INEQUALITIES FOR MEROMORPHIC FUNCTIONS WITH PRESCRIBED POLES Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 16. № 1 (2024). С. 126-136.

INEQUALITIES FOR MEROMORPHIC FUNCTIONS WITH PRESCRIBED POLES

M.Y. MIR, W.M. SHAH, S.L. WALI

Abstract. The extremal problems for functions of complex variables, as well as approaches for obtaining classical inequalities on the base of various methods of the geometric function theory, are known for various norms and for many classes of functions such as rational functions with various constraints and for various domains in the complex plane. It is important to mention that different types of Bernstein-tvpe inequalities appeared in the literature in more generalized forms in which the underlying polynomial was replaced by a more general class of functions. One such generalization is the passage from polynomials to rational functions. In this paper, we prove some inequalities for meromorphic functions with prescribed poles and restricted zeros. These results not only generalize some Bernstein-tvpe inequalities for rational functions, but also improve and generalize some known polynomial inequalities. These inequalities have their own importance in the approximation theory.

Keywords: polynomials, Blaschke product, inequalities, rational functions.

Mathematics Subject Classification: (2020) 30A10, 30C10, 30C15

1. Introduction

n

Let Vn denote the class of all complex polynomials P(z) := ^ Cjzj of degree at most n.

j=o

Assume that D- represents the set of all points which lie inside Tk := {z : \z| = k} and D+ is the set of the points which lie outside Tk. Concerning the estimate of \P'(z)\ in terms of \P(z)\ for z G Ti, Bernstein [1] proved the following that if P G Vn, then

max\P'(z)\ ^ nmax\P(z)\.

zeT-i zeT-i

The result is sharp and the equality holds for the polynomials of the form P(z) = azn,a = 0. This inequality can be sharpened under additional conditions on the zeros of P(z). In fact, if P(z) = 0 in D-, then

max\P'(z)\ max\P(z)\, (1.1)

zeTi \ v 2 zeTi \ v v '

whereas if P(z) = 0 in D+, then (1.1) can be replaced by

max\P'(z)\ max\P(z)\. (1.2)

zeTi \ w\ 2 zeTi 1 y n v ;

Both these inequalities are sharp and equality in each case holds for the polynomials of the form P(z) = azn + b, where \a\ = \b\.

M.Y. Mir, W.M. Shah, S.L. Wali, Inequalities for meromorphic functions with prescribed poles.

© Mir M.Y., Shah W.M., Wali S.L. 2024.

The first author is highly thankful to the funding agency DST-INSPIRE for their financial support, IF: 190129.

Submitted January 10, 2023.

Inequality (1,1) was conjectured by Erdos and latter it was verified by Lax [2], whereas inequality (1,2) is due to Turan [3]. Both these inequalities were generalized by Malik [4] as

follows: if P(z) is a polynomial of degree n, which does not vanish in D-, where k ^ 1, then

n

max lP'(z)l ^ -- max |P(z)| (1.3)

M=i 1 W| 1 + k M=i | W| v ;

and if does not vanish in D+, k ^ 1, then

n

max lP'(z)l ^ -- max |P(z)|. (1.4)

zen 1 v J1 1 + k zeTi v 1

These inequalities were refined and generalized by various authors (for the references see [5], [6], [7]) for the operators besides the ordinary derivative and in some cases underlying polynomials were replaced by a more general class of functions.

For a polynomial P(z) of degree at most n, the polar derivative with respect to a point a E C, denoted by DaP(z), is defined as

DaP(z) := nP(z) + (a — z)P'(z).

Here DaP(z) is a polynomial of degree at most n — 1 and it generalizes the ordinary derivative in the sense that

lim ^ = P{;).

H^^ a — z

Aziz [8] extended inequality (1,1) to the polar derivative of a polynomial and proved that If P(z) is a polynomial of degree n, which does not vanish in D-, then for every real or complex number a with |a| ^ 1 and for z E T U D-

max №(¿)| ^ —(|azra_1| + 1)m^x |P^

In the same paper Aziz proved that if P(z) is a polynomial of degree n, which does not vanish in D- ,k ^ 1, then for every real or complex number a with | a | ^ 1,

max lDaP(z)\ ^ nl ] max \P(z)\. (1.5)

zETi " K J> 1 1 + H zETi V '

Shah [9] extended inequality (1,2) to the polar derivative and under the same assumption observed for every a E C, with |a| ^ 1 that

v

^D^P(z^ > 2(|«| — 1)|P(*)|, for * E Tx.

These results were further extended and generalized in various ways by various authors (for references see [10], [11] ),

E, P. Bose proposed to obtain Bernstein-tvpe inequalities for the rational functions instead of polynomials and accordingly over the past few decades, many inequalities for rational functions were established and also used in rational approximation theory. In particular, during the last few decades Bernstein-tvpe inequalities for polynomials were extended to a class of rational functions where

f P(z) " \

= 'Rn(al,... ,an) := j : P E Vn,w(z) = - aj) j

with poles Be TZn, where

with

, an and with finite limit at infinity. We observe that the Blasehke product

B(z) :=

n (V^)

w*(z) w(z) '

w*(z) = znw(1) = Y[(l - a]z)

Z 3 = 1

and satisfying \B(z)\ = 1 for z G T\. Through out this paper we assume that all poles a\,a2,... ,anY\e in D+.

Li, Mohapatra and Rodriguez [12] obtained Bernstein-tvpe inequalities for rational functions r G 'Rri with prescribed poles a\,a2,..., an replacing zn by B(z). In particular, they proved that if all the zeros of r G 'Rri lie in Ti U D+, then for z G Ti

1

\r'(Z)\ ^\B'(z)\\r(z)\.

(1.6)

2

The result is sharp and equality holds for the rational function

r(z) = aB(z) + b, \a\ = \b\ = 1.

In the same paper they proved that if all the zeros of a rational function r G fcnHe in Ti U D-, then for z G Ti

y(Z)\ > 1 \B'(z)\\r(z)\.

The result is sharp and equality holds for the rational function

r(z) = aB(z) + b with \a\ = \6\ = 1.

These results were further improved and generalized in various ways from time to time, see [13], [14], [5]. In this paper we prove some results which generalize the known inequalities for rational functions and thereby deduce generalizations of the known estimates for the maximum modulus of the polar derivative as well as the derivative of a polynomial on the disk.

2. Main results

We first prove the following comparison inequality, which gives a rational analogue of a result due to Dewan et al. [15].

Theorem 2.1. If r G 'Rri has all zero s in Tk U D- ,k ^ 1, then for every ¡3 with \ ^ 1, and for z G T\, we have

,, s fiis^u m 2m - n(1 + k) . . . zr'(z) + 2l \B'(z)\ +-^-)r(z)

>

^ + +"2m - "(1+k)

2T v"" '2 1 + k

Here m ^ n denotes the number of zeros of r(z).

inf \r(z)\.

zeTk

Remark 2.1. For k = 1 and m = n, Theorem 2.1 reduces to a result due to Hans, et al. [16, Thm. 1].

If in Theorem 2,1 we assume that r(z) has a pole of order n at z = a, \a\ ^ 1, then we can write

P (z)

so that

Also we have in this ease

r(z) r'(z) =

" (z — a)n'

-DgP (Z) (z — a)n+l'

B(z)

-i^ /l — az\ /l — az\ -ly*- \ z — a I \ z — a I

This gives

B '(z)

n(1 — az)n-1(\a\2 — 1) (z — a)n+1

Using these facts, we immediately get from Theorem 2,1 for z E T\

(z — a)n+1 2

—zDaP (z) + ^ 2m — n(1 + k) +

1 + k

n(1 — az)n-1(\a\2 — 1)

(■+s)

n(1 — äz )n-1(\a\2 — 1)

(z — a)n+1

+

(z — a)n+1 s 2m — n(1 + k)

)

P( )

1 + k

(z — a)'

\P (z)\

zeTk \z — a\n

mm

This gives

—* DaP (z) + S( 2m — n(1 + fc) + z — a 2 ^ 1 + k

n(\a\2 — 1)

0+s)

n(\a\2 — 1)

( — a)2

+

( — a)2

s 2m — n(1 + k)

)

P( )

2

1 + k

\z — a\n min

\ P( )\

ziTk \z — a\

Therefore from Theorem 2,1, we have the following corollary.

Corollary 2.1. If all zeros of a polynomial P(z) lie in Tk UDfc , then for every a with |a| ^ 1 and ¡with IP| ^ 1, we have for z E T

—zDaP(z) S ( 2m — n(1 + k)

z — a + 2 y 1 + k +

n(\a\2 — 1)

('+s)

n(\a\2 — 1)

( — a)2

+

( — a)2

s 2 m — n(1 + )

)

P( )

2

1 + fc

\z — a\n min

\ P( )\

ziTk \z — a\n'

If in Corollary 2,1 we let |a| ^ <x>, then we have the following statement.

Corollary 2.2. If all zeros of a polynomial P(z) lie in Tk UD-, then for 3 E C with we have for z E T\

mfi

z p'(Z)+m~kP (z)

>

n +

1 + k

min IP(z)\.

ziTk

Bv taking m = n,k = 1 in Corollary 2,2, we get a result due to Dewan et al, [15]. We next prove the following generalization of a result due to Li [13],

2

n

Theorem 2.2. Let r,s G Kn and assume that all zeros of s(z) lie in Tk U Dk ,k ^ 1. If

| r(z) | ^ | s(z) | Jor zgTu (2.1)

then for every real or complex number ft with ^^ ^ 1 and for z G T

,, , [/,„// m 2m — n(1 + k)\ . , z r'(z) + 2 i |B'(z^ +--r(z)

<

,, , [/,„// m 2m— n(1 + k)\ . , * s'(*) + 2 i |B'(z^ +-^-j s(z)

where m ^ n are the zeros of s(z).

Remark 2.2. A result recently proved by Mir [14, Thm, 3] follows from Theorem 2.1 once we take m = n. Also for k = 1 and m = n, Theorem 2.2 reduces to a result due to Hans, et al.\ 16, Thm. 2].

If s(z) = B(z)||r||, where \\r\\ := sup |r(z)^ then from Theorem 2.2 we get the following

zeTi

corollary.

Corollary 2.3. Let r G Kn, then for every real or complex [ with ^^ ^ 1

\r(z)

,, , [(inl, M 2m — n(1 + k)\

zr(z)+4|B(z)| + 1+k /

(i + ^B w + f" — n(1 + k)

1 + k

sup | ( )|.

zeTi

We also prove the following theorem.

Theorem 2.3. If r G Kn, then for [ G C with ^| ^ 1,

n \ m 2m — n(1 + k)\ . . z r'(z) + H ^' (z^ +-^-r(z)

+

<

| B (

| B ( | B ( | B (

zr*'(z) + (Z)^^ —1fk+kA r*(z)

1 + (z) +

\B> (z)\ + 2m —Ifk

(2.2)

for z G T\, where r*(z) = B(z)r(1

Remark 2.3. For k = 1 and m = n, Theorem 2.3 reduces to a result due to Hans, et al. [16, Thm. 3].

We consider r G Kn having all its zeros in Tk U D+,k ^ 1 and let m' = inf |r(z)^ then

zeTk

m' ^ |r(z^ for all z G Tk. By Rouehe's theorem for some complex 5 with |5| < 1 all the zeros of R(z) = r(z) — 5m' He in Tk U D+ and therefore all the zeros of

5 (z) = B (z)R{ 1) = r*(z) — Sm'B(z)

lie in Tk U D- . Therefore from Theorem 2,2 we get

(\ B' (

z r'(z) + ß-[ \B'(z)\ + 2m — n(1+fcM (r(z) — m)

1 + fc )

<

z(r*'(z) — SB'(z)m') + 2 (\B'(z)\ + 2m 1+(1fc +fc))

(r *(z) — ¿B (z)m')

This in particular gives for z E Ti

2 m n

(1 + )

, n sr ni 2m — n(1 + fc)\ r(z) — m' i \ B'(z) \ +--^

<

zr*' (z) + s ( \B' (z)\ + 2m in(,+ fcM — ^B' (z)m'

1 + fc /

sTns , n„ M 2m — n(1 + fc)\ — £ SB (z)m \ B' (z) \ +--—^-- I

2

1 + fc /

Now dividing both sides bv B(z) and using the fact that for z E Ti

I B (z) | = 1, ^ = | B' (z) |,

we get

2 m n

zr'(z)+2(ib'1+fc )

2 m n

— £* m (b (z)\ + 1 + fc

(1 + fc) \

*'/ n m 2m — n(1 + fc)\

z r* (z) + 2 i \B'(z)\ + v

^ s 2m — n(1 + fc)\ /T

ft*— )r*( -->— (\B' w\ (1 + 2) + 2'"'

Choosing suitably the argument of S in right hand side of above inequality and letting ^ 1, we get

s (,B'(J)\ + 2m -n(1 + fc)^

z r* (z) + ^ \ B' (z) \ +

2

1 + fc /

*( )

(2.3)

1 + 0\B' W\ + f2^^^^

B (

— B' ( -) TLi^m.

m .

Now combining (2.2) and (2.3), we have the following corollary.

Corollary 2.4. Let r G Kn has all zeros in Tk U D+, k ^ 1, then for [ G C with |[| ^ 1, we have for z G T1

// \ [/m// m 2m — n(1 + k)\ . . z r'(z) + 2 i ^' (z^ +--r(z)

1

< -2

+

| B ( ) | +

2 m — n(1 + ) 1 + k

| |

— I (| b (z}¡ + 2m-**l||ní

By taking m = n,k = 1 and [ = 0, Corollary 2,4 reduces to a result due to Aziz and Shah [19, Thm. 3].

If we assume that r(z) has a pole of order n at z = a, ^ ^ 1, then

P (z)

so that

( ) = r'(z) =

(z — a)n' —DaP (z)

(z — a)n+1'

Using the same procedure as in Corollary 2.1, we get from Corollary 2.4 for z G T1 the following statement.

Corollary 2.5. Let P G Vn has all zeros in Tk U D+, k ^ 1, then for any a with |a| ^ 1 and [with H | ^ 1, we have

—zDaP (z) + [ 2m — n(1 + k) +

(z — a)n+1 2

1+

n(1 — azy^M2 — 1)

(z — a)n+1

P( )

(z — a)

^ 1

+

[

)

n(1 — az)nk1 (|a|2 — 1)

[ (

2 I

(z — a)n+1 n(1 — az T^M2 — 1)

+

[ 2m — n(1 + k)

2

1 + k

1

2

\[

2

(z — a)n+1 n(1 — az^^a2 — 1)

2m — n(1 + k)

+-m-I I max

1 + k / / zeTi

P( )

(z — a)

)

(z — a)n+1 n(1 — az^^2 — 1)

+

[ 2m — n(1 + k)

2

1 + k

(z — a)n+1

2m — n(1 + k) +--^-- I I min

1 + k / / zeTk

- | | mil

e T

P( )

(z — a)

In Corollary 2.5, if we make a ^ <x and on simplification, we get the following corollary.

Corollary 2.6. Let P GVn has all zeros in Tk U D+, k ^ 1, then for [ G C with m ^ 1 and G T1

m[ 1 + k

P( )

^ 1 ( (n + \(m[ + n|[|)^ max ^(z^

2

e Ti

1

— ( n + -¿(m[ — nH|) I min ^(z^

2 j Z eTk

0 mz| P W|)

By taking 3 = 0, m = n and k = 1, Corollary 2,6 reduces to a result due to Aziz and Dawood [17, Thm, 2] .

3. Lemmas

For the proof of these results we need the following lemmas. The first lemma is due to Li, Mohapatra and Eodgriguez [12],

Lemma 3.1. Let A and B = 0 he two complex numbers, then |A| £ |B| if and only if A = vB for any complex number v with, |i>| < 1 .

The next lemma which we also require is due to Aziz and Shah [18],

Lemma 3.2. Suppose r E be such that all the zeros of r(z) lie in Tk U D-, k ^ 1, then E Ti ,

№)| £ UIB'(z)I + 2m—Ir(z)I, (3.1)

where m is the number of zeros of r, with m ^ n.

4. Proofs of Theorems Proof of Theorem 2.1. If r(z) has any zero on Tk, then inf | r(z)| = 0 and the statement becomes trivial. Now we assume that r(z) has all zeros in D-, If m' = inf \r(z)\, then m' > 0

v y k zeTk 1 v y 1

and |r(z)| ^ m' or z G Tk. Also IB(z)| ^ 1 for z G Ti (see [5, p.40]), therefore |B(z)| ^ 1 for z G Tk,k ^ 1, Hence, for each complex ó with |¿| < 1, by the Rouché theorem we see that

F (z) = r(z) — óm'B(z)

has all zeros in D-. Applying Lemma 3.2 to the rational function F G we get

, _i, . i m 2m — n(1 + k) \ . ..

\zF (z)| ^ ^\B'(z)| +-1-+-k-n \F(z)|.

Since F(z) = r(z) — 8m'B(z) = 0 in Tk U D+, therefore for each complex number / such ^ 1 by Lemma 3,1 we get:

T(z) =(/(z) - Sm'B(z)) + |B(;)| + 2m 1+(1k +k)j (r(z) - im'B(;))

=-'(;) + |(|B(,)\ + 2m -1+(k +k))r(.-) - JzB' (,) + 2 B(z) (\B (,)| +2m -1+k +k) = 0

in Tk U D+. In particular, for |¿| < 1 we obtain

// N 2m-n(1 + k)\ . . Í ß\t. zr'(z) + 2 I \B'(z)\ +-1-+-k-nr(z) > ^1 + 2J \B'

i+ßw (z)\+ß 2m -n(i+k)

inf |r(z)I.

zeTk

2 1 + k

This completes the proof of Theorem 2,1, □

Proof of Theorem 2.2. Since s(z) has all to zeros in Tk U D-, k ^ 1 and \r(z)\ < |s(z)| for z G T]_, by Rouehe theorem for |X| < 1 we see that Ar(z) + s(z) has the same number of zeros in Tk U D- as s(z). Hence, applying Lemma 3,2, for z G T1 we get

|z(Ar'(z) + s'(z))\ > \B'(z)\ + KAr(z) + s(z))l

Since (Ar(z) + s(z)) has no zero in Tk U D+, by using Lemma 3,1 for every real or complex [ with \ [\ ^ 1 we get

z(Ar'(z) + s'(z)) + 2 (B'(z)\ + 2m —1n(1k +k)^j (Ar(z) + s(z)) = 0.

This implies that

2m — n(1 + k)\ 2\ ^ (*)\ ' 1 + k )

A zr'(z) + - \B'(z)\ + 2m — "(1 + ") \r(z)

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

( f/ \ [ f,n„ M 2m — n(1 + k)\ . A [z s'(z) + ^ \B (z)\ + 1+(k ^j.

= —I* s'(z) + --(\B' {z)\ + 2m — n({+k))s(z) A | A| < 1 G T1

2 m n

( \ b' +2m+fc)U)l < U)+-L' +2m

This completes the proof of Theorem 2,2, □

Proof of Theorem 2.3. Let M = sup |r(z)\, then |r(z)\ ^ M for z G T1m Therefore, for > 1,

e Ti

the function

H (z) = r(z) — 7M

has no zero in D- . Hence, the function

G(z) = B(z)H (1) = r*(z) — 7B (z)M

has all its zeros in Tk U D-k. Applying Theorem 2.2 to the rational function G(z) which has m

n

H (,) + 2(\B' (z)\+ 2m ~lfk+kAH (z) « *ff M + 2(\B' (z)\ + 2m ^W)

This gives

//x [ / m^// n i 2m — n(1 + k) [ , . [/,„,/ m 2m — n(1 + k) . zr'(z) + -r^B'(~)| +-^--r(z) — 7-1 B(z)| +-^-I M

(\B(z)\ + 2m )

, [ *f wn'f w , 2m — n(1 + k)[ , , .

zr (z) + ^r (z)\B (z)\ +--2r (z) (4A)

— y(zB' (z) + - U (znB (z) + 2m ~lfk +k) Biz)) M

Now choosing suitably the argument of 7, which is possible by Corollary 2,3, from inequality (4,1) by using triangle inequality in the left hand side we find

// \ ß / \I 2m - n(l + k) ß . . zr'(z) + |ф)|Б' (z)| +-i+k-2r(z)

<

zB (z) + ß

^zB'

( ^) + 2m -l+(lfc +fc)B(z)

— iti

M

IB' (z)l +

2 m - n(l + )

l + fc

M

— ItI

ß ß 2 m - n(l + ) (z) + 2|B'(z)|r*(z) + 2-i+fc-r*(z)

Letting |tI —^ l, we Set

// \ ß / мп// \ I 2m — n(l + fc) ß . . zr'(z) + 2 r(z)|B' (z)| +-l+fc-^ r(z)

+

*'( \ I ßlRV M * / N i ß 2m — n(l+fc) */ ч

zr (z) + 2|B U)|r (Z) + 2-l+fc-Г (z)

IB' (z)I +

2m — n(l + fc)

l+

+

I b( ,)| + f(|B'(,)| + 2m —I IM.

!)l)

for z G Ti. This allows to conclude that, for z G Тъ

ß 2 m — n(l + ) zr'(z) + |i |B'(z)| +-l+fc->- )r(z)

+

<

*'( \ I ß(\U'i M I ß 2m — n(l + k)\ *( ч

zr (z) + 2l |B (z)I +2-TTk- 'r (z)

IB' (z)I +

2 m — n(l + )

l + fc

+

(l+2 )ib W|+f2^—l^

^Jm.

This completes the proof of Theorem 2,3,

Acknowledgements The authors are deeply thankful to the referee for the valuable remarks.

□

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M.Y. Mir,

Department of Mathematics, Central University of Kashmir, Tulmullah Ganderbal, 191131 E-mail: myousf Scukashmir. ac . in

W.M. Shah,

Department of Mathematics, Central University of Kashmir, Tulmullah Ganderbal, 191131 E-mail: waliScukashmir. ac . in

S.L. Wali,

Department of Mathematics, Central University of Kashmir, Tulmullah Ganderbal, 191131 E-mail: shahlwSyahoo. co . in