GENERALIZATIONS OF CERTAIN WELL-KNOWN INEQUALITIES FOR RATIONAL FUNCTIONS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 12 (30), No 1, 2023, pp. 25-33

DOI: 10.15393/j3.art.2023.11810

25

UDC 517.53

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

GENERALIZATIONS OF CERTAIN WELL-KNOWN INEQUALITIES FOR RATIONAL FUNCTIONS

Abstract. In this paper we generalize and refine a result of Wali and Shah concerning the estimate of the derivative of the maximum modulus of rational functions with prescribed poles and restricted zeros. The obtained results generalize and sharpen some well-known inequalities for the derivative of rational functions besides the refinement of some polynomial inequalities.

Keywords: polynomial, rational function, s-fold zeros, Bernsteintype inequality

2020 Mathematical Subject Classification: 30A10, 30C10, 26D10

1. Introduction. For each positive integer n, let Vn denote the linear

n

space of all polynomials p{z) := a,jzj of degree at most n over the field

j-0

C of complex numbers. If p e Vn and p' is its derivative, then we have the following:

Let p e Vn and suppose \p{z)\ ^ M on \z\ = 1. Then, for \z\ = 1:

\p' {z)\ ^ nM. (1)

Inequality (1) is referred to as Bernsteins's inequality. Riesz [11] (see also [12, p. 557]) was probably the first to formulate this inequality in the present form. However, in a stronger version it was first proved by Smirnoff [13] (see also [7]). Equality holds in (20) if and only if p has all its zeros at the origin. However, if we impose restrictions on the location of zeros of p, then Erdos conjectured and latter Lax [8] proved the following: If p e Vn has all zeros in \z\ ^ 1, then

n

max \p'{zq\ ^ - max \p{z)\.

\z\ = l 2 \z\ = l

© Petrozavodsk State University, 2023

On the other hand, if p e Vn has all zeros in \z| ^ 1, then Turan [14] proved:

n

max \p'{z)\ ^ — max \p{z)\.

\z\ = l 2 \z\ = l

The above inequality of Turan was further refined by Dubinin [5], who obtained, under the same assumptions:

max \p'(z)\ ^ Un + H , \a°\) max \Ppz)\.

\z\-i 2 V \an\ + \anW \z\"i

Recently, Kompaneets et al. [7] considered the problem for polynomials with zeros outside a disk and obtained some results related to classical inequalities of Bernstein and Smirnoff. These inequalities were further generalized in another paper of Kompaneets et.al [6], where it is assumed that all but one zero of p(z) lie inside the disk. We write

( p(z) )

Rm,n = Mm,n(ai,«2, ...,®n) := ] —r^r: p e Vm\,m ^ n,

iw(z) )

where

n

w(z) " ri(z _ a3), \ai \> 1, j " 1, 2,...,n. j-i

Thus, MTO,n is the set of all rational functions with poles a1,a2,... ,an and with finite limit at 8. Throughout this paper, we shall assume that all poles a1,a2,... ,an lie in \z\ > 1. We also observe that the Blaschke product B(z) e Rn,n, where

B M - ff (—) " ^ ■

x | V z — a, / w(z) 3=1 J y '

with w*(z) = znw(I) = n(1 — ®jz), satisfying \B(z)\ = 1 for \z\ = 1 z 3=1

zB'jz) '

and zB(z) " \^'(z)\. Li, Mohapatra, and Rodriguez [9] proved the following results for a rational function r(z) e ^n,n with prescribed poles a1, a2,..., an, replacing zn by B(z):

Theorem 1. If r e Mn,n has all the n zeros in \z\ ^ 1, then, for \z\ = 1, we have:

\r'(z)\ ^ 2\B'(z)\Hz)l (2)

The result is sharp and the equality holds for r{z) = aB{z) + b, with

M = \b\ = 1.

In the same paper, they proved the following:

Theorem 2. If r p Kra,ra has all n zeros in \z\ ^ 1, then, for \z\ = 1, we have:

\r'(z)\ > 2\B\z)\\r(z)\. (3)

The result is sharp and the equality holds for r{z) = aB{z) + b, with

\a\ = \b\ = 1.

In this paper, we relax the condition that all zeros of a rational function r{z) lie in \z\ ^ 1 and prove the following results, more general than those proved by Wali and Shah [15]. 2. Main results.

Theorem 3. If v £ n has a zero of order s at z0 with \z°\ > 1, and the remaining m — s zeros are in \z\ ^ 1, then, for any \z\ = 1 :

max \r'{z)\ > M) *x

\z\ = 1 \ p q\ 21V1 + \ Z°\)

x (\B'{z)\ — {n — m + s)+ ^\ —\C°\) — j^r-} max\r(z)\. (4) v \Cm-s\ + \c°\/ 1 + \Z°\) \z\ = 1

For s = 0, (4) reduces to a result by Wali and Shah [15].

Corollary 1. If we assume r{z) has a pole of order n at z = a, then { )

rPz) " 7-r~, where p{z) is a polynomial of degree m. Then

{z — a)n

, p{z) V {n — m)p{z) + Dap {z)

v ' \(z-a\nJ

{z — a)nJ {z — a)n+1

where Dap{z) = mp{z) + {a — z)p'{z) is the polar derivative of p{z) with respect to the pole a.

Since B{z) = ^r = (^V, therefore, B'{z) = 1 (1.

' w(z) y z—a I ' ' \ / fz—a)2 y z—a j

Also for \z\ = 1, \B'{z)\ = a—1). Now for \z\ = 1 and m = n, from Theorem 1 we get:

Dap {z) > 11 — \ z°\y^ n{\a\2 — 1) — s+\ cn—s\ — \ c°\y

{z — a)n+1 2U1 + \z°\J V \z — a\2 \cn—s\ + \c°

2s

p(z)

1 + \zo\) (z — a) Now, letting \a| ^ 8, we get, from (5):

max w(Z)\ > 1 {()'(n — s + jin^^j^

\z\=^ 2 IV1 + \zo\J \ \cn-s\ + \co\

max \p(z)\.

—

1 + \zo\i \z\ = 1

Remark. If we put s = 0 in (6), we get the result by Dubinin [5].

Next, we obtain the following generalization of Theorem 3:

Theorem 4. If v e M.m n has zeros at z0 and z1 of order s and d with \z0\ > 1, \z1\ > 1, and remaining m — s — d zeros lie in \z\ ^ 1, then, for

W = 1:

x f\B'(z)\ — (n — m + s + d)+ \\Cm-s-d\\ \Co\)

^ \ cm-s-d \ + \ c0 \ '

— ~r—l—i — ^—"T—11 max \r(z) \. (7)

1 + \ Zo \ 1 + \ Z1 \ J \z\ = 1

For d = 0, Theorem 4 reduces to Theorem 1; for s = d = 0, (7) reduces to the result by Wali and Shah [15].

Again, if we consider r(z) with a pole of order n at z = a, then

r(z) = 7-r-. Now, using the same procedure as in Corollary 1 and

(z — a)n

letting \a\ ^ 8, we get:

Corollary 2. If p e Vn has zeros zo and z1 of order s and d, respectively, with \zo \ > 1, \z1 \ > 1, and remaining n — s — d zeros lie in \z \ ^ 1, then, for \ z \ = 1:

^ \ > 2 {(^ —- - fedrR)—

max \p(z) \. (8)

1 + \ Zo \ 1 + \ Z1 \) \z\ = 1

Using the similar technique, we also have:

Corollary 3. Let r(z) = (z — zo)S0(z — z1)S1 ...(z — zt)Sts(z) e Rm,n.

t

Let s(z) be a rational function of degree m — si = I (say), with each

i=o

t

\zi\ > 1,i = 0,1, 2,... ,t and 0 ^ XI s i ^ m — 1, and let the remaining I

i=o

zeros lie in \z \ ^ 1. Then, for \ z \ = 1, we have:

SH"«'* H f+H! r-( t+3)

\ c \ — \ Co \

\ Ci \ + \ Co \ 2 s o 2 S1___ 2 st |

x(\B' «\ —<"—™ + 0 + ^)

max ( ) .

II II ... ill -L-L-LCUA

1 + \Zo \ 1 + \Z1 \ 1 + \Zt \) \z\"1

For the proof of these theorems, we need following lemma. 3. Lemma and Proofs of Theorems.

Lemma 1. Suppose r e ETO,n has all zeros in \ z \ ^ 1; then, for \z \ = 1, we have:

\r'(z)\ * 2{ \B'(z)\ — (n — m) + }\r(z)\.

This lemma was proved by Wali and Shah [15].

Proof of Theorem 3. Since r(z) has a zero of order s at z = zo, therefore, r(z) = (z — zo)st(z), where t e ETO-s,n. This implies

r'(z) = (z — zo)st'(z) + s(z — Zo)s-1 t(z).

Hence,

\ r'(z)\ * \ (z — zo)st'(z)\ — 8\(z — zo)s-1t(z)\.

This implies

max \r'(z) \ * \ 1 — \zo\ \smax\t'(z) \ — s|1 + \Zo\ \S-1 max \t(z)\. (9) \z\"1 \z\"1 \z\"1

Now, using Lemma 1 for the rational function t(z), we have

\ t'(z)\ * 1 { \B'(Z)\ — (n — m + s) + ^^}\t(z)\.

Therefore, from (9) we get:

max |r'(z)\ > \b'(z)\_ pn _m + s) +

\z\=i 2 I

+ ^ — ^ ) miax\t(z)\-s|1 + |zcll^-1 mmax \t(z)\. (10)

|Cm-s\ + \C°\) \z\ = 1 \z\ = 1

Now, for \z\ = 1 we have:

^ " H—^^ > !Г+^°!|S^

This implies

max\t(z)\ ^ --:—— max\r(z)\.

M" \1 + \¿°\\s k\=i \ ( )\

Therefore, we get, from (10):

^ max \r(z)\.

1 + \Z°\) \z\ = 1

This completely proves Theorem 3. □

Proof of Theorem 4. Since r(z) has two zeros of order s and d at z° and z1, respectively, with \z°\ > 1, \z1\ > 1, and remaining m — s — d zeros lie in \z\ ^ 1, therefore: r(z) = (z — z°)s(z — z1)dt(z), where t e M.m-s-d,n has all zeros in \z\ ^ 1; then

r'(z) = (z — z°)s(z — z1)dt'(z) + s(z — z°)s-1(z — zx)dt (z) + + d(z — z°)s(z — Z1 )d-1 t(z).

This implies

max \r'(z)\ ^ max\(z — z°)s(z — z1)dt'(z)\ —

\z\-1 \z\-1

— max \ (z — z°)s-1(z — Z])d-1{s(z — z1) + d(z — z°)}t(z)\.

\z\"1

Also, for \z\ = 1:

\1 — \^H ^ \z — ^ ^ 1 + \^

and

\ 1 — \ Z1 \ \ ^ \ z — Z1 \ ^ 1 + \ Z1 \,

we have then:

max\r'(z)\ * \ 1 — \zo \ \* \ 1 — \^ \ \ d max\t'(z)\ —

— \ 1 + \Zo\ \S-1 \ 1 + \^ \\ d-1{s|1 + \^ \ \ + d\ 1 + \^\} max \t(z)\. (11)

\z\"1

Applying Lemma 1 to the rational function ( ), we have

\ t'(z)\ * 2{ \B'(z)\ — (n — m + s + d)+ ^^ ++ \C°\}\t(z)\. Therefore, from inequality (11) we get:

max \r'(z) \ * ^-^j1-^{ \B'(z) \ — (n — m + s + d) +

\z\"1 2 i

+ \ Cm--d\ — \Co \ } max \t(z) \ — \ Cm-s-d \ + \ Co \> \z\"1

— \ 1 + \Zo\ \S-1 \ 1 + \^ \ \ d-1{s|1 + \^ \ \ + d|1 + \Zo\ \ } max \t(z) \. (12)

\z\"1

Also,

( )

t(z) =

(z — zo)s(z — Z1)d' This implies:

max \t(z) \ = max \z\"1 \z\"1

, , max \r(z) \

r(z) 11 -

* \z\"1

(z — zo)*(z — Z1)d " \ 1 + \Zo \ \* \ 1 + \Z1 \ \ Therefore, we get from (12):

1 (71 -\ --o \V(1 -\

sx \ \* ^irrfei) ir+R) ( \ (z)\—(i—m+s+d)+

Cm-s-d \ — \ Co \ \ 2 s 2 d

+

\Cm-s-d \ + \ Co \y 1 + \ Zo \ 1 + \ Z1 \ ) \z\" This completes the proof of Theorem 4. □

max ( ) . J \z\"1

d

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

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

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[15] Wali S. L., Shah W. M. Applications of the schwarz lemma to inequalities for rational functions with prescribed poles. J. Anal., 2017, vol. 25, no. 1, pp. 43-53. DOI: https://doi.org/10.1007/s41478-016-0025-2

Received May 04, 2022. In revised form,, November 25, 2022. Accepted November 29, 2022. Published online Decrmber 9, 2022.

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

M. Y. Mir

E-mail: myousf@cukashmir.ac.in, W. M. Shah

E-mail: wali@cukashmir.ac.in S. L. Wali

E-mail: shahlw@yahoo.co.in