AN INTEGRAL ESTIMATE FOR AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 9 (27), No3, 2020, pp. 137-147

DOI: 10.15393/j3.art.2020.8250

137

UDC 517.53

S. L. WALI

AN INTEGRAL ESTIMATE FOR AN OPERATOR PRESERVING INEQUALITIES BETWEEN POLYNOMIALS

Abstract. Let Pn be the class of polynomials of degree at most n. Rahman introduced a class Bn of operators B that map Pn into itself. In this paper we prove some results concerning the integral estimates of such operators and, thereby, obtain generalizations, as well as improvements, of some well known Lp inequalities.

Key words: Polynomials, B-Operator, Inequalities in the complex domain, Zeros.

2010 Mathematical Subject Classification: 30A06, 30A64,30E10

n

1. Introduction. Let Pn be the class of polynomials P(z) := ajzj

j=o

of degree at most n with complex coefficients, and P'(z) be its derivative. Let B be a linear operator mapping polynomials from Pn to polynomials in Pn. We say that B is a Bn-operator if for every polynomial P of degree n having all its zeros in the closed unit disk, B[P] has all its zeros in the closed unit disk. Rahman [10] (see also Rahman and Schmeisser [11, Lemma 14.5.7]) introduced this class Bn of operators B and observed that if A0, Ai, and A2 are such that

n!

Ao + C (n, 1)Aiz + C (n, 2)A2Z2 = 0, C (n, r) := '

r!(n — r)!

for Re(z) ^ n, then the operator B, which maps a polynomial P of degree at most n to the polynomial

^rm/ N > n/ N \ (nz\ P'(z) a (nz\2 P"(z) B [P] (z) := A0P(z) + —) ^ + —)

© Petrozavodsk State University, 2020

is a Bn-operator. This can also be deduced from a result of Marden [9, Corollary 18.3]. Concerning this class of operators, Rahman [10] observed:

|B[P]| ^ M|B[En]|, |z| ^ 1, (1)

whenever

|P| ^ M, |z| = 1,

where |B[P]| = |B[Pz]| and E„(z) = zn.

The famous Bernstein inequality [5] is: for every P G

max |P'(z)| ^ n max |P(z)|; (2)

|z| = 1 |z|=1

it is a special case of Inequality (1) and follows from it with a suitable choice of Aj, i = 0,1, 2.

As an improvement of Inequality (1), Shah and Liman [14] proved: Theorem A. If P G and P(z) = 0 in |z| < 1, then, for |z| ^ 1,

|B[P]| ^ 2 i{|B[En]|+|Ao|} max |P(z)|- {|B[E]|-|Ac|} min |P(z)| 2 l lz|=i |z|=i

(3)

where En(z) := zn.

The result of Erdos-Lax [8], Ankeny-Rivlin [2], and Aziz-Dawood [3] follows from Inequality (3) with a suitable choice of A',i = 0,1, 2 (see for reference [14]). For every f G Pn, from now onwards we write:

2n 1

Lp norm of f =| f ||p= j^y |f (e'0)|pd^j , 1 ^ p < ro, whereas

0

2n i

Lp mean norm of f =|| f ||*= j — J |f(e)|pd0j , 0 < p < œ, and

o

1 f max|z|=i|f(z)|.

Concerning the integral mean inequalities for polynomials, there exists several results proved recently. In this direction, the interested readers may see [16], [13]. In particular, in connection to Lp norm of a B-operator, Shah and Liman [15] (see also [17]) proved the following:

Theorem B. If P G Pn be such that P(z) = 0 in |z| < 1, then for every R ^ 1 and p ^ 1,

Il B[P](R .) Bp i |B 1E1(R„)| + |Ao' II P lip, |z| = 1, (4)

1 1 + En BP

where B e Bn and En(z) := zn.

Recently Rather and Shah [12] obtained the following result and showed that Theorem A is valid for 0 < p < 1 as well.

Theorem C. Let P e Pn and P(z) = 0 in |z| < 1, then for every R ^ 1,0 ^ p < w,

Rn A z + A, 1 + z

b[p](r .) lip ^ " ......„;Hp ii p "p,

p

where B e Bn and A := Ao + Ai+ A2n-in-).

In this paper, we extend Theorem A to the Lp mean norm and, thereby, obtain a refinement of Theorem C. In fact, we prove:

Theorem 1. If P e Pn and P(z) does not vanish in |z| < 1, then, for every P e C with |P| ^ 1, R > 1, and 0 ^ p < to,

" b[p](r.)+(r'az^) m|P|"; s " r'"a + +"Ao " P ";, (5)

where

nn,2 n^ (n_1)

B e Bn, m := min |P(z)| and A := Ao + Ai — + A2—--. (6)

|z|=i 2 8

The result is sharp and the extremal polynomial is P(z) = zn + 1. If we make p ^ to in (5), we get the following sharp result:

Corollary 1. If P(z) is a polynomial of degree n, which does not vanish in | z| < 1, then, for | z| = 1 ,

" B[P](R .) "U^ 1|(Rn| A | + |Ao|) " P(z) "U —(Rn| A | — |Ao|)mj, (7)

where B, A, and m are defined in (6).

The following result, which is an improvement of Theorem B, follows from Theorem 1 by using the triangle inequality.

Corollary 2. If P e Pn and P(z) does not vanish in |z| < 1, then for every P eC with |P| ^ 1, R > 1 and 0 ^ p < to,

" B[P](R .) + (RnAzn-iM) m|P| "p « " P "p, (8)

where B G Bn, A and m are defined in (6).

The result is sharp and the equality holds for the polynomial P(z) =

= zn + 1.

Corollary 2 also shows that Theorem A is valid for 0 ^ p < 1 as well. Substituting for B[P](z) and assuming that A', i = 0,1, 2, are same for all polynomials P(z), we obtain from inequality (5):

,nz^DP'(Rz) | ,nzN2d2P''(Rz) 1 ^2 (

{Ao P (Rz) + Ai( f tf^ + A2( f )2 R2

1 (( n2 n3(n — 1)) ^ *

+ (Ao + Ai ^ + A2 )Rnzn — |Ao|} m|

p

^ ll(Ao + Ain2 + A2^)Rnzn + Ao||* (9)

^ ^ l|P llp" (9)

Suppose that A1 = 0 = A2, which is possible, as in this case it is easy to see that

n

Ao + C(n, 1)Aiz + C(n, 2)A2z2 = 0 for Re(z) .

We get from (9) the following refinement of a result earlier proved by Boas and Rahman [6] for p ^ 1 and Rahman and Schemessier [11] for 0 ^ p < 1.

Corollary 3. If P G Pn and P(z) does not vanish in |z| < 1, then for every P gC with |P| ^ 1, R > 1 and 0 ^ p < ro,

Rnzn - 1 || Rnzn + 1 ||p

ll P(Rz) + —^m|P| ||p ^ " ^ z + 1||;Hp || P ||*,

where m = min|z|=1 |P(z)|.

Again, assuming that Ao = A2 = 0, which is also possible, we get the following refinement of de Brujin's theorem [7] for p ^ 1 and a result of Arestov [1] for 0 ^ p ^ ro.

Corollary 4. If P G Pn and P(z) does not vanish in |z| < 1, then for

every P G C with |P| ^ 1, R > 1 and 0 ^ p < ro,

n

llzP'(z) + nmP||* ^ -||P||*"

H1 + z Hp

Remark 1. Similarly, if we choose P and A' = 0,1, 2 suitably in Corollary 2, many known polynomial inequalities can be obtained by a fairly uniform procedure.

Remark 2. Apparently it seems that Theorem 1 follows from a result earlier proved by Wali and Liman [16]. But the parameter of zn in this theorem makes it different and more interesting. The comparison between the two is also obvious by the corollaries deduced and the lemmas used in the two papers, besides the methods used to prove the various results.

2. Lemmas. For the proof of the theorem, we need the following Lemmas. The first Lemma follows from [9, Corollary 18.3]:

Lemma 1. If all the zeros of a polynomial P(z) of degree n lie in a circle |z | ^ 1, then all the zeros of the polynomial B [P](z) also lie in |z| ^ 1.

The next Lemma we require is due to Aziz and Shah [4].

Lemma 2. If L, M, N are non-negative real numbers, such that M + N ^ L, then for every real number a

|(L — N) + eia(M + N)| ^ |L + eiaM|.

Lemma 3. If P e Pn and P(z) = 0 in |z| < 1, then for |z| ^ 1,

|B[P]| ^ |B[P;]|,

where B e Bn and P;(z) := znP(±).

The above lemma is due to Shah and Liman [14, Lemma 2.2].

Lemma 4. If P e Pn and P;(z) := znP(i), then we have

|B[P;](Rz)| = |B([P;](Rz));| := |(B[P;](Rz));|, for |z| = 1.

Proof. For P e Pn, we have P;(Rz) = RnznP(). This gives B [P ;](Rz) = B[P ;(Rz)] =

+ Aif nz + 2! I 2

nz

2

AoRnznP(1/Rz) + Ai(y) nRnzn-iP(1/Rz) — Rn-izn-2P'(1/Rz)

+

+ Rn zn-4P "(1/Rz)

n(n — 1)Rnzn P (1/Rz) — 2(n — 1)Rn z n-3P '(1/Rz) + n2 n3(n — 1)

(Ao + y Ai + n(n8 1) A2) RnznP (1/Rz ) —

— I nAi + n2(n 1) A2}Rn-izn"iP '(1/Rz) + ^ A2Rn"2zn"2P "(1/Rz).

(10)

Also,

(B[P*](Rz))* = znB[P*(R/z)] = (Ao+yAi+n (n8 )A2)RnP(z/R)_ _ InAi + n2(n4- A2}Rn-1zP'(z/R) + £A2Rn-2z2P"(z/R). (11)

From (10), (11) it follows that |B[P*](Rz)| = |(B[P*](Rz))*| for |z| = 1. □ Lemma 5. If P G Pn, then for every p ^ 0, R ^ 1 and 0 ^ a < 2n,

2n 2n

J j|B[P](Rei0) + eia(B[P*](Rei0))*|pd0da ^ 0 0

2n 2n

^ j |Rn A eia + Ao|pd^y |P(ei0)|pd0, 00

where B and A are defined in (6).

The above Lemma is due to Rather and Shah [12, Lemma 6]. 3. Proof of the Theorem 1. If P(z) has a zero on |z| = 1, then m = min|z|=i |P(z)| = 0 and the result follows from Theorem B. We suppose that P(z) has no zeros on |z| = 1, so that m > 0, and we have |mPzn| < |P(z)| for |z| = 1 for any P with |P| < 1. By Rouche's theorem, the polynomial F(z) = P(z) + Pmzn has all its zeros in |z| > 1. By Lemma 3, we have

|B[F](z)| ^ |B[F*](z)| for |z| ^ 1,

where F*(z) = znF(1). In particular, for every R > 1 we have

|B [P(Rz) + PmRnzn] | ^ |B [P*(Rz) + pm] |. (12)

Since P(z) and zn are of same degree and B is linear in this case,

B[P (Rz) + PmRnzn] = B[P ](Rz) + PmB [En(Rz)], where [En](z) = zn. Also, it can be easily verified that B[P *(Rz) + Pm] = B [P *](Rz) + PmAo

for |z| = 1. Therefore, we have, from inequality (12),

|B[P](Rz) + mPB[En](Rz)| ^ |B[P;](Rz) + mpAo|, |z| = 1. (13)

Choosing the argument of P suitably in the left-hand side of (13) and using the triangle inequality in the right-hand side, we get from (13):

|B[P](Rz)| + m|P||B[En](Rz)| ^ |B[P;](Rz)| + m|P||Ao|.

Equivalently,

|B[P](Rz)| ^ |B[P;](Rz)| — m|P|{|B[En](Rz)| — |Ao|>. (14) Since |B[En](Rz)| = Rn| A | for |z| = 1, this gives

|B[P](Rz)| + |^^}m|P| ^ (15)

^ |B[P;](Rz)| — {JA|R2—M}m|P|.

If we take L = |B[P;](Rz)|, M = |B[P](Rz)| and N = { |a|r"2-|Aq| }m|P|, so that M + N ^ L — N ^ L and we get, by using Lemma 2, for every real a and | z| = 1 ,

|B[P;](Rz)| — Big{|A|Rn2 — N }m|P| + | A |Rn — |Ao|

+eia

|B[P](Rz)| + { |A|R2 |Ao^m|P|]| ^ |B[P](Rz)|+eia|B[P;](Rz)|

This gives, for each p > 0, 0 ^ 9 < 2n,

2n 2n

J |G(9) + eiaF(9)|pd9 ^ J oo

p

|B[P ](Rei0 )| + eia |B [P ;](Rei0 )|'

d9, (16)

where I }

F(9) = |B[PjCRe-)| + | |A|R"2 — ^ }m|

and I }

G(9) = |B [P ;](Re-)|^|A|Rn2—|Ao^ m|

Integrating both sides of (16) with respect to a from 0 to 2n and then using Lemma 3, we get

2n 2n

|G(9) + e'aF(9)|pd9da ^

oo

2n 2n

|B[P ](Re'g )| + e'a|B [P ;](Re'g )|

oo

2n 2n

oo

d9da

|B[P ](Re'g )| + e'a|(B[P; ](Re'g ));|

d9da.

Using Lemma 5, this gives for every real a and 0 ^ 9 < 2n,

j J |G(9) + e'aF(9)|pd9da ^ J |Rn A e'a + Ao|pdaj |P(e'g)|pd9. (17) o o o o

Now, we know that for every real a and r ^ 1

|r + e'a| ^ |1 + e'a|. This implies, for each p > 0,

2n 2n

/|r + e'a|pda + eTda (18)

oo

Now, if F(9) = 0, then from (15) for r = ^gj1 ^ 1, we have, by using (18),

2n

2n

J |G(9) + e'aF(9)|pda = |F(9)|p J

oo

|| p

G(9)+ e<"

F(9)

da =

2n

|F (9)|p

o

G(9)

F(9)

+ e'

B[P ](Re'g) +

Rn| A | — |Ao|

m

2n

p |1 +

J o

2n

|P | p /|

J o

|1 + e'a|pda. (19)

p

p

p

Clearly, Inequality (19) is trivial in the case F(0) = 0. Now, integrating the two sides of (19) with respect to 0 and then using (17), we get

2n 2n

' ,Rn|A|-|Ao| } p

B [P ](Re-) + { R |A 2-|Ao|| m|

d0 / |1 + eia|pda ^

o

2n 2n

^ J |Rn A eia + Ao|pda J |P(ei0)|pd0, o o

for every P with |P | < 1, R > 1, and p > 0, where A is defined by (6), a is real, and 0 ^ 0 < 2n. This, in particular, gives

2n

B№) + {}m| p

d0 <

2n

j |Rn A eia + Ao|pda 2n

^ ^- i |P(e*)|pd0.

/|1 + eia|pda o

o

And this, in turn, gives the desired result.

For the equality case, note that for the polynomial P(z) = zn + 1, m = 0, and || 1 + zn ||p=|| 1 + z ||p; so that

B [P](R.) + (RnAzn - |Ao|)m|

p

= || (Ao + Aiy + A2n2(n8 1))Rnzn + Ao ||p=

= || RnAzn + Ao ||p= H RnA1z++Ao "p || P Hp .

H 1 + z Hp

References

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DOI: https://doi .org/10.15393/j3.art.2016.3250.

Received March 22, 2020. In revised form, September 15, 2020. Accepted October 14, 2020. Published online November 15, 2020.

Central University of Kashmir,

Ganderbal - 191201, Jammu and Kashmir, India

E-mail: shahlw@yahoo.co.in