ON ZEROS OF POLYNOMIAL Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 11. № 1 (2019). С. 113-119.

ON ZEROS OF POLYNOMIAL

SUBHASIS DAS

Abstract. For a given polynomial

P (z) = zn + an-izn-1 + On-2Zn~2 +-----h aiz + ao

with real or complex coefficients, the Cauchv bound

< 1 + A, A = max |a7-1

0^j^n-1 J

does not reflect the fact that for A tending to zero, all the zeros of P (z) approach the origin 2 = 0. Moreover, Guggenheimer (1964) generalized the Cauchv bound by using a lacunarv type polynomial

p (z) = zn + an-p zn-p + an-p-izn-p-1 +-----h ai z + ao, 0 < p < n.

In this paper we obtain new results related with above facts. Our first result is the best possible. For the case as A tends to zero, it reflects the fact that all the zeros of P(z) approach the origin z = 0; it also sharpens the result obtained by Guggenheimer. The rest of the related results concern zero-free bounds giving some important corollaries. In many cases the new bounds are much better than other well-known bounds.

Keywords: zeroes, region, Cauchv bound, Lacunarv type polynomials. Mathematics Subject Classification: 30C15; 30C10; 26C10

1. Introduction and statement of results

Let

P (z) = zn + an-izn-1 + an-2Zn-2 +-----+ aiz + a0

be a polynomial of degree n. A classical result by Cauchv [6, Ch. VI, Sect. 27, Thm, 27.2] concerning the bounds for the moduli of the zeros of a polynomial can be stated as follows.

Theorem A. AH the zeros of P(z) lie in the disc

\z| < 1 + A, (1.1)

where

A = max la J .

Joyal, Labelle and Rahman [1] improved Cauchv bound (1.1) and proved the following theorem.

Theorem B. If B = max \ak\, then all the zeros of P (z) lie in the disc

0< k^^ffi—2

N« 2

i + k.-i| + \/(i -k.-il)2 + 4B

S. Das, On zeros of polynomial.

©S. Das 2019.

Поступила 30 августа 2017 г.

Datt and Govil [3] improved Cauchy bound (1,1) and obtained the following result. Theorem C. All the zeros of P (z) lie in the ring-shaped region

-N-^ w ^ i+a( i .

2(1 + Af-1 (1 + nAp '' V +

One more improvement of Cauehv bound (1,1) was made by Jain [8], who proved the following statement.

Theorem D. AH the zeros of P(z) with an-l = an-2 = 0 lie in the disc

|z| < 22 (1 + B)3 ,

except for B > 1, a| = B for some j, 0 ^ j ^ n — 3 and |a^| ^ a = 23 — 1 for all i = j, i E {0,1, 2,... ,n — 3}. In the latter case, all the zeros of P (z) lie in the disc

M < (1 + B)3.

Guggenheimer [5] generalized the Cauehv bound (1,1) by using a class of laeunarv type polynomial

p (z) = zn + an-pzn-p + an-p-izn-p-1 + • • • + alz + a0, 0 < p < n, and proved the following theorem,

( )

W < S,

where 5 > 1 is the only positive root of the equation

tp — tp-1 = Qn

and

Qn = max a |.

O^k^n—p

In this paper, we obtain three bounds of Cauehv type. The bound in Theorem 1,1 is best possible and sharpen of the Theorem E, Also, in many eases, the bound in Theorem 1,1 is better than some other known bounds. The bounds in Theorem 1,2 and Theorem 1,3 are zero free. More precisely, we prove

( )

n

M ^ <Wp,

where 50 E (1, 2) provided Q ^ 1, otherwise, 50 E (1, ro) is the greatest positive root of the equation

/ n \™+l / n \n ( n \™+l

qn+l(Q — qn(Q — qn-p+l(Q ') + Qn = 0. Remark 1.1. As Q ^ 0, all zeroes of p(z) approach the origin z = 0.

n

Remark 1.2. The hound 80Qp in Theorem 1.1 is the best possible and it is attained at the polynomial

p (z) = zn — Qn (zn-p + zn-p-1 + ••• + Z + 1) .

Remark 1.3. Theorem 1.1 is an improvement of Theorem E, which can be seen by observing that

„ \ra+l / „ \ra+l / „ \n / „ \n / „ \ n-p+l / „ \ "+1

5Q-(Q ^J — (5Q-^ (Q ^ — (¿Q-^J [Q ^ +Qn -_ sn+l — 8n — 6n-p+l (QP + Qn = Sn-p+l (8P — 8p-1 — Qn) +Qn = Qn > 0,

s

Phc. 1. Variation of as p varies from 1 to 99,

which implies

80 < SQ p, i.e. 80Qp <8.

Remark 1.4. In many cases, the Theorem 1.1 gives a better bound than those in 'previous results. In order to demonstrate this, we consider the polynomial

p (z) = z5 + a2z2 + a1z + a0,

with |a0| = 8, |ai| = 2, |a21 = 6 Here n=5, p=3 and Qn = 8. By Theorem 1.1, we obtain 80 = 1.17174 and all the zeros of p (z) lie in |z| ^ 2.3434^, whereas all the zeros of p (z) lie in the regions

< 9,

< 3.37,

< 8.989,

< 2.426,

< 2.3948,

< 8.999,

< 6.75,

by Theorem A by Theorem B by Theorem G by Theorem D by Theorem E by |4, Thm, 1|

.

Example 1. Theorem 1.1 gives an idea for finding the bound of zeros for a class of lacunary type of polynomials

{

n—p

K (Qn) = <j p (z) = zn + J2 aJ: 0<m<ax lak I ^ , 0 <P<n-

3=0

}

By Theorem 1.1, all the zeros of each polynomial of the class Ypn (Qn) always lie in the region

n

M ^ 80Q p . It is very difficult to find the value of 80 for each non-negative real value of Qn, as n is a fixed natural number and p varies from 0 < p < n. Here we consider an example by choosing Qn = 10 and n = 100 and we draw a picture for the variation of 8, when p varies from 1 to 99, see Figure 1. Once we determine the value of 80 for a particular valu,e of p (0 < p < 100), we can easily obtain the bound of each polynomial from the class of polynomials r^00 (10).

In particular, for p = 17, the value of 80 is 1.09786. Using Theorem 1.1, we see that all zeros of each polynomial in the class r^ (10) lie in the region

IzI ^ 1.25709.

Theorem 1.2. All the zeros of p(z) with |a0| = 0 never vanish in the region

w ^1,

to

where t0 (> 1) is the greatest positive root of the equation

|ao| tn+l — (|ao| +Qn) tn — t + 1 = 0. With the help of Theorem 1,1 and Theorem 1,2, we can easily obtain the following Corollary, Corollary 1.1. All zeroes of the polynomial p(z) with |a0| =0 lie in the ring-shaped region

1 < |z| ^ SoQ?, to

where 80 and t0 are the greatest positive roots of the equations

. , f n \™+l / n \ n . , f n \™+l

qn+l(Q ') — qn(Q ') — qn-p+l(Q') + Qn = 0

and

|ao| tn+l — (|ao| +Qn) tn — i + 1 = 0,

respectively.

Theorem 1.3. All the zeros of p(z) with |a0| = 0 never vanish in the disc

w < T+V"

1 + M

provided

Q < min {1, 2P |ao|2} . One can easily obtain the following corollaries by using Theorem 1,3, Corollary 1.2. If

| an_3 — an_lan_2 + an_ l (an-l — ara_2) | = 0, | a0 (an_l — ara_2) | = 0

and

pra+2 = max 1 dn — ara-lara-„+l + ara_p+2 l — an-2) 1

with a_l = a_2 = 0, then the polynomial P (z) of degree n > 3 never vanish in the region

M < -~T-

| | 1+ r

provided

|ao(a^_1_«n-2)

r < min {1, 23 |ao {a2n_l — ara_2) ^ . The Corollary 1,2 can be easily obtained by using Theorem 1,3 on the polynomial

R (z) = (z2 — an_lZ + a2n_l — ara_2) P (z). Corollary 1.3. The polynomial P (z) of degree n > 4 never vanish in the region

M < —--T-

1+

|ao(2an_ian_2_an-3_a®_1)

provided

where

Y < min{1, 24 |ao (2an_lan_2 — an_3 — ain_l) ^ , |ao (2an_lan_2 — an_3 — ^ | = 0, |(| = 0, Yra+3 = max |r^

C = an-4 — an-ian-3 + an-2 (a>n-1 — an-2) + an-1 (2an-ian-2 — an-3 — ^ ,

Vp = an-p — tin- 1^n-p+1 + 0>n-p+2 1 — an-2) + 0>n-p+3 {2tin- 2 — Q"n-3 — 1)

with, a_ 1 = a_2 = a_3 = 0.

The Corollary 1,3 obtained by using Theorem 1,3 on the polynomial

R (z) = — an-1 z2 + (-1 — an-2) z + 2an- 1an-2 — an-3 — -J P (z).

2. Proof of theorems Proof of Theorem 1.1. As |z| > 1, we have

IIU-P+1 _ 1 1

IP (z)I > IzIn — Qn IZ' , 1 = n^-T {Mra+1 — M™ — Qn + Qn} .

Iz I — 1 H — 1

We introduce a function

f (t) = tn+1 — tn — Qntn~p+1 + Qn.

For q > 0 we get

n+1 / „ \n / „ \ n-p+1

'p - aO p) -On oO p

/ n\ / n\ n\ n / n

\ n-p+1

f{qQ >) = [gQ >) — {qQ >) — Qn {qQ >) + Qra

. / n \ K+1 f n \n / „ N H+1

=q (Qp) — qn [Qp) — qn~p+1 [Qp) + Qn (2.1)

i n ( 1 1 \

^ (Q'') {" — Qf — + Qr

Denote

(n \ / n \ n+1 / n \ n / n \ n+1

qQ >) = qn+1 {Q >) — qn {Q >) — qn~P+1 {Q >) +

- S / " \P

9 (0)= (Q*)

and

/ s / n\P ( n\n

g (1) = [Q >) — [Q >)

which is negative as Q > 1 and is positive as 0 < Q < 1. In particular, as Q = 1, the value g (q) can be written as

9 (q) = (q — 1) r (q) ,

where

r (q) = qn — (1 + q + ■ ■ ■ + qn~p)

with

r (1) = — (n — p) < 0, r (2) = 2ra — 2n~p+1 + 1 > 0, which shows that g (q) = 0 has two positive roots, one of which is 1 and other, say 50 in (1, 2). Therefore,

g (q) > 0

if q > 50. Hence,

f (t) > 0

n

as t > 50Q ^d Q =1. This implies the desired result.

n

As Q > 1, the equation g (q) = 0 has two positive roots, one of which is Q~ p E (0,1) and

the other, say 80, ^^togs to (1, 2) bv (2.1). In this case,

g (q) > 0

if q > <50, which implies that

f (t) > 0

if t > S0Qp. This leads us to the desired result,

_n

As 0 < Q < 1, the equation g (q) = 0 has two positive roots, one of which is Q p > 1 and

the other, say t0, lies in (1, ro). Let £0 be the greatest positive root of g (q) = 0, Then

g(q) > 0

> o

f(t) > 0

n _

> 6oQ p and this leads us to the desired result, □ Proof of Theorem 1.2. Consider

R (z) = a0zn + alzn~l + a2zn~2 +-----+ an_pzp + 1.

As | z | > 1, we have

\ z\n~p - 1 Qn \z\n

|R (*) | ^ |ao| M™ — Qra |z|p |Z' , 1 1 — 1 > |ao| M™ — — 1

z'- 1 10111 'z' - 1

1 {'a0'k'ra+1 - (M +Qra) 'z'n -k' + 1}

W — 1

The equation

|ao| tn+l — (|ao| + Qn) tn — i + 1 = 0 obviously has exactly two positive roots, one lies in (0,1) and the other is in (1, ro). Let t0 > 1 be the greatest positive root of the above equation. Then

|ao| tn+l — (|ao| + Qra) tn — t + 1 ^ 0 for all t ^ to.

So, |R (z)| > 0 if | z| ^ t0 and this proves the desired result, □

Proof of Theorem 1.3. Consider

R (z) = aozn + alzn~l + a2zn~2 +-----+ an-pzp + 1

As |z| > 1, we get

\ z\n~p - 1

|R (*)| ^ |ao| \z\a — Qn MP | | , . — 1

|z| — 1

1 {|ao| k|ra+l — (|ao| + Qra) |z|ra — Qn |z|p — |z| + 1} .

W — 1

We consider the function

(.) f (t)

9(t) = —1,

where

f (t) = |ao| tn+l — (|ao| + Qra) tn — Qntp — t + 1.

We obtain

» 0 ++ R) = |aol(1 + ^ J " Q-l M {(l + 1°)" — (l + ^)'} - ^

" - Qn~l laJ + " + 0,l~l laJ (1 + "

' "oil 1 + Rj -0-1'«ol(1 + +Q-1|ao'^1 + i^j - 1

2 / I ~ \\P

- 1

(1+ái) - «"''•o' (1+© + f (1+B)

(

' "0 ' l1 + S" f1 ') + - n > 0

provided

Q ^ min{1, 2P |ao|2} . Clearly, f(t) = 0 has exactly two positive roots with

/(0) > 0, /(1) < 0, /(ro) > 0.

In this case,

f(1 + Q)>0 if Q < min I1'2PIßol2}

V N1/

g(t) > 0 for all t^ 1 + Q

I ao|

which implies that

provided Q ^ min {1, 2P '"0'2}. Therefore,

'R (z)' > 0 for all tS 1 + if Q < min {1, 2P '"o'2>

' 0'

and this completes the proof, □

REFERENCES

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B. Datt, N. K. Govil. On the location of the zeros of polynomial // J. Approx. Theory. 24:1, 78-82 (1978).

F. G. Boese, W. J. Luther. A note on a classical bound for the moduli of all zeros of a polynomial // IEEE Trans. Automat. Contr. 34:9, 998-1001 (1989).

H. Guggenheimer. On a note by Q. G. Mohammad // Amer. Math. Monthly. 71:1, 54-55 (1964). M. Marden. Geometry of polynomials. Amer. Math. Soc. Math. Surveys 3. Amer. Math. Soc. Providence R.I. (1966).

S. Das, S. K. Datta. On Cauchy's proper bound for zeros of a polynomial // Int. J. Math. Sci. Eng. Apple. 2:4, 241-252 (2008).

V. K. Jain. On Cauchy's type bounds for zeros of a polynomial // Ann. Univ. Mariae Curie-Sklodowska Lublin Polonia XLIX:8, 109-116 (1995).

V. K. Jain. On the location of zeros of a polynomial // Bull. Math. Soc. Sci. Math. Roum. Nuov. Ser. 54(102): i. 337-352 (2011).

Subhasis Das,

Department of Mathematics, Kurseong College, Dow Hill Road, 734203, Kurseong, India E-mail: subhasis091@yahoo. co . in