LP-INEQUALITIES FOR THE POLAR DERIVATIVE OF A LACUNARY-TYPE POLYNOMIAL Текст научной статьи по специальности «Математика»

UDC 517.5 Вестник СПбГУ. Математика. Механика. Астрономия. 2021. Т. 8 (66). Вып. 3

MSC 30C10, 26D10, 41A17

Lp-inequalities for the polar derivative of a lacunary-type polynomial

N. A. Rather1, L. Ali1, S. Gulzar2

1 Department of Mathematics, University of Kashmir, Srinagar, 190006, India

2 Department of Mathematics, Government College for Engineering & Technology, Safapora, Ganderbal, Kashmir, 193504, India

For citation: Rather N. A., Ali L., Gulzar S. Lp-inequalities for the polar derivative of a lacunary-type polynomial. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8(66), issue 3, pp. 502-510. https://doi.org/10.21638/spbu01.2021.311

In this paper, we extend an inequality concerning the polar derivative of a polynomial in Lp-norm to the class of lacunary polynomials and thereby obtain a bound that depends on some of the coefficients of the polynomial as well. Keywords: Lp-inequalities, polar derivative, polynomials.

1. Zygmund type inequalities for polynomials. Let f (x) be a real polynomial of degree at most n then according to a well-known classical result in approximation theory due to A. Markov [1],

max If'(x)| < n2 max If(x)|.

The above inequality is best possible because for Chebyshev polynomial Tn(x) = cos(narccosx), max_i<x<i |Tn(x)| = 1 and \T!n(±1)| = n2. This inequality has been generalized in several ways, in particular, S. Bernstein (for details see [2] or [3]) obtained its extension to complex polynomials. Let Pn denote the space of all complex polynomials P(z) = Y^n=o ajzj of degree at most n. According to Bernstein's inequality, if P € Pn then

IIP'IL < n ||PIL , where ||P:=max |P(z)|.

|z|=1

Define the standard Hardy space Hp norm for P € Pn by

,-2n \ 1/P

„¿0\|P

/1 r'2n N

1И1 P:={^J0 We)\PM) , 0 < p < oo.

It is well known that the supremum norm of the space Hsatisfies

||P= l_im |P|p .

The other limiting case, also known as Mahler measure of a polynomial P(z), is

\\P\\0:=exp^j"\og\P(ei9)\dey

© St. Petersburg State University, 2021 502 https: / / doi.org/10.21638/spbu01.2021.311

If P 6Pn, then

IIP 'lip < n||P lip. (1)

Inequality (1) is due to Zygmund [4]. Zygmund obtained this inequality as an analogue to Bernstein's inequality. Arestov [5] showed that the inequality (1) remains valid for 0 < p < 1 as well. Equality in (1) holds for P(z) = azn, a = 0.

For the class of polynomials P € Pn having no zero in |z| < 1, inequality (1) can be sharpened. In fact, if P € Pn and P(z) = 0 for |z| < 1, then

n

imi^^iia.^i. (2)

Inequality (2) was found out by De Bruijn [6]. Rahman and Schmeisser [7] proved the inequality (2) remains true for 0 < p < 1 as well.

The estimates is sharp and equality in (2) holds for P(z) = azn + b, |a| = |b| = 0. Govil and Rahman [8] generalized inequality (2) and proved that if P € Pn does not vanish in |z| < k where k > 1, then

n

(3)

As a refinement of inequality (3), it was shown by Rather [9] that if P € Pn and

P(z) = En=o ajzj = 0 for |z| < k, k > 1, then

n

\\P'L < ne , .1, \\PL,P>0. (4)

11*1 + Zl

where is defined by

i ai I i

51 = k -. (5)

+1

n |ao|

2. Extension of Zygmund type inequalities to polar derivatives. By Gauss — Lucas theorem (see [10]), if all the zeros of a polynomial P € Pn of degree n lie in a half plane then its critical points are also contained therein. Since we may map a half plane onto a closed disk through a bilinear transformation z = 4>{w) = a,b,c,dG C

with ad — bc = 0. Let g(w ) = (cw + d)nP (f^) be transformation of P( z) under then if £ is a critical point of g, then is either to or a zero of the polynomial nP(z) + — z) P'(z). This property guides us to the polynomial

Da[P](z) := nP(z) + (a — z)P'(z),

called the polar derivative of P with respect to a complex number a (for details see [10, p. 44]). Note that Da[P](z) is of degree at most n — 1 and it generalizes the ordinary derivative P'(z) of P(z) in the sense that

a ^^o a

uniformly with respect z for |z| < R, R > 0.

p

Aziz and Rather [11] extended inequality (2) to the polar derivative of a polynomial and proved that if P £?„ and P(z) does not vanish in |z| < 1, then for a e C with |a| > 1, and p > 1,

/ H + i N |l + *l

11^]||Р<П|7ГГГГ1Г11И1Р- (6)

II p

Aziz et al. [12] also obtained an analogue of inequality (3) to the polar derivative and proved that if P eP„ and P(z) = 0 for |z| < k where k > 1, then for a G C with |a| > 1 and p > 1,

\\Da[P]\\<n(P±^-) ||P|I. (7)

llk-

z

p

Later N. A. Rather [13, 14] showed that inequalities (6) and (7) remain valid for 0 < p < 1

as well.

Recently, Rather et al. [15] extended (4) to the polar derivative which among other things also include a refinement of (7) and proved if P ePn and P(z) does not vanish in |z| < k where k > 1, then for a e C with |a| > 1 and 0 < p < to,

»^-"(m)11^1" (8)

where is given by (5).

Let P„iM C Pn be a class of lacunary type polynomials P(z) = a0 + aizj,

where 1 < ^ < n.

As a generalization of inequality (8), they [15] also proved that if P e P„iM and P(z) does not vanish in |z| < k where k > 1, then for a e C with |a| > 1 and 0 < p < to,

' |a| + ^

||Да[Р]||р<п| „V ; ^ ) ||P||p, (9)

where

"p" Vll^ + zip

3. Main results. Our main result is a compact generalization of all the above results for the class of polynomials not vanishing in | z| < k, k > 1. Here, we present our main result:

Theorem. If P e Pnand P(z) does not vanish in |z| < k where k > 1, then for a e C with |a| > 1, 0 < p < to and 0 < t < 1,

where m = minu|=k |P(z)| and

Г (я) + i I

p

For t = 0, (10) reduces to (9). If in above theorem, we let p ^ to, we obtain the following Corollary.

Corollary 3.1. If P € Pn,M and P(z) does not vanish in |z| < k where k > 1, then for a € C with |a| > 1 and 0 < t < 1,

1+ *

■i(M + <W)l|PlU - (|a| - 1)tm}

where is given by (11).

If we divide both sides of inequality (10) by |a| and let |a| ^ to, we obtain the following refinement of inequality (4).

Corollary 3.2. If P € Pn,M and P(z) does not vanish in |z| < k where k > 1, then for 0 < p < to and 0 < t < 1,

|P 'I +

nmt

1 + *

<

|z + *

Hp

-l|P Hp

(12)

where m = min|z|=k |P(z)|. The result is best possible as shown by the polynomial P(z) = (z^ + k^)n/^.

Inequality (12) also includes a refinement of (3). By taking k = 1 and m = 1 in (12), the following improvement of inequality (2), which holds uniforms for 0 < t < 1, follows immediately.

Corollary 3.3. If P € Pn and P(z) does not vanish in |z| < 1 then for 0 < p < to and 0 t 1,

IP 'I +

nmt

2

<

1 + z|

IPII

(13)

where m = min|z| = 1 |P(z)|. The result is sharp and equality in (13) holds for P(z) =

zn + 1.

4. Lemmas. For the proof of above theorem, we need the following lemmas. The first lemma is due to [16].

Lemma 4.1. If P(z) = a0 + ajzj, 1 < M < n, is a polynomial of degree n

having no zeros in |z| < k where k > 1, then for 0 < t < 1,

<W|P '(z)|<|Q'(z)|— tmn for |z| = 1

and 6> k^1 > 1 where 6is given by (11), Q(z) = znP(l/z) and m = min|z|=fc|P(z)|.

Lemma 4.2. If A, B and C are non-negative real numbers such that B + C < A, then for every real number 0,

|(A - C) + eie(B + C)| < |A + eieB|.

n

n

p

n

p

p

p

The above lemma is due to Aziz and Rather [17] and the next lemma is due to [15].

Lemma 4.3. If a, b are any two positive real numbers such that a > bc where c > 1, then for any x > 1,p > 0 and 0 < 0 < 2n,

(a + |c + eiß|P dß < (c + x)p / |a + beiß|P dß.

./0 ./o

We also need the following lemma due to Aziz and Rather [18].

Lemma 4.4. If P £ Vn and Q(z) = znP(l/~z), then for every p > 0 and ß real, 0 < ß < 2n,

/ / |P'(eiö)+ eißQ'(eiö)|P dödß < 2nnM |P(eie)|P d<9. 0 0 0

5. Proof of Theorem. By hypothesis P G Vn does not vanish in \z\ < k where k > 1 and Q(z) = znP(l/z), therefore, by Lemma 4.1, we have for \z\ = 1,

|P'(z)| < |Q'(z)| - tmn = |Q'(z)| - tmn

1 + <W

, 1 + Equivalently,

(V'WI + Yrf^) - |i3'(i)l ~ irf^ N-I. (")

i i mnt Setting A = \Q'(eie)\ , B = |P'(ei0)| and C = --— in Lemma 4.2 and noting by (14)

1 + Vt

that B + C < ¿M,t(B + C) < A — C < A since ¿Mjt > 1. Therefore, by Lemma 4.2 for each real 0, we get

I9V)I - YT^j + ^ (|p,(eiö)l + iTJ^Tt

This implies for each p > 0,

< |IQ'(eiö)| + eie|P'(eie)

2п л 2n

/ |F(0)+ eieG(0)|P dö < / ||Q'(eiö)| + eie|P'(eiö)||P d0 (15)

J 0 ./q

where

F(0) = \Q\e^)\- and G{B) = |P'(e^)| + (16)

1 + Vt 1 + Vt

Integrating (15) both sides with respect to 0 from 0 to 2n and using properties of definite integrals, we get

f 2n f2n f 2n 2n

/ / |F(0)+ e^G(0)|P d0d, < / / ||Q'(eie)| + e^|P'(eie)||P d0d, = Jo Jo Jo Jo

= / |P'(eie) + e^Q'(eie)|P d0d0.

oo

By using Lemma 4.4 this implies,

,-2n 2n 2n

nzn f 2n

|F(0)+ < 2nnM |P(ei0)|P d<9. (17)

./0

Since S^t > 1, we have

mnt „ „-д,, mnt

1 + Vi 1 + Vt

( mnt \

On adding t |P'(e*e)| -|---— on both sides, where 0 < t < 1, we get

V 1 + SM,t J

mnt „-д,, mnt

1 + SM,t V 1 + SM,t

This further implies for each p > 0,

Now for a e C with | a| > 1 and 0 < t < 1, we have

|I?a[P](e«)| +nmi ) < + |Q'(ei9)| + nmt {^^f

n„¿0 M , mnt ^ , Z'i^w„¿0- mnt

= H [\Р'(П\ + —— + |Q'(0|-

1 + sM,t7 v 1 + sM,t

By integrating both sides with respect to d from 0 to 2n, for each p > 0, we get

Multiply both sides by /02n |SM,t + e®^ |pd,0, we obtain

IS^t + eVfdpJ*" | \Da[P](ei9)\ + nmt (') ^ <

£ jf (^)l+i=fc) + (m* - ^

(18)

Further, since (5Mji > 1, 1 < ¡1 < n, by Lemma 4.3 with a = |Q'(e*e)| — t b = |P'(ei0)| + t, c = and x = |a|, we get for every a G C with |a| > 1,

|Q'(e

mnt

1 + *

+ |a| |P '(ei0 )| +

mnt

1 +

P /*2п

J ) Jo

|<W + eie |pde <

л2п

< (N + <W)P

1 + *

1 + *

¿в.

Again, integrating both sides with respect to 0 from 0 to 2n, we obtain

jf { (ЮУ)1 - + M (|i>V)l + jff;

P t-2n

o

Ml |<W + eie |pde <

л2п л2л

< (|a| + / |F(0) + eieG(0)|P ded0

oo

where F(0) and G(0) are given by (16). Using this in inequality (18), we get

J |<W + eie |de J ||Da[P](eie )| + nmi|

|a| - ^ P

d0 <

1 +

2n 2n

< (|a| + ¿M,t)p / / |F(0)+ eieG(0)|pded0. (19)

oo

By using (17) in (19), we obtain for each p > 0 and |a| > 1

r-2n

|<W + eieИв/ ] |Da[P](e!S)| + nmt

r-2n

H ~

\ \ ¿в <

1 + V«

p

/*2п

< (|a| + *Mji)P2nnM |P(eie)|P d0 o

Equivalently,

л2п

2п

|Da[P ](eie )| + nmt

<

H - <W 1 + <W

n(\a\ +

1/p d0 1 <

(¿/o^ + e*)

1/p

which immediately leads to (15) and this completes the proof of Theorem for p > 0. To obtain this result for p = 0, we simply make p ^ 0+. □

в

p

o

o

o

1

o

References

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2. Milovanovic G.V., Mitrinovic D.S., Rassias Th. Topics in polynomials: Extremal properties. Inequalituies, Zeros. Singapore, World scientific (1994).

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4. Zygmund A. A remark on conjugate series. Proc. London Math. Soc. 34, 392-400 (1932).

5. Arestov V. V. On integral inequalities for trigonometric polynomials and their derivatives. Math. USSR-Izv. 18, 1-17 (1982). http://dx.doi.org/10.1070/IM1982v018n01ABEH001375 [Russ. ed.: Ob in-tegral'nykh neravenstvakh dlia trigonometricheskikh mnogochlenov i ikh proizvodnykh. Izv. Akad. Nauk SSSR Ser. Mat. 45, 3-22 (1981)].

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9. Rather N.A. Extremal properties and location of the zeros of polynomials. PhD thesis. University of Kashmir (1998).

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11. Aziz A., Rather N.A. On an inequality concerning the polar derivative of a polynomial. Proc. Indian Acad. Sci. (Math. Sci.) 117, 349-357 (2007). https://doi.org/10.1007/s12044-007-0030-0

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Received: August 30, 2020 Revised: November 24, 2020 Accepted: March 19, 2021

A u t h o r s' i n fo r m a t i o n:

Nisar Ahmad Rather — Dr. Sci., Professor; [email protected] Liyaqat Ali — PhD; [email protected] Suhail Gulzar — PhD; [email protected]

¿^-неравенства для полярной производной полинома лакунарного типа

Н. А. Ратхер1, Л. Али1, С. Гульзар2

1 Кашмирский университет, Индия, 190006, Сринагар

2 Государственный инженерный и технологический колледж, Индия, 193504, Кашмир, Гандербал, Сафапора

Для цитирования: Rather N. A., Ali L., Gulzar S. Lp-inequalities for the polar derivative of a lacunary-type polynomial // Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия. 2021. Т. 8(66). Вып. 3. С. 502-510. https://doi.org/10.21638/spbu01.2021.311

В настоящей работе мы распространяем неравенство относительно полярной производной многочлена в Ьр-норме на класс лакунарных многочленов и тем самым получаем оценку, которая также зависит от некоторых коэффициентов многочлена. Ключевые слова: Ьр-неравенства, полярная производная, многочлены.

Статья поступила в редакцию 30 августа 2020 г.;

после доработки 24 ноября 2020 г.; рекомендована в печать 19 марта 2021 г.

Контактная информация:

Ратхер Нисар Ахмад — [email protected] Али Лийакат — [email protected] Гульзар Сухайл — [email protected]