INEQUALITIES FOR A CLASS OF MEROMORPHIC FUNCTIONS WHOSE ZEROS ARE WITHIN OR OUTSIDE A GIVEN DISK Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 9, No. 1, 2023, pp. 104-112

DOI: 10.15826/umj.2023.1.008

INEQUALITIES FOR A CLASS OF MEROMORPHIC FUNCTIONS WHOSE ZEROS ARE WITHIN OR OUTSIDE A GIVEN DISK1

Mohd Yousf Mirt, Shah Lubna Wali^, Wali Mohammad Shahttt

Department of Mathematics, Central University of Kashmir, Ganderbal-191201, India

tmyousf@cukashmir.ac.in ttshahlw@yahoo.co.in tttwali@cukashmir.ac.in

Abstract: In this paper, we consider a class of meromorphic functions r(z) having an s-fold zero at the origin and establish some inequalities of Bernstein and Turan type for the modulus of the derivative of rational functions in the sup-norm on the disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases. In particular, our results generalize as well as refine a result due Dewan et al. [6].

Keywords: Polynomial, Rational function, s-fold zeros, Bernstein inequality.

1. Introduction

Let Pn denote the class of all complex polynomials

n

p(z) :=Y1 azj j=0

of degree at most n and p'(z) denote the derivative of p(z). Let Dk denote the region inside Tk := {z : |z| = k} and D+ denote the region outside Tk. For aj € C, we write

n / -

1 — CtiZ

w(z):=l\(z-a3y, B(z):=H

j=i j=i ^ J

and

Tin = Kn{ai,a2, otn) := \ : .

[w(z) j

the set of rational functions with poles ai,a2,...,an, such that aj € D+ and with finite limit at infinity. A famous result due to Bernstein states that:

Theorem 1 [5]. If p € Pn, then for any z € C

max |p'(z)| < n max |p(z)|.

zeTi v/l zeTi

1The first author is highly thankful to the funding agency DST-INSPIRE for their financial support.

n

If p(z) = 0 for z € D- then it was conjectured by Erdos and latter proved by Lax [9] that

n

maxli/fz)! < — max biz) I, where as, if p(z) = 0 for z € D+, then Turan [11] proved:

n

maxln'iz)! > — max biz) I.

Li, Mohapatra and Rodriguez [10] obtained Bernstein-type inequalities for rational functions r € Rn with prescribed poles a\,a2, ...,an replacing zn by Blashke product B(z). Among other things they proved the following results for rational functions with prescribed poles.

Theorem 2. If r € Rn has n zeros all lie in

Ti U D+, then for

z € Ti, we have

\r'(z)\<-2\B'(z)\\r(z)\.

The result is sharp and equality holds for r(z) = aB(z) + b, with |a| = |b| = 1.

As a refinement of Theorem 2, Aziz and Shah [2] proved the following:

Theorem 3. Let r € Rn be such that all the zeros of r(z) lie in T1 U D+. If t1,t2,tn are the zeros of B(z) + A and s1, s2,..., sn are the zeros of B(z) — A, A € T1, then for z € T1

\r'(z)|<^M{( max \r{tj)\)2 + { max |^()|)2\. (1.1)

2 1<j <n 1<j<n J

In this paper we prove some results which infact strengthen certain known inequalities for rational functions with prescribed poles and inturn produce refinements of some known polynomial inequalities. We first prove the following generalization as well as a refinement of a result due to Wali and Shah [12].

Theorem 4. Let

2. Main results

rw=44 6J,

w(z)

where

p(z) = z^ao + ajzj^j

j=1

is a polynomial of degree m, having all zeros in Tk U D+, k > 1 except an s-fold zero at the origin. If t1,t2,...,tn are the zeros of B(z) + A and s1,s2, ...,sn are the zeros of B(z) — A,A € T1} then for z € T1

\r'(z)\ < j ( max |r(ij)l)2 + ( max |r(Sj)l)2

—4

k I |ao| — km slam-s|\ sk 2m — n(1 + k)

2 ^ 1/2

,1 + k^ |ao| + km~slam-sIJ 1 + k 2(1 + k)

If we take k = 1 and m = n, in Theorem 4, we get the following:

IK*)|2y

\B'(z) 1/

Corollary 1. Let

where

rw=44 6J,

w(z)

p(z) = zs (ao + ^ aj zj j=i

is a polynomial of degree n, having all zeros in T U D+ except a zero of multiplicity s at origin. If ¿i, t2, are the zeros of B(z) + A and si,s2, ...,sn are the zeros of B(z) — A, A € Ti, then z € Ti

|r'(z)| <

\B'{z) |f,

\i max V i<j<

max |r(tj)|)2 + ( max |r(sj)|)2 — 2

i < j <n

I<3-01 - K-g|\ \rjz) I2 _ I<3,o| + \an-s\) \B'(z)\ S

i/2

• (2.1)

On comparing inequalities (1.1) and (2.1) and noting that |a0| > |an—s|, it is easy to see that for s = 0, Corollary 1 is an improvement of Theorem 3 which is a result due to Aziz and Shah [2].

n— s

Remark 1. For s = 0, k = 1 and m = n, Theorem 4 reduces to a result due to Wali and Shah [12, Theorem 1].

It is to be noted that in the paper of Wali and Shah [12] an advanced tool (Osserman's lemma) has been used for its proof. However, we here use a simple application of mathematical induction to prove a more general result from which the result of Wali and Shah follows as special case. If we take s = 0, m = n in Theorem 4, we have the following:

Corollary 2. Let

rw=44 6J,

w(z)

where

n

p(z) = (ao + ^ aj zj) j=i

is a polynomial of degree n, having all zeros in Tk U D+, k > 1. If ti,t2, ...,tn are the zeros of B(z) + A and si, s2,..., sn are the zeros of B(z) — A, A € Ti, then for z € Ti

\r'(z)\ < maxJr^OI)' + ( maxJr(Sj)l)'

, 2 , / i / M\2

—4

n(k-1) k f\a0\-kn\an\\\r(z)\2

i/2

2(k + 1) k + 1V|ao| + kn|an|y |B'(z)|_ If we consider that r(z) has a pole of order n at z = a, then

, ^ P(z)

w (z — a)n

where p(z) is a polynomial of degree m. Therefore, we have

p(z) A (n — m)p(z) + Dap(z)

r'(z) =

(z — a)n I (z — a)n+i

where for any a € C, Dap(z) denotes the polar derivative of the polynomial p(z). Also

B (z) =

1 — az

z — a

w*(z) w(z)

with B(z) ^ zn as a ^ ro, and

B '(z) =

n{\a\'2 - 1) / 1 - az (z — a)2 1 ¿-a

n- 1

Further for z € T1,

|B'(z)| =

n(\a\2 - 1) | z — a| 2

Using these observations with m = n in Theorem 4 and letting |a| ^ to, we get the following: Corollary 3. Let p € Pn be such that all the zeros of

p(z) = zs ^ao + ^ ajzj)

j=i

lie in Tk U D+ except an s-fold zero at the origin. If t1,t2, ■■■,tn are the zeros of zn + A and s1, s2,sn are the zeros of zn — A, A € T1, then for z € T1

n ( 2 2

b'(*)| < öi (max b(ij)l) + ( max \p(sj)\)

2 1 1<j<n 1<j<n

—4

k f\ao\ — kn s|an_s|\ sk n(l — k) lTk V|oo| + kn-s\an-s\) ~ 1 + k ~ 2(1 + k)_

|p(z)|2

n

1/2

By taking k = 1 in Corollary 3, we get the following:

Corollary 4. Let p € Pn be such that all the zeros of

n— s

p(z) = zs(^ao + ^ ajzj^j

j=i

lie in T1 U D+ except an s-fold zero at the origin. If t1,t2,...,tn are the zeros of zn + A and s1, s2,sn are the zeros of zn — A, A € T1, then for z € T1

\r'(z)\ < ( max \p(tj)\)2 + ( max b(Sj)l)2 - 2

|fl'o| ~ \p(z)\' |a0| + \an-s\J n

—s

1/2

Taking s = 0, and noting that |a0| > |an_s|, it can easily be seen that Corollary 4 is an improvement

of a result due to Aziz [1, Theorem 4]. We next prove the following:

Theorem 5. Let r € Rn be such that all zeros of r(z) lie in Tk U D_, k < 1 with an s-fold zero at the origin, then for some y with |yI < 1 and for any z € T1

zr'{z) + 1-{\B'{z)\ +

2ks + n(l -k)-1 + k

inf |r(z)|.

z€Tk

n

n— s

By taking s = 0, k = 1, Theorem 5 reduces to the result due to Hans et al. [8, Theorem 1].

Again substituting for r(z),r'(z) and |B'(z)| the values as in Corollary 3 and letting |a| ^ to, we get the next property from Theorem 5.

Corollary 5. Let p € Pn be such that all the zeros of a polynomial p(z) lie in Tk U D— except an s-fold zero at the origin, then for some 7 with |y| < 1 and for any z € Ti

. 7 ( 2ks + n(1 - k) , , ,

zp(z) + j U +-Y+k— }p{z)

>

min |p(z)|.

zeTk

(2.2)

Remark 2. For s = 0, k = 1, (2.2) reduces to a result due to Dewan and Hans [6, Theorem 1].

3. Lemmas

For the proof of these theorems we need the following lemmas.

Lemma 1. If

Then for z € T

U -

*<«> = n 'z

• 1 z - aj

Re

,zw'(z) x n — |B'(z)|

w(z) 2

The above lemma is due to Aziz and Zargar [3].

Lemma 2. If (xj)°=i be a sequence of real numbers such that Xj > 1, j € N. Then

yl -O 1 li; 1

+ - 1 • li; ,.r?

for all n € N.

The proof of Lemma 2 is a simple consequence of the principle of mathematical induction.

Lemma 3. Suppose r € Rn and if ti,t2 ,...,tn are the zeros of B (z) + A and si,s2,...,sn are the zeros of B(z) — A, A € Ti, then for z € Ti

№)|2 + |r*'(z)|2 < Wtjtf + i max |r(Sj)l)'

2 i < j < n i< j< n

The above lemma is due to Aziz and Shah [2].

Lemma 4. Let r € Rn be such that all zeros of r(z) lie in Tk U D—, k < 1 with s-fold zeros at the origin, then for z € Ti

\zr\z)\>-2^\B\z)\ + T^(2ks + n(l-k))yr(z)\.

The above lemma follows from a result due to Akhter et al. [4]. Next lemma is due to Li, Mohapatra and Rodgriguez [10].

Lemma 5. If A and B are two complex numbers, then

(i) if |A| > |B| and B = 0, then A = vB for some complex number v with |v| < 1;

(ii) conversely, if A = vB for some complex number v, with |v| < 1, then |A| > |B|.

4. Proofs of Theorems

Proof of Theorem 2. Since

where

This implies

Equivalently, we get

r(z) =

zsh(z) w(z) :

h(z) = a0 + E ajzj.

j=1

zr'(z) zh'(z) zw'(z)

r(z)

h(z) w(z)

Re =s + Re - Re

r(z)

w(z)

Let z1, z2,..., zm—s be the zeros of h(z), such that |zj| > k > 1. In particular for z € T1 , we get by using Lemma 1.

Re

zr'(z) r(z)

= s

+ Re E

j=1

z — z.

- Re

zw' (z) w(z)

< s + E

j=1

= s +

m — s

1 + k

+ E

j=1

1

— Re

1

zw'(z) w(z)

1 + |zj | 1 + k

n — |B '(z)|

m—s

= s + TTk +

m—s < S + -r—r +

k

E

k — |zj|

1 + k k + |zj|k

m—s , | | k — |zj |

n—

1 + k l + k^k + |z7-| j=1

n-TOiy

Now using Lemma 2 with |zj |/k > 1, we get

Re

zr'(z) r(z)

<s+

m—s 1 + k

i + Hi + n-Lri^lAv V

'n — |B '(z)|

m—s

= S + ——r +

k / km

1 + k 1 + k\km-s |am-s| + |ao|

|am_s| - |a0|

2

2 V Ujl 1+k 1 + k 1+k V K| + km-8\Om-s\

Now

therefore using the fact that

r*{z) =B{z)r(l/z), zB '(z)

B (z)

= |B '(z)|,

(4.3)

z

m— s

1

j

2

2

(see also [10]) we get for any z € T1

|r*' (z)| = ||B'(z)|r(z) — zr'

This implies for z € T1

zr*' (z) 2 ||

r(z) |

|B'(z)| —

Now using (4.3) in (4.4), we get

zr'(z)

r(z)

zr* (z)

= |B'(z)|2 +

(4.4)

r(z)

>|B '(z)|2 +

zr'(z)

— |B '(z)|(jB '(z)| + This gives for z € T1

2m-n(l + k) 2 sk

r(z)

2k (lao |— km—s | am—s |

1 + k

1 + k 1 + k\ |ao | + km—s l am—s |

1 ( jl -1' ( jl 11 + ^ Vlao| + km~s\am-s\

1 + k

2m — n(1 + k)\ 1 ' jl B

1+k

This implies

21 '( . / ( la,ol ~ km~s\am-s\ \ _ 2sk_ _ 2m - 77,(1 + k) \

17 Uj| + 11 + k V |aoI + km~s\am-s\ J 1 + k

1 + k

/

|B '(z)||r(z)|2

<|r'(z)|2 + |r*' (z)|2

Using Lemma 3, we get

2\r'{z)\2 + [ ^ (\ao\~km- _ 2sk_ _ 2m-»(l + k))| ■ s

\ 1 + k \ |a,o| + km~s\am-s\ J 1 + k 1 + k

<

1 + k

^{(^JrfeJD' + i^JrC,)!)5

On simplification, it follows that

\r'(z)\ < |r(ij)l)2 + (,max |r(Sj)l)

" k

—4

2 1<j<n

|ao| — km—s l am—s |\ sk

,1 + k V|ao| + km_sIam_sI

This completely proves Theorem 2.

1<j<n

2m — n(1 + k)

1 + k

2(1 + k)

\r(z) I2) 1/2 \B'(z)\i ■

□

2

2

2

2

2

Proof of Theorem 3. Suppose r(z) has a zero on Tk, then

m = inf |r(z)| = 0

z€Tk

and the result holds trivally. We assume all the zeros of r(z) lie in D_,k < 1 with an s-fold zero at the origin. So that m > 0 and for z € D_, |r(z)| > m.

Since |B(z)| < 1 for z € T UD_ (see [7, p. 40]), therefore |B(z)| < 1 for z € Tk, k < 1 . Hence it follows by Rouche's theorem that for some 5 with |5| < 1,

F (z) = r (z) — 5mB(z)

has all zeros in D- , k < 1. Applying Lemma 4 to F(z), we get for z € Ti

That is for z € Ti

zr'(z) — ÄmzB'(z)

> l{2k* T+k~ k) + №)l\ - SmB(z) I

Since F (z) =0 in Tk U D+, therefore for any complex number 7 with |y | < 1, we have from (i) of Lemma 5,

T(z) = zr'(z) - ömzB'(z) + l{2kS^fk) k) + (r(z) - 5mB(z)) / 0.

This gives for z € Ti

t(z)=+|{2fcsv+rfc)+|i3'w|}rw

—¿m

= 0.

Now using (ii) part of Lemma 5, we get for < 1, 171 < 1 and k < 1

'. . 7 f2ks + n(1 — k) ,.) . .

zr'(z) + ^ j-^-+ \B'(z)\jr(z)

>m

Equivalently for z € T1 , we have

'. . 7 f2ks + n(1 — k) ,.) . .

zr'(z) + ^ j-^-^ + \B'(z)\jr(z)

>

l + + )\B>(z)\ +

n( 1 + k 7 / '2ks + ??.(1 - k)

2 V 1 + fc

inf |r(z)|.

zeTk

This completely proves Theorem 3.

□

Acknowledgements

The authors are highly grateful to the referee for his/her useful suggestions.

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