CC BY
3Probl. Anal. Issues Anal. Vol. 13 (31), No3, 2024, pp. 101-117
DOI: 10.15393/j3.art.2024.16050
101
UDC 517.53
N. A. Rather, N. Wani, A. Bhat
INTEGRAL MEAN ESTIMATE FOR POLYNOMIALS WITH
RESTRICTED ZEROS
Abstract. In this paper, we present certain sharp Lp-inequalities for polynomials with restricted zeros. Our results improve and generalize some known integral inequalities for polynomials in the complex domain.
Key words: polynomials, inequalities, complex domain
2020 Mathematical Subject Classification: 26D10, 41A17, 30C15.
1. Introduction and statements of the main results. Let Pn
denote the set of all complex polynomials P(z) = YTj=0 bjof degree n. The subset P°(p) consists of polynomials whose zeros all lie within the disk defined by \z\ ^ p. Specifically, = P°(1) represents polynomials with zeros inside the unit disk. The set P8 includes polynomials whose zeros are located in the region \z\ ^ 1.
For a polynomial P e Pn, the p-norm in the Hardy space is defined as
/ 1 f \ Vp
\\p||P = J \P(ez&)\p M) , 0 < p <8.
0
It is not hard to observe that lim \\PL = max \ P(z)\. For this reason,
the uniform norm max\P(z)\ of P(z) is denoted by \\P\\8. On the other
M=i
/ 27r \ hand, lim \\P\\p = exM 2L ^ ln\P(eid)\dO) (see [15, p. 139], [21]). This P^o+ V 2 0 /
is known as the Mahler measure of P(z) and is denoted by \\P\\0. As an
© Petrozavodsk State University, 2024
application of Jensen's inequality, the Mahler measure of the n-th degree
n
polynomial P(z) = b ^ (z ~ zv) can be explicitly given by
V=i
\P||o = \b\ nmax(1, \zv|).
V = i
If Pe P„, then
and
max\P'(z)\ ^ nmax\P(z)\
H=i M=i
max \P(z)\ ^ Rn max\P(z)\. (3)
!z!-R>i !z!=i
Inequality (2) is an immediate consequence of S. Bernstein's Theorem [8] on the derivative of a trigonometric polynomial. Inequality (3) is a simple deduction from the maximum modulus principle. The equality in (2) and (3) holds for P(z) = azn, a ^ 0.
If we restrict ourselves to the class of polynomials P e P8, then inequalities (2) and (3) can be, respectively, replaced by
max\P'(z)\ ^ -max\P(Z)\ (4)
M"1 2 !^!"1
and
Rn + 1
max \P(z)\ ^ —-— max\P(z)\. (5)
!z!-R>i 2 M=i
Inequality (4) was conjectured by P. Erdos and later verified by P. D. Lax [14]. Ankeny and Rivlin [1] used (4) to prove inequality (5). The equality in (4) and (5) holds for P(z) = azn + b, \a\ = \6\ ^ 0.
As an analogue of Bernstein's inequality in the Hardy space norm, Zygmund [23] proved that if P(z) is a polynomial of degree n, then
\\P'\\P < n\\P\\p, p ^ 1. (6)
De Bruijn and Springer [10] and later Mahler [21] proved that this inequality also holds for p = 0, but for the case 0 < p < 1 its validity remained an open question for quite a long time. Finally, Arestov [3] obtained an inequality concerning the Schur-Szego product of polynomials, which among other things also answered the question.
n
The Schur-Szego composition of a polynomial P(z) = bjzj e Pn
j-0
n n
with another polynomial Q(z) = (ra) jjzj is defined as P*Q = jjbjzj.
3=0 3 j=0
For Q * P, Arestov [2] (see also [3]) proved the following inequality, which also includes the case 0 ^ p < 1 of (6) as a special case:
| | Q * P||p ^ ||Q\\oIIPIIp, for p > 0. (7)
Inequality (6) follows at once from (7) by taking Q(z) = nz(z + 1)ra~1 =
n
= Zj (™) Jzj • For the class of polynomials P e Vf, inequality (6) can be
3=0 3
sharpened. In fact, in this case inequality (6) can be replaced by
IIP'IIp ^ n ,1|P|py , P > 0. (8)
11 1 ^ znp
This inequality is due to N. G. De Bruijn [10] for p ^ 1, whereas Rahman and Schmeisser [18] extended it for 0 ^ p < 1.
As a generalization of (8) and in the spirit of (7), Arestov [4] also
n
proved that if P(z) e V8 and Q(z) = ^ ljz3 e then
3=0
IIP * Q\\P ^ IIPII» P > 0. (9)
1 1 1 ' ^11 p
n
Inequality (8) follows from (9) by choosing Q(z) = nz(z+1)n~1 = (ra) jz^.
3=0 3
For polynomials P e Pf, Boas and Rahman [9] established an analogue of inequality (5) in the Lp-norm for p ^ 1:
I I * + Rn I I P
I I P (Rz) 11 p ^ 1 1 |U ■ HI1 1 p I I P (z) 11 p , R > 1. (10)
1 1 z ~r 1 11 p
Equality in (10) holds for P(z) = azn + b, with |a| = |6| ^ 0. By letting p in (10), one can recover inequality (5).
Rahman and Schmeisser [18] (see also [4]) later showed that inequality (9) also holds for 0 ^ p < 1. The above inequality has been generalized in several ways and a good number of papers are available (see, for example, [7], [6], [19], [20]). By applying inequality (10) to the polynomial znP (1),
we immediately deduce that for P e Pn and 0 < r ^ 1, the following inequality holds for each p > 0:
I rn 7 + 111
I I P M }} p ^ 11 P (z) }} p. (11)
I 1 Z + 1 11 p
This result is sharp and equality in (11) holds for P(z) = azn + b with \a\ = \b\ ^ 0.
By letting p ^ œ in (11), we obtain the following sharp inequality under the conditions of (11):
rn + i
+ 1 1 '12)
ML < IIP(z
In this paper, we present the following result, which is a generalization as well as a refinement of inequality (11). More precisely, we prove
n
Theorem 1. For any polynomial P (z) = bj ze P°, 0 ^ p < œ,
j-0
0 < r < 1, and 0 ^ t < 1, we have
.. _ r.n \\rn + zI \p(rz)\+ tm^—- ^ "y—f IIP(z)II„ , (13)
1 + pr p II^r + ZI
up
where m = min \p(z)\ and
\z\ = l
_ ( r\60\ + rtm + \bn\ \ ^r ^ \bo\ + tm + r\bn\ J
The result is sharp and equality in (13) holds for P(z) = azn + b, \a\ = \b\ ^ 0.
n
Remark 1. Since P(z) = £ b3zj e P0,
j-0
n
Q(z) = znnW) = 2 b:i^ e Pn .
3=0
By Lemma 4, we have \bn\ ^ \&0\+ m, wherem = min\Q(z)\ = min\P (z)\.
\z\ = l \z\ = l
This implies for 0 ^ t ^ 1 \bn\ ^ \b0\ + tm, which gives for 0 < r ^ 1
(1 - r)\bn\ ^ (1 - r)\b0\ + tm(1 - r),
or, equivalently,
|bA + Abo| + rtm > 1 |bo| + Ab,n\ + tm ^ ' That is, Tr ^ 1 for 0 < r ^ 1.
Since | | fir + z 11 p ^ | | 1 +z | | p , p ^ 0, inequality (13) refines inequality (11). For t = 0, inequality (13) reduces to the following refinement of inequality (11).
n
Corollary 1. If P(z) =2] bvzv e P0, then for each r < 1, 0 ^ p <8:
v=0
I I + z 11 p
11P M 11 p ^ lTT—r 11P (z) 11 p , (14)
1 1 °r + ^1 1 p
where
0 = A M +1 U r | b01 + A M.
The result is sharp and equality in (14) holds for P(z) = azn + b with |a| = ^ ^ 0.
By letting p ^ 8 in (13) and noting that tr ^ 1, we obtain the following refinement of inequality (12):
Corollary 2. For any polynomial P(z) = bjZJ e 0 < r < 1, and
3=0
0 ^ t < 1, we have
1 1 p ^ I-«()lP WI L -tm (T+w), (15)
where m = min ^(z)| and Tr is given by (13). The inequality is sharp
M"1
and equality in (15) holds for P(z) = azn + b, |a| = |6| ^ 0.
If p(z) = Y, bjZj e P-, then the polynomial P*(z) = znP(1/z) e
3=0
Applying Theorem 1 to the polynomial P*(z) with r = , we obtain the following refinement of inequality (10):
n
Corollary 3. IfP(z) = 2 bjZj e P-, then for each R > 1, 0 ^p <8,
3=0
and 0 ^ t ^ 1:
Rn5n- 1 lRn + z ||p
^ (RzK+ tm—^— ^ " + } I IP (z)I Ip , (16)
1 + O r p 11 Or + Z11 p p
where m = min \ P(z)\ and
\z\ = l
R\bo\ + \ bn\ + tm R " \bo\ + R\bn\ + Rtm U ).
The bound is sharp and equality in (16) holds for P(z) = azn + b, \a\ = \6\ ^ 0. Since \\6R + z\\p ^ \ \ 1 + z\ \ p, p ^ 0, inequality (16) refines inequality (10). Fort = 0, inequality (16) also refines inequality (10).
A. Aziz [5] proved that if P(z) = YTj=0 e Pn(p) where p ^ 1, then for 1 ^ p <8 and 0 ^ t ^ 1:
\ \ P'(Z) \\ 8 ^ J^-^ \\ P(*) \\P . (17)
Now, we will show that the bound in (17) can be improved by using Corollary 3. More precisely, we prove the following result:
Theorem 2. If P(z) = Xlj=0 bjZj e P°(p) where p ^ 1, then for 0 ^ p <8 and 0 ^ t ^ 1 :
n \\((p) + z\\p . , tmpn((p) — 1
\ \ p'(z)\\ »^ t^T i,:;1 , \p(z)\ +
\ \ pn + Z \ \ p \ \ 1 + Z \ \ p
where
pn 1 + ((p)
:18)
¡I x \6o\ + Pn+1\bg\ + tm .
((p) = ——:-—- and m = min\F (z)\. (19)
( ) Pn\bn\+p\bo\+ptm \z\-p \ ( y '
The result is sharp and equality in (18) holds for P(z) = zn + pn.
Remark 2. Since all the zeros of P(z) are in \z\ ^ p, p^ 1, it can be easily seen that ((p) ^ 1. In view of this, Theorem 2 is a refinement of the inequality (17).
The following result is obtained by letting p ^ 8 in the Theorem 2.
Corollary 4. If P(z) = YTj-0 bjzj e P(0(p), where p ^ 1, then for 0 ^ p <8 and 0 ^ t ^ 1
!№)!!» s {l?^ + , (20)
where ( ( ) is given by (19). The result is the best possible as shown by p (z) = zn + pn.
p
Since <fr(p) > 1, inequality (20) improves the result by N. K. Govil [11], which states that if P(z) e P%(p), p> 1, then
IIP'II >-^l|P|| I I P I I 8 > i + pn\ I P 11 8
2. Lemmas. For the proof of our results, we need the following lemmas. The first lemma is a well-known generalization of the Schwarz lemma by Osserman [16].
Lemma 1. Let F(z) be analytic in \z\ < 1 with F(0) = 0, and \F(z)\ < 1 for \ z\ < 1; then
Lemma 2. Let a, b be complex numbers independent of a, where a is real. Then for each p > 0 :
2n 2-k
V i lll_l , w\„ia\P
\a + beia\P da = J \\a\ + \b\eia[ da.
o o
Using periodicity, it is easy to verify the lemma, so we omit the details. The following Lemma is by Aziz and Rather [6]:
Lemma 3. If A, B, C are non-negative real numbers and B + C ^ A, then for every real number a
\(A - C)eia + (B + C)\ ^ \Aeza + B\.
The next lemma is by Gulzar and Rather [12]:
n
Lemma 4. If P(z) = bjZJ e and m = min \P(z)\, then
j-o \z\"1
\bn\ > \bo\ + m.
The next lemma is a consequence of the result by Arestov [ [3], Theorem 4]. Yet, here we deduce it from inequality (7) due to Arestov [2].
n
Lemma 5. If P(z) = bjZJ e P8, then for every p > 0, r < 1 and
j-0
real ft:
J\P(reie) + eifirnP(eie/r)\P d0 ^ \rneifi + 1\P J\P(eie)\P d9. oo
Proof. For 0 < r ^ 1 and \z\ > 1, \z + r\> \rz + 1|. This gives,
\z + r\n >\rz + 1\n, \z\ > 1,
which implies that the polynomial ei3 (z + r)n + (rz + 1)n has no zeros in \z\ > 1. Hence, all the zeros of ei3(z + r)n + (rz + 1)n lie in \z\ ^ 1 for
0 < r ^ 1. Setting Q(z) = ei3(z + r)n + (rz + 1)n and noting that by (1),
1 I Q||o = \rnei3 + 1\, we obtain by invoking inequality (7),
| | P(rz) + ei3rnP(z/r)IIP ^ \rnei3 + 1\|}P(z)}}p, p^ 0. That proves Lemma 5. □
Definition 1. [17, pp. 36]. Let f and g be analytic in \z\ < 1. We say that the function f is subordinate to g, if there exists a function w, analytic in \ z\ < 1 with w(0) = 0 and \w(z)\ < 1 for \z\ < 1, such that
f(z)=g(w(z)) (\z\ < 1).
Lemma 6. [17, pp. 36]. Let f and g be analytic for \ z\ ^ 1 and such that f is subordinate to g. In addition, if g is univalent in the same disc, then for each p > 0 we have:
j\f(eid)\pd9 ^ J \g(eid)\>d9. oo
3. Proofs of the theorems.
Proof of Theorem 1. By the assumption, all the zeros of polynomial
n
P(z) " X! bvzv lie in \z\ ^ 1; therefore, the conjugate polynomial
u-0
__n _
Q(z) = znP(1/z) = bjZn~j has all its zeros in ^ 1
j-0
and m = min \ P(z)\ = min \ Q(z)\, which implies
\z\ = l \z\ = l
\z\nm ^ \Q(z)\ for \z\ = 1. By the Maximum Modulus principle, we have
\z\nm <\Q(z)\ for \z\ < 1. (21)
So, for any a e C with \a| ^ 1, the polynomial G(z) = Q(z) + amzn does not vanish in \z\ < 1. Indeed, if G(z) = Q(z) + amzn has a zero in \z\ < 1 at z = z0, then
G(zo) = Q(zo) + amz" = 0, \zo\ < 1
This implies
\Q( Z0)\=m\a\\Z0\n < m\Z0\n for \ Z0\ < 1,
contradicting (21). Hence, we conclude that @a e C with \a| ^ 1, the polynomial G(z) = Q(z) + amzn = (b0 + am)zn + ^"¡=1 bjZn~j has all its
zeros in \z\ ^ 1. Let H(z) = znG(1/~z) = P(z) + am, then the function
z H(z) . „ . . ,T . . b0 + am
r (z) = satisfies the assumption of Lemma 1 with r '(0) =
G( )
and, therefore,
\F(z)\ ^ \ 4
W +
bn + am
n
1
bn + am
n
| |
This gives
iu-/M/\bn\\z\ + \b0 + am\ , ,
\H(z)\ ^ ' , .. —_ ,, \G(z)\ for \z\ < 1.
| n| + | 0 + am | |
n
(22)
Setting z = re% where 0 ^ 9 ^ 2n and r < 1 in (22), we get
e* ^ rlbn + \b, + am\ eid
\ bn \ +r \ b0 + am\
(23)
The function f(x) = ^ \ ^ \ + X is non-decreasing for x ^ 0. Using the fact
that for any a e C
\ bn \ + rx
\b0 + am\ ^ \b0\ + \a\m,
we get from inequality (23) that for every a e C with \a\ ^ 1, r < 1 and \ * \" 1:
\ H(re«)\ < M +j'°\\+ \G(re«)\.
\ bn \ + r \ b0 \ + r \ a\m
Equivalently,
Pr\P(reid) + am\ ^ \rnP(ete/r) + amrn\. (24)
where pr = \ ^—\. 0\ + \. \.—. Choosing the argument of a in the left-r\bn\ + \ bo \ + \a\m
hand side of (24) such that
\P (r eid) + am\ = \P (r eld )\ + \a\m,
we get
Pr{\P(reid)\ + \a\m} ^ \rnP(eid/r)\ + \a\rnm.
This gives
Pr\P(rel9)\ + \a\m(nr - rn) ^ \rnP(e/r)\,
equivalently,
pA\P(reie)\ + \a\m^r—-} ^ \rnP(el9/r)\- \a\mPr—-. (25) ( 1 + pr ) 1 + pr
Since, by Remark 1, pr > 1, therefore, we have
n n
\P(re10)\ + \a\mpr—- ^ \rnP(et&/r)\- \a\mpr—-. (26) 1 + pr 1 + pr
Taking A = \rnP(e%d/r)\, B = \P(re%d)\, and C = \a\m—-in Lemma 3
1 + pr
and noting by (26) that B + C ^ A - C ^ A, we get for every real ft:
(\rnP(eie/r)\- + (\P(rel°)\ + )
1 + pr ) V 1 + pr
^ \\rnP(e/r)\e113 + \P(rei&
This yields, for each > 0,
+ e13N(0)\pd9 ^ f \\rnP(e*/r)\e13 + \P(re10)\\Pd9, (27)
o
where
II _ iy*'° II _ iV>'</
M(0) = \P(reie)\ + \a\m——-, N(0) = \rnP(eid/r)\- \a\m-—-.
1 + -r 1 + -r
Integrating both sides of (27) with respect to 3 from 0 to 2ir and using Lemma 2, we get
2k 2K
\ M(0) + eil3N(0)\pd0d3 ^
00
2K 2K
r.np(J0 lr\\J3 , \p(rJ8\I \pr
^ | J \ \ rnP(eM/r)\eil3 + \P(re )\\Pd0dp = 0
2k
| f \ \ rnP(eM/r)\ei3 + \P(reiB)\\Pdp^d0 =
00
2k 2K
\ P (r e *) + rn e i33P (eie/r)\Pd0dp. 00
Combining this with Lemma 5, we have
2K 2K 2K 2K
\M(0) + ei33N(0)\Pd0dp ^ J\rnei33 + 1\Pdp J \P(eid\vd0. (28) 0 0 0 0
Now, for every real 3 and r0 ^ r1 ^ 1, we have
\ r0 + ei3 \ ^ \ r1 + ei3 \ , which implies, for each > 0,
2K 2K
r0 + ,3\>da >J\n + e«rda.
00
If \ M(0)\ * 0, we take ro = \ N(0)\/\M(0)\ and n = -r; then, by (25), t0 ^ r1 ^ 1, and we get, by using Lemma 2:
ift
ift
J | M(9) + el"rnN(9)\J dp = \M(9)\ 0 0
1 + e ^
N( )
ift
= \ m (6) r
e ^ +
N( )
ift
M( )
dp = \m (d) f
M( )
dp = n(d) p
M( )
dp ^
ift
^ \ M(9) \PJ | e%p + Pr \1Jdp. 0
For \M(9)\ = 0, this inequality is trivially true. Using this inequality in (28), we obtain for each p > 0, r < 1, and real P:
ift
e 13 + pr Y dp
1ft r*
! \p (
j 0
ift
0
\ P (r eie) \ + \ a\m
^ | \rnel3 + 1\P dp j \P(e19)\J d0 = 0
pr — r'~
1 + pr
de <
ift
\p
ift
ift
= J \rn + e113\P dp J \P(e19)\Pdd. 00
This gives
\ P (re19) \ + \ a\m
Pr — r
1 + pr
^ -n-1
p \\ Pr + z\\
P( )
(29)
which proves the desired result for p > 0. To prove the result for p = 0, we simply let p ^ 0+ in (29). □
Proof of Theorem 2. By the assumption, all the zeros of P(z) lie in \z\ ^ p, where p ^ 1. Therefore, all the zeros of T(z) = P(pz) are in \z\ ^ 1 and, consequently, the zeros of polynomial R(z) = znT(1/z) are outside \z\ < 1. If z1, z2, ..., zn are the zeros of R(z), then \Zj\ ^ 1, j = 1, 2,... ,n, and
zR' (z)
R(z) j" Z — Zj
2
p
rn + z
p
p
p
This gives, for the points eie, 0 ^ 9 < 2n with R(eie) ^ 0:
Re e-RM^ = £ Re
Ad
R{e * )
£
1 n
3 = 1
eid - z^ H 2 2 "
3 = 1
This implies
ezdR' Peld )
nR{ eid )
ezdR' Peld )
nR{ eid )
R(e10) ^ 0.
Equivalently,
| R' pe10) | ^ | nR{e10)- e10R' {ety)|
(30)
for the points e%d, 0 ^ 9 < 2n, which are not zeros of R(z). This inequality is also true, even if e%e is a zero of R(z). It follows that
| R'Pz) | ^ | nR{z) - zR'Pz) | for |z | = 1.
(31)
Since all the zeros of TPz) are in \z\ ^ 1, by the Gauss-Lucas theorem the zeros of T'Pz) also lie in \z\ ^ 1. This implies that the polynomial
zn-iT' ^ =nRPz)-zR' Pz)
(32)
does not vanish in \z\ < 1. Therefore, in view of (30), we conclude that the function
fPz) =
R P )
n RP ) - R P )
is analytic for \z\ ^ 1 and \fPz)\ ^ 1 for \z\ = 1. Moreover, fP0) = 0. Therefore, it follows that the function 1 + fPz) is subordinate to the univalent function 1 + z for \z\ ^ 1. Hence, by Lemma 6, we obtain
1-K
|1 + fPeie)|pd9 ^J |1 + ewfd9, p > 0. 0 0
Now,
1 + fPz) =
n RP )
n RP ) - R P )
1
This gives, for \z\ = 1, with the help of (22), for each p > 0
np\R(etd)\p = \1 + f(eie)\p\nR(ez&) — et&R'(ez&)\p = = \1 + f(e * )\p\e <n+iqeT{^\p =
= \1 + f(etd )\P\T' (eld )\p. (34)
inequality (33) in conjunction with (34) gives, for each p > 0:
1-K ift p
np [\R(eie)\pdd ^ J\1 + e*\pdd ^max \T(35) 0 0
Equivalently, for each p > 0:
n\\R(z)\\p ^ \\1 + z\\p max \T(z)\. (36)
\z\ = l
As the polynomial R(z) does not vanish in \z\ < 1, we can apply Corollary 3 to R(z) with R = p and obtain
\\\ R(p z)\ + t m* pl~ 11 ^ + \\ R(z)\\p, (37)
1 + 9(p) W(p) + 4j}
where
\ 6o\+ pn+l \ bn\+ tm* *
Q(p) = —-;—:-r—- and m = mm\R(z)\.
pn\bn\+p\bo\+tpm* \*\=i \ ( )\
Again, since R(z) = znT(1/z) = znP(p/z), we see that for 0 ^ 9 < 2n \R(peid)\ = pn\P(eid)\ and m* = min\R(z)\ = min \T(z)\ = min\P(z)\.
\z\"1 \z\"1 \z\=p
Combining this with (36) and (37), we get:
nilpn\P(z)\ + tmp]m 1 \\p ^ \\pn + ZW*max \T'(Z)\. (38)
1 + 9(p) W(p) + A\p M=1
Applying inequality (2) to the polynomial T'(z) = pP'(pz) of degree at most n — 1, where p ^ 1, we have
max \T'(z)\ = pmax\P'(pz)\ = pmax\P'(z)\ ^ pn max\P'(z)\. (39)
\z\"1 \z\"1 \z\=p \z\"1
By using inequality (39) in (38), we finally obtain
-IN < —11 + Z
pn y 1 + 111 < _
n
trni pnM - 1 \
^p z)| + - V 1 + M )
\ \ np) + z \ \ p
□
Acknowledgment. The authors are extremely grateful to the anonymous referee for valuable suggestions regarding the paper, which helped us to improve the quality of the manuscript.
[1
[2
[3
[4
[5
[6 [7
[8 [9
References
Ankeny N. C., Rivlin T. J. On a theorem of S. Bernstein. Pacific J. Math.,
1955, vol. 5, no. 6, pp. 849-852.
DOI: https://doi .org/10.2140/pjm.1955.5.849
Arestov V. V. On integral inequalities for trigonometric polynomials and their derivatives. Math. USSR-Izv., 1982, vol. 18, no. 1, pp. 1-17. DOI: https://doi.org/10.1070/IM1982v018n01ABEH001375
Arestov V. V. Integral inequalities for algebraic polynomials on the unit circle. Math. Notes Acad. Sci. USSR, 1990, vol. 48, no. 4, pp. 977-984. DOI: https://doi .org/10.1007/BF01139596
Arestov V. V. Integral inequalities for algebraic polynomials with a restriction on their zeros. Anal. Math., 1991, vol. 17, no. 1, pp. 1-20. DOI: https://doi .org/10.1007/BF02055084
Aziz A. Integral mean estimates for polynomials with restricted zeros. J. Approx. Theory, 1988, vol. 55, no. 2, pp. 232-239. DOI: https://doi .org/10.1016/0021-9045(88)90089-5 Aziz A., Rather N. A. Lp inequalities for polynomials. Glas. Math., 1997, vol. 32, pp. 39-43. DOI: https://doi.org/10.4236/am.2011.23038 Aziz A., Rather N. A. Some new generalizations of Zygmund-type inequalities for polynomials. Math. Inequalities Appl., 2012, vol. 15, no. 2, pp. 469-486. DOI: https://doi.org/10.3103/S1068362322040021 Bernstein S. Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une fonction réelle. Paris, 1926, vol.30. Boas R. P., Rahman Q. I. Lp inequalities for polynomials and entire functions. Arch. Rational Mech. Anal., 1962, vol. 11, no. 1, pp. 34-39. DOI: https://doi .org/10.1007/BF00253927
De Bruijn N. G., Springer T. A. On the zeros of composition-polynomials. Proc. Sect. Sci. Kon. Ned. Akad. Wet. Amst., 1947, vol. 50, no. 7-8, pp. 895-903.
CO
[11] Govil N. K. On the derivative of a polynomial. Proc. Amer. Math. Soc., 1973, vol. 41, no. 2, pp. 543-546.
[12] Gulzar S., Rather N. A. On a Composition Preserving Inequalities between Polynomials. J. Contemp. Math. Anal., 2018, vol. 53, no. 3, pp. 21-26. DOI: https://doi.org/10.3103/S1068362318010041
[13] Hardy G. H., Littlewood J. E., Polya G. Inequalities. Cambridge University Press, 1988. DOI: https://doi.org/10.1112/blms/bdv008
[14] Lax P. D. Proof of a conjecture of P. Erdos on the derivative of a polynomial. Bull. Amer. Math. Soc., 1944, vol. 50, pp. 509-513.
DOI: https://doi.org/10.1090/S0002-9904-1944-08177-9
[15] Mahler K. On the zeros of the derivative of a polynomial. Proc. R. Soc. Lond. A., 1961, vol. 264, no. 1317, pp. 145-154.
DOI: https://doi.org/10.1098/rspa.1961.0189
[16] Osserman R. A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc., 2000, vol. 128, no. 12, pp. 3513-3517.
DOI: https://doi.org/10.48550/arXiv.math/9712280
[17] Rahman Q. I., Schmeisser G. Analytic Theory of Polynomials. Oxford University Press, 2002.
DOI: https://doi.org/10.1093/oso/9780198534938.001.0001
[18] Rahman Q. I., Schmeisser G. Lp inequalities for polynomials. J. Approx. Theory, 1988, vol. 53, no. 1, pp. 26-32.
[19] Rather N. A., Bhat A., Shafi M. Integral inequalities for the growth and higher derivative of polynomials. J. Contemp. Math. Anal., 2022, vol. 57, no. 4, pp. 242-251. DOI: https://doi.org/10.3103/S1068362322040021
[20] Rather N. A., Gulzar S., Bhat A. Lp inequalities for the growth of polynomials with restricted zeros. Arch. Math. (Brno), 2022, vol. 58, no. 3, pp. 159-167. DOI: https://doi.org/10.5817/AM2022-3-159
[21] Smyth C. The Mahler measure of algebraic numbers: a survey. Lond. Math. Soc. Lecture Note Ser., 2008, vol. 352, pp. 322-349.
DOI: https://doi.org/10.48550/arXiv.math/0701397
[22] Turan P. Uber die Ableitung von Polynomen. Compositio Math., 1940, vol. 7, pp. 89-95. http://eudml.org/doc/88754
[23] Zygmund A. A remark on conjugate series. Proc. Lond. Math. Soc., 1932, vol. s2 -34, no. 1, pp. 392-400.
DOI: https://doi.org/10.1112/plms/s2-34.1.392
Received April 25, 2024. In revised form, September 17, 2024. Accepted September 19, 2024. Published online October 22, 2024.
Department of Mathematics, University of Kashmir, Srinagar-190006, India N. A. Rather
E-mail: [email protected] Naseer Wani
E-mail: [email protected] Aijaz Bhat
E-mail: [email protected]
