 1 ) / 2 + l + 1 2 l + 2 4 l + 1 = f ( ( d + 3 ) / 4 , ( d + 1 ) / 2 ) = 9 d + 3 d + 3 f ( ( d − 1 ) / 4 , ( d − 1 ) / 2 ) .

Using the upper bound (26) on f ( ( d − 1 ) / 4 , ( d − 1 ) / 2 ) we deduce

∣ a ( d + 1 ) / 2 ∣ H ( Q ) < ( d + 1 ) 0.47 ( 9 d + 3 ) d + 3 ⋅ 3 3 d / 4 − 3 / 4 ( d − 1 ) / 2 .

Here, the right hand side is less than 3 3 d / 4 7 d , for each d ≥ 5 , because

( d + 1 ) ( d + 1 / 3 ) < ( d + 3 ) d 2 − d

and 0.47 ⋅ 9 ⋅ 3 − 3 / 4 ⋅ 2 < 2.63 < 7 . Hence, ∣ a ( d + 1 ) / 2 ∣ < 3 3 d / 4 7 d H ( Q ) . Since we already proved that the moduli of all other coefficients a k , k ≠ ( d + 1 ) / 2 , are less than 2 d ⋅ 3 3 d / 4 H ( Q ) , this completes the proof of the inequality H ( P ) < 3 3 d / 4 7 d H ( Q ) for each d of the form 4 l + 1 . This finishes the proof of the theorem.□

Acknowledgements

The author thank the referees for careful reading and pointing out some misprints.

  1. Funding information: This research has received funding from the European Social Fund (project no. 09.3.3-LMT-K-712-01-0037) under grant agreement with the Research Council of Lithuania (LMTLT).

  2. Conflict of interest: Author states no conflict of interest.

References

[1] M. Mignotte , An inequality about irreducible factors of integer polynomials, J. Number Theory 30 (1988), 156–166. 10.1016/0022-314X(88)90014-5Search in Google Scholar

[2] P. Batra , M. Mignotte , and D. Ştefănescu , Improvements of Lagrange’s bound for polynomial roots, J. Symbolic Comput. 82 (2017), 19–25. 10.1016/j.jsc.2016.10.001Search in Google Scholar

[3] B. Beauzamy , Products of polynomials and a priori estimates for coefficients in polynomial decompositions: a sharp result, J. Symbolic Comput. 13 (1992), 463–472. 10.1016/S0747-7171(10)80006-9Search in Google Scholar

[4] P. Borwein and S. K. -K. Choi , The average norm of polynomials of fixed height, Trans. Amer. Math. Soc. 359 (2007), 923–936. 10.1090/S0002-9947-06-03952-3Search in Google Scholar

[5] D. W. Boyd , Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39 (1992), 341–349. 10.1112/S0025579300015072Search in Google Scholar

[6] D. W. Boyd , Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 26 (1994), 449–454. 10.1112/blms/26.5.449Search in Google Scholar

[7] A. Dubickas , Three problems for polynomials of small measure, Acta Arith. 98 (2001), 279–292. 10.4064/aa98-3-5Search in Google Scholar

[8] K. Mahler , On two extremum properties of polynomials, Illinois J. Math. 7 (1963), 681–701. 10.1215/ijm/1255645104Search in Google Scholar

[9] M. Mignotte and P. Glesser , On the smallest divisor of a polynomial, J. Symbolic Comput. 17 (1994), 277–282. 10.1006/jsco.1994.1017Search in Google Scholar

[10] L. Panaitopol and D. Ştefănescu , Inequalities on polynomial heights, JIPAM J. Inequal. Pure Appl. Math. 2 (2001), 7. Search in Google Scholar

[11] I. E. Pritsker and S. Ruscheweyh , Inequalities for products of polynomials. I, Math. Scand. 104 (2009), 147–160. 10.7146/math.scand.a-15091Search in Google Scholar

[12] H. Robbins , A remark on Stirling’s formula, Amer. Math. Monthly 62 (1955), 26–29. 10.1007/978-1-4612-5110-1_36Search in Google Scholar

[13] M. Abramowitz and I. A. Stegun , Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, D.C. 1964. Search in Google Scholar

[14] T. J. Rivlin , The Chebyshev Polynomials, Wiley-Interscience, New York-London-Sydney, 1974. Search in Google Scholar

[15] Q. I. Rahman and G. Schmeisser , Analytic Theory of Polynomials, Clarendon Press, Oxford, 2002. 10.1093/oso/9780198534938.001.0001Search in Google Scholar

[16] S. N. Bernstein , On a theorem of V. A. Markov , in: Collected Works , Vol. II, Izd. AN SSSR, Moscow, 1954, pp. 281–286 (in Russian). Search in Google Scholar

[17] N. N. Osipov and N. S. Sazhin , An extremal property of Chebyshev polynomials, Russian J. Numer. Anal. Math. Modelling 23 (2008), 89–95. 10.1515/rnam.2008.006Search in Google Scholar

[18] G. V. Milovanović and D. D. Tošić , An extremal problem for the length of algebraic polynomials, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 31–36. Search in Google Scholar

[19] H.-J. Rack , On the length and height of Chebyshev polynomials in one and two variables, East J. Approx. 16 (2010), 35–91. Search in Google Scholar

[20] M. J. Ablowitz and A. S. Fokas , Complex Variables: Introduction and Applications, Cambridge University Press, Cambridge, 1997. Search in Google Scholar
Received: 2021-01-20
Revised: 2021-05-07
Accepted: 2021-05-20
Published Online: 2021-07-05

© 2021 Artūras Dubickas, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0055
Keywords for this article
height of a polynomial; Chebyshev polynomials
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
Downloaded on 13.4.2026 from https://www.degruyterbrill.com/document/doi/10.1515/math-2021-0055/html
Scroll to top button