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7URAL MATHEMATICAL JOURNAL, Vol. 7, No. 1, 2021, pp. 87-95
DOI: 10.15826/umj.2021.1.007
ON ZYGMUND-TYPE INEQUALITIES CONCERNING POLAR DERIVATIVE OF POLYNOMIALS
Nisar Ahmad Rather
University of Kashmir, Hazratbal, Srinagar, Jammu and Kashmir 190006, India dr.narather@gmail.com
Suhail Gulzar
Government College of Engineering and Technology, Safapora, Ganderbal, Jammu and Kashmir 193504, India sgmattoo@gmail.com
Aijaz Bhat
University of Kashmir, Hazratbal, Srinagar, Jammu and Kashmir 190006, India aijazb824@gmail.com
Abstract: Let P(z) be a polynomial of degree n, then concerning the estimate for maximum of |P'(z)| on the unit circle, it was proved by S.Bernstein that ||P< n||PLater, Zygmund obtained an Lp-norm extension of this inequality. The polar derivative Da[P](z) of P(z), with respect to a point a € C, generalizes the ordinary derivative in the sense that Da[P](z)/a = P'(z). Recently, for polynomials of the form
P(z) = ao + n ajzj, 1 < u < n and having no zero in |z| < k where k > 1, the following Zygmund-type inequality for polar derivative of P(z) was obtained:
/ |a| + k^ \
ll^Q[P]||P< n(^-p-^J||P||p, where |a|>l, p > 0.
In this paper, we obtained a refinement of this inequality by involving minimum modulus of |P(z)| on |z| = k, which also includes improvements of some inequalities, for the derivative of a polynomial with restricted zeros as well.
Keywords: Lp-inequalities, Polar derivative, Polynomials.
1. Zygmund type inequalities for polynomials
Let Pn denote the space of all complex polynomials of degree at most n. Define
/i r2n \ 1/p
||P||p := i — J |P{eld) |Pde) , 0 < p < oo.
It is well known that the supremum norm satisfies
IP||„ := max |P(z)| = lim HP||p .
|z| = 1 p^^ llp
It is also known [11] that lim ||P||p = ||P||o, where
"p
IP ||o
:= exp J^ log \P(eie)\d9^j .
Let Da[P](z) denote the polar differentiation (see [12]) of a polynomial P(z) of degree n with respect to a complex number a, then
D«[P](z) := nP(z) + (a - z)P'(z).
Note that Da[P](z) is a polynomial of degree at most n — 1 and it generalizes the ordinary derivative P'(z) of P(z) in the sense that
lim = P>(z)
a y^o a
uniformly with respect to z for |z| < R, R > 0. If P eP,n, then
||P'||p < n|P||p. (1.1)
Inequality (1.1) is due to Zygmund [21] for the case p > 1. In its proof, he uses M. Riesz's interpolation formula by means of Minkowski's inequality and obtained this inequality as an Lp-norm analogue of Bernstein's inequality (for details see [13] or [20]). A natural question was raised here: whether the restriction on p was indeed necessary? The question remained open for quite a long time despite some partial answers. Finally, it was Arestov [1] came up with some remarkable results which among other things proved that the inequality (1.1) remains valid for 0 < p < 1 as well. This result is sharp as shown by P(z) = azn, a = 0. Arestov [2] also obtained some sharp Bernstein-Zygmund type inequalities for the Szego composition operators on the set of algebraic polynomials with restrictions on the location of their zeros.
For the class of polynomials P € Pn having no zero in |z| < 1, inequality (1.1) can be sharpened.
In fact, if P € Pn and P(z) = 0 for |z| < 1, then
n
P' —-n-ll-PlL, p> 1- (1.2)
II Hp - ||l+2||p 11 "P' y >
Inequality (1.2) is due to De Bruijn [7]. Later Rahman and Schmeisser [16] followed Arestov's technique and proved that this inequality remains true for 0 < p < 1 as well. The estimates is sharp and equality in (1.2) holds for P(z) = azn + b, |a| = |b| = 0.
Govil and Rahman [10] generalized inequality (1.2) and proved that if P € Pn does not vanish in |z| < k where k > 1, then
n
p' —-n-ll-PIL, p> i- (i-3)
Let Pn>M cPn be a class of lacunary type polynomials
P (z) = a0 + ^ aj zj,
where 1 < ^ < n.
As a generalization of inequality (1.3), it was shown by Gardner & Weems [8] that if P € Pn>M and P(z) = 0 for |z| < k, k > 1, then
n
P' ——n-||-P|L, (1.4)
Aziz and Rather [5] extended inequality (1.2) to the polar derivative of a polynomial and proved that if P ePn and P(z) does not vanish in |z| < 1, then for a € C with |a| > 1, and p > 1,
|Da[P]||p < n
M +1 I1+-II,-
IPII
(1.5)
Concerning the concept and properties of the polar derivative refer to [14].
Aziz et. al [6] also obtained an analogue of inequality (1.3) to the polar derivative and proved that if P € Pn and P(z) = 0 for |z| < k where k > 1, then for a € C with |a| > 1 and p > 1,
11 Da [P ]L < n
|a| + k \\k + z\L
PI
(1.6)
Rather [17, 18] showed that inequalities (1.5) and (1.6) remain valid for 0 < p < 1 as well.
Recently, as a generalization of inequality (1.6), Rather et. al [19] proved that if P € Pn,U and P(z) does not vanish in |z| < k where k > 1, then for a € C with |a| > 1 and 0 < p < to,
|Da[P]L < n
|a| + ku
I|kU + zyp
2. Main results
PI
(1.7)
In this paper, we obtain a refinement of inequality (1.7) by involving the minimum modulus of a polynomial. We prove the following main result.
Theorem 1. If P € Pn,U and P(z) does not vanish in |z| < k where k > 1, then for a € C with |a| > 1, 0 < p < to and 0 < t < 1,
|Da[P ]| + nmt
M-i
1 + ku
<„( H+fcAt )||P|,
-n| '\z + k%
(2.1)
where m = min|z|=k |P(z)|. Since
nmt(|a| — 1)
1 + ku
> 0 for |a| > 1,
then one can easily observe that
I|Da[P]|p <
|Da[P ]| + nmt
N-1 1 + №
and this implies that the Theorem 1 is a refinement of inequality (1.7).
If we divide both sides of inequality (2.1) by |a| and let |a| ^ to, we obtain the following refinement of inequality (1.4).
Corollary 1. If P € Pn,u and P(z) does not vanish in |z| < k where k > 1, then for 0 < p < to,
|P '| +
nmt
<
n
|z + ku|
IP ||p,
1 + ku
where m = min|z|=k |P(z)|. The result is best possible as shown by the polynomial
P (z) = (zu + ku)n/u,
where ^ divides n.
(2.2)
p
p
p
p
p
Inequality (2.2) also includes a refinement of (1.3). By taking k = 1 and ^ = 1 in (2.2), the following improvement of inequality (1.2) follows immediately.
Corollary 2. If P € Pn and P(z) does not vanish in |z| < 1 then for 0 < p < ro,
|P'| + nmt
n
< ,M , \\P\\P, (2.3)
p II1 + zii P
2
where m = min^^ |P(z)|. The result is sharp and equality in (2.3) holds for P(z) = zn + 1.
3. Lemmas
For the proof of above theorem, we need the following lemmas. Lemma 1. If
n
P(z) = a0 + ^ ajzj, 1 < ^ < n,
j=M
is a polynomial of degree n having no zeros in |z| < k, where k > 1, then
kM|P'(z)| < |Q'(z)| for |z| = 1,
w/iere Q(^) = znP(l/z).
The above Lemma 1 is implicit in Qazi [15] and the proof of next lemma is implicit in [9].
Lemma 2. If P(z) is a polynomial of degree n having no zero in |z| < k, k > 1, then for every A € C with |A| < 1,
|Q'(z)| > |A|mn for |z| = 1,
where
m = minw=fc|P(z)|, Q(z) = znP(l/z).
Lemma 3. If
P(z) = a0 + ^ ajzj, 1 < ^ < n,
j=M
is a polynomial of degree n having no zeros in |z| < k, where k > 1, then for 0 < t < 1,
kM|P'(z)| < |Q'(z)| — mnt for |z| = 1, (3.1)
where
Q(z) = znP(l/z), m = mm]zl=k\P(z)\.
Proof. By hypothesis, the polynomial P(z) has no zero in |z| < k, k > 1. We first show for a given A € C with |A| < 1, the polynomial F(z) = P(z) — Am does not vanish in |z| < k. This is clear if m = 0, that is if P(z) has a zero on |z| = k. We now suppose that all the zeros of P(z) lie in |z| > k, then clearly m > 0 so that m/P(z) is analytic in |z| < k and
m
P(z)
< 1 for | z| = k.
Since m/P(z) is not a constant, by the Minimum modulus principle, it follows that
m < |P(z)| for |z| < k. (3.2)
Now, if F(z) = P(z) — Am has a zero in |z| < k, say at z = zo with |z01 < k, then
P (z0) — Am = 0.
This gives
|P(z0)| = |Am| = |A|m < m, where |z01 < k, which contradicts (3.2). Hence, we conclude that in any case, the polynomial
F(z) = P(z) — Am
does not vanish in |z| < k, k > 1, for every A € C with |A| < 1. Applying Lemma 1 to
F(z) = P(z) — Am,
we get
\Q'(z) - \mnzn~l\ > ¥l\P'{z)\ for \z\ = l. (3.3)
Now choosing the argument of A so that on |z| = 1,
|Q'(z) - ~Xmnzn~11 = \Q'(z)\ - |A|mn (3.4)
which is possible due to lemma 2. By combining (3.3) and (3.4), we obtain
|Q'(z)| > kU|P'(z)| + tmn for |z| = 1, (3.5)
where t = |A| and 0 < t < 1. For the case t = 1, the inequality (3.1) follows immediately by letting t ^ 1 in (3.5) and this completes the proof.
The following lemma is due to Aziz and Rather [3].
Lemma 4. If A, B and C are non-negative real numbers such that B + C < A, then for every real number 5,
|(A — C) + e^(B + C)| < |A + e^B|.
Lemma 5 [19]. 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 < 5 < 2n,
f-2n f-2n
(a + bx)p / |c + e^|pd5 < (c + x)p / |a + be^|pd5 00
We also need the following lemma due to Aziz and Rather [4].
Lemma 6 [4]. If P e Vn and Q{z) = znP(l/z), then for every p > 0 and (5 real, 0 < /3 < 2ir,
f2n /*2n f2n
/ / |P'(ei0) + e^Q'(eie< 2nnp / |P(ei0)|pd0. 0 0 0
4. Proof of Theorem 1
Proof. By hypothesis P € Pn>M and does not vanish in |z| < k, where k > 1 further if
Q(z) =znP(l/z),
then, by Lemma 3, we have for |z| = 1,
P'(z)| < |Q'(z)| — mnt = |Q'(z)| — mnt
1 + V1 1 + №
Equivalently,
fcMIP'^l+T^UlQ'^l-T^ for \z\ = 1.
Setting
A =
1 + k^
QW B = |P'(eiy)|, C =
1 + k^
mnt
1 + fc'4
in Lemma 4 we note by (4.1) that
B + C < kM(B + C) < A — C < A, since k > 1. Therefore, by Lemma 4 for each real we get
\Q'{eie)\ -T^TTJ ) ( \P'(ew)\ +
mnt
1 + k^
This implies for each p > 0
■¿M l
mnt
1 + fc'4
<
|Q' (ei0 )| + e^ |P' (ei0 )|
r 2n
p i'2n
F(0)+ e^G(0) d0 < / |Q'(eie)| + eip|P'(ei0)| Jo
d0,
where
1 + k^
Let P'(0) = |P'(0)|e^ and Q'(0) = |Q'(0)|ei0, then
r2n
Q' (eie )e^ + P' (eie)
r 2n
r2n
d£ =
|Q' (eie )|e^+<« + e^ |P' (ei0 )|
|Q' (eie + |P' (eie )|
Putting ft + 0 — — = then we obtain,
r-2n
Q' (eie )e^ + P' (eie)
d£ =
|Q' (ei0 )|ei# + |P' (ei0 )|
d$.
Since the function
T ($) = |Q' (eie )|ei# + |P' (eie )|
(4.1)
(4.2)
(4.3)
o
o
0
o
0
is periodic with period 2n, hence we have Q' (eie )e^ + P' (eie)
/• 2n
0
r-2n
=
|Q' (ei0 )|ei# + |P' (ei0 )|
0
|Q' (ei0 )|e^ + |P' (e^ )|
Integrating (4.2) both sides with respect to 5 from 0 to 2n and using (4.4), we get
r2n r2n p /-2n /• 2n
n2n p /*2n /*2n
F(0)+ e^G(0) d0d5 < / / |Q'(eie)| + e^|P'(ei0)|
00
n2n
Q' (eie) + e^ P' (eie)
n2n
P' (eie) + e^ Q' (eie)
_
p
d5d0
d5d0
d0d5.
By using Lemma 6 this implies,
r2n /• 2n
n2n
F (0) + e^ G(0)
r2n
d0d5 < 2nnp
P (eie)
d0.
Now for |z| = 1, 0 < t < 1 and a € C with |a| > 1 and using the fact that
|nP (z) — zP' (z)| = |Q' (z)|
for z with unit modulus, we have
\Da[P\{e»)\ + nmt (y^) = InP(z) + (a - z)P'(z)| + nmt (y^ )
< \a\\P'(z)\ + |nP{z) - zP'(z)| +nmt (yq^)
= \a\\P'(e*d)\ + \Q'{eid)\ + nmt (yy-^) By integrating both sides with respect to 0 from 0 to 2n, for each p > 0, we get
mnt 1 + №
\Da[P]{e»)\+nmt№^\
d0
r2n
<
|a| ( |P'(ei0)| +
i^M'^'-I^)
d0.
Multiply both sides by
/•2n
/ |ku + e^ |pd5, 0
we obtain
/*2n /*2n
/ |ku + e^ |pd5 / 00
r2n
<
lA^Ke*^ nmt
p r 2n
d0
M (V(^)l + ryfr) + (lQ'(e")l - ryfr)\ddlW\k" + e*Fdp
(4.4)
(4.5)
(4.6)
0
0
p
p
0
0
Further, since kM > 1, 1 < ^ < n, and if
a=
Q' (e*')
mnt 1 + №
, b =
P' (e*')
+
mnt
1 + №'
c = kM, x = |a| ,
then from (4.1) one can observe that a > bc. Using Lemma 5, we get for every a € C with |a| > 1,
1 + k^
c2n
1 + k^
P /-2n
|k^ + e^ |Pd^
< (|a| + k^)P
/0
|Q' (e* )| —
mnt
1 + №
+ e^ |P' (e*' )| +
mnt
1 + fc^
Again, integrating both sides with respect to 0 from 0 to 2n, we obtain
r2n
|Q' (e* )| —
mnt
mnt
1 + W+W 1^)1 +i + t.
< (|a| + k^)P
r-2n /-2n
/0 JO
F (0) + e^ G(0)
P /• 2n
d0 / |kM + e^ |Pd^ Jo
p
d^d0,
where F(0) and G(0) are given by (4.3). Using this in inequality (4.6), we get
r-2n
r2n
/ |k^ + e^ |d£ 10 Jo
|Da[P ](e*' )| + nmt
|a| — 1
1 + k^
< (|a| + k^)M / |F(0) + e^G(0)|Pd^d0. jo Jo
By using (4.5) in (4.7), we obtain for each p > 0 and |a| > 1
d0
(4.7)
/•2n /*2n
/ |k^ + e^ |d£ Jo Jo
Equivalently,
|Da[P ](e*' )| + nmt
M~1
1 + fc^
r2n
d0 < (|a| + k^)P2nnP / P(e*') Jo
d0.
r2n
2n
|Da [P ](e*' )| + nmt
M-1
1 + A;/*
d0
1/P
<
n(|a| + kM)
1/(2n)/02n |k^ + e^|d£
1/p
p \1/p d0
which immediately leads to (2.1) for 0 < p < oo and the cases p = 0 and p = oo follow by respectively taking the limits p ^ 0+ and p ^ ro. This completes the proof of Theorem 1. □
Acknowledgement
We are thankful to the referee for useful comments and suggestions.
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