On The Zeros of Certain Class of Polynomials

International Journal of Modern Engineering Research (IJMER)
www.ijmer.com Vol.2, Issue.6, Nov-Dec. 2012 pp-4363-4372 ISSN: 2249-6645

On The Zeros of Certain Class of Polynomials
B.A. Zargar
Post-Graduate Department of Mathematics, U.K, Srinagar.

Abstract: Let P(z) be a polynomial of degree n with real or complex coefficients . The aim
of this paper is to obtain a ring shaped region containing all the zeros of P(z). Our results not only generalize some known
results but also a variety of interesting results can be deduced from them.

I. Introduction and Statement of Results
The following beautiful result which is well-known in the theory of distribution of zeros of polynomials is due to Enestrom
and Kakeya [9].
Theorem A. If P(z) =∑ 𝑗𝑛=0 aj zj is a polynomial of degree n such that
(1) an ≥ an-1 ≥…..≥ a1 ≥ a0 ≥ 0,
then all the zeros of P(z) lie in 𝑧 ≤ 1.
In the literature ([2] ,[5] –[6], [8]-[11]) there exists some extensions and generalizations of this famous result. Aziz
and Mohammad [1] provided the following generalization of Theorem A.
Theorem B. Let P(z) =∑ 𝑗𝑛=0 aj zj is a polynomial of degree n with real positive coefficients. If t1 ≥ 𝑡2 ≥ 0 can
be found such that
(2) 𝑎 𝐽 t1 t 2+ aj−1 t1 − t 2 a j−2 ≥ 0, j=1, 2, … ;n+1, 𝑎−1 = 𝑎 𝑛 +1 = 0 ,
then all the zeros of P(z) lie in 𝑧 ≤ 𝑡1 .
For 𝑡1 = 1, 𝑡2 = 0, this reduces to Enestrom-Kakeya Theorem (Theorem A).
Recently Aziz and Shah [3] have proved the following more general result which includes Theorem A as a special
case.
------------------------------------------------------------------------------
Mathematics, subject, classification (2002). 30C10,30C15,
Keywords and Phrases , Enestrom-Kakeya Theorem, Zeros ,bounds.

Theorem C. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n. If for some t > 0.
(3) 𝑀𝑎𝑥 𝑧 =𝑅 𝑡𝑎0 𝑧 𝑛 + 𝑡𝑎1 − 𝑎0 𝑧 𝑛 −1 + ⋯ + 𝑡𝑎 𝑛 − 𝑎 𝑛−1 ≤ 𝑀3
Where R is any positive real number, then all the zeros of P(z) lie in
𝑀 1
(4) 𝑧 ≤ 𝑀𝑎𝑥 𝑎 3 , 𝑅
𝑛
The aim of this paper is to apply Schwarz Lemma to prove a more general result which includes Theorems A, B
and C as special cases and yields a number of other interesting results for various choices of parameters 𝛼, 𝑟, 𝑡1 𝑎𝑛𝑑 𝑡2 . In
fact we start by proving the following result.
Theorem 1. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n, If for some real numbers 𝑡1 , 𝑡2 with 𝒕 𝟏 ≠ 𝟎, 𝒕 𝟏 >
𝒕𝟐 ≥ 𝟎
(5) 𝑀𝑎𝑥 𝑧 =𝑅
𝑎 𝑛 𝑡1 − 𝑡2 + 𝛼 − 𝑎 𝑛−1 + ∑ 𝑗𝑛=0 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1 − 𝑡2 − 𝑎𝑗 −2 𝑧 𝑛−𝑗 +1 ≤ 𝑀1

(6)
𝑀𝑖𝑛 𝑧 =𝑅 𝑎0 𝑡1 − 𝑡2 + 𝑎1 𝑡1 𝑡2 + 𝛽 +
∑ 𝑗𝑛+2 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1 − 𝑡2 −
=2 (𝑎 𝑗 −2 𝑧 𝑗 ≤ 𝑀2

Where R is a positive real number, then all the zeros of P(z) lie in the ring shaped region.

𝑡1 𝑡2 𝑎0 𝑀1 + 𝛼 1
(7) 𝑀𝑖𝑛 𝑧 =R 𝛽 +𝑀2 , 𝑅 ≤ 𝑧 ≤ 𝑀𝑎𝑥 𝑧 =R 𝑎𝑛
, 𝑅

Taking t2 =0, we get the following generalization and refinement of Theorem C.

Corollary 1. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n. If for some t > 0.
𝑀𝑎𝑥 𝑧 =𝑅 a n t + α − a n−1 + ∑n j=0 a j−1 t − a j−2 z
n−j+2
≤ M1
n+2 j
𝑀𝑖𝑛 𝑧 =𝑅 a 0 t + β + ∑j=0 a j−1 t − a j−2 z ≤ M2
Where R is a real positive number then all the zeros of P(z) lie in

www.ijmer.com 4363 | Page


𝑀1 + 𝛼 1
z ≤ 𝑀𝑎𝑥 𝑧 =𝑅 𝑎𝑛
, 𝑅

In case we take t=1=R in corollary 1, we get the following interesting result.

Corollary 2. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n,
𝑀𝑎𝑥 𝑧 =𝑅 𝑎 𝑛 + 𝛼 − 𝑎 𝑛−1 +
𝑎 𝑛 −1 − 𝑎 𝑛 −2 𝑧 + ⋯ + 𝑎1 − 𝑎0 𝑧 𝑛−1 + 𝑎𝑜 𝑧 𝑛 ≤ M,

M+ α
then all the zeros of P(z) lie in the circle z ≤ Max an
,1

If for some α > 0, 𝛼 + 𝑎 𝑛 ≥ 𝑎 𝑛 −1 ≥ ⋯ ≥ 𝑎1 ≥ 𝑎0 ≥ 0, then ,

𝑀 ≤ 𝑎 𝑛 + 𝛼 + 𝑎 𝑛 −1 + 𝑎 𝑛 −1 − 𝑎 𝑛−2 + ⋯ + 𝑎1 − 𝑎0 + 𝑎0

= 𝑎 𝑛 + 𝛼 − 𝑎 𝑛 −1 + 𝑎 𝑛−1 − 𝑎 𝑛−2 + ⋯ + 𝑎1 − 𝑎0 + 𝑎0

= 𝑎𝑛 + 𝛼

Using this observation in Corollary 2, we get the following generalization of Enestrom-Kakeya Theorem.

Corollary 3. If, P(z)= 𝑎 𝑛 𝑧 𝑛 + 𝑎 𝑛 −1 + 𝑧 𝑛 −1 + ⋯ + 𝑎1 𝑧 + 𝑎0 , is a polynomial of degree n, such that for some
𝛼 ≥ 0,
𝛼 + 𝑎 𝑛 ≥ 𝑎−1 ≥ ⋯ . ≥ 𝑎0 ≥ 0,
then all the zeros of P(z) lie in the circle
2𝛼
𝑧 ≤ 1+ ,
𝑎𝑛

For 𝛼 = 0, 𝑡ℎ𝑖𝑠 𝑟𝑒𝑑𝑢𝑐𝑒𝑠 𝑡𝑜 𝑇ℎ𝑒𝑜𝑟𝑒𝑚 𝐴. 𝐼𝑓 𝑤𝑒 𝑡𝑎𝑘𝑒 𝛼 = 𝑎 𝑛−1 − 𝑎 𝑛 ≥ 0, then we get Corollary 2, of ([4] Aziz and
Zarger).
Next we present the following interesting result which includes Theorem A as a special case.
Theorem 2. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n, If for some real numbers with 𝒕 𝟏 ≠
𝟎, 𝒕 𝟏 > 𝒕 𝟐 ≥ 𝟎
𝒏
(𝟖) 𝑴𝒂𝒙 𝒛 =𝑹 ∑ 𝒋=𝟎 𝒂 𝒋 𝒕 𝟏 𝒕 𝟐 + 𝒂 𝒋−𝟏 𝒕 𝟏 − 𝒕 𝟐 − 𝒂 𝒋−𝟐 𝒛 𝒏−𝒋 ≤ 𝑴3
then all the zeros of P(z) lie in the region
(9) 𝑧 ≤ 𝑟1,

Where ,
2𝑀3
(10) 𝑟1 =
𝑎 𝑛 𝑡 1 −𝑡 2 𝑎 𝑛 −1 2 +4 𝑎 𝑛 𝑀3 1/2 − 𝑎
𝑛 𝑡 1 −𝑡 2 − 𝑎 𝑛 −1

Taking 𝑡2 = 0, we get the following result

Corollary 4. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n, If for some 𝑡 > 0,

𝒏
∑ 𝒋=𝟎 𝒂 𝒋−𝟏 𝒕 − 𝒂 𝒋−𝟐 𝒛 𝒏−𝒋 ≤ 𝑴3,
𝑴𝒂𝒙 𝒛 =𝑹

then all the zeros of P(z) lie in the region

𝑧 ≤ 𝑟1,

Where ,
2𝑀3
r= 𝑎 𝑛 𝑡− 𝑎 𝑛 −1 2 +4 𝑎 𝑛 𝑀3 1/2 − 𝑎 𝑡−𝑎
𝑛 𝑛 −1

Remark. Suppose polynomial P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 , satisfies the conditions of


Theorem A, then it can be easily verified that from Corollary 4.
M3 =an-1 ,
then all the zeros of P(z) lie in

2𝑎 𝑛 −1
𝑧 ≤
𝑎 𝑛 − 𝑎 𝑛 −1 2 + 4𝑎 𝑛 𝑎 𝑛−1 − 𝑎 𝑛 − 𝑎 𝑛 −1

2𝑎
= 2𝑎 𝑛 −1 = 1,
𝑛 −1

which is the conclusion of Enestrom-Kekaya Theorem.

Finally , we prove the following generalization of Theorem 1 of ([2], Aziz and Shah).

Theorem 3. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n, If𝛼, 𝛽 are complex numbers and with 𝑡1 ≠
0, 𝑡2 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑤𝑖𝑡ℎ 𝑡1 ≥ 𝑡2 ≥ 0 and
(11) 𝑀𝑎𝑥 𝑧 =𝑅 𝑎 𝑛 𝑡1 − 𝑡2 + 𝛼 − 𝑎 𝑛 −1 𝑧 + ∑ 𝑗𝑛=0 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1 −𝑡2 − 𝑎𝑗 −2 𝑧 𝑛 −𝑗 −2 ≤ 𝑀4

and,

(12) 𝑀𝑎𝑥 𝑧 =𝑅 𝑎 𝑛 𝑡1 − 𝑡2 + 𝛽 − 𝑎 𝑛 −1 𝑧 + ∑ 𝑗𝑛+2 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1 −𝑡2 − 𝑎𝑗 −2 𝑧 𝑗 ≤ 𝑀5
=2

Where 𝑎−2= 𝑎−1 = 0 = 𝑎 𝑛+1 = 𝑎 𝑛+2 and R is any positive real number, then all the zeros of P(z) lie in the ring shaped
region.
1
(13) Min (r2 , R) ≤ 𝑧 ≤ max 𝑟1 ,
𝑅

Where

2[ 𝑎 𝑅 2 𝑎 𝑛 𝑡 1− 𝑡 2 −𝑎 𝑛 −1 +𝛼 +𝑀4 ] 2
(14) 𝑟1 = 2
𝛼 𝑅 2 𝑀4 +𝑅 2 𝑀4 − 𝑎 𝑛 𝑎 𝑛 𝑡 1− 𝑡 2 +𝛼−𝑎 𝑛 −1 2 +4 𝛼 𝑅 2 𝑎
𝑛 𝑡 1− 𝑡 2 +𝛼−𝑎 𝑛 −1 +𝑀4 𝑎 𝑛 𝑅 2 𝑀4 1/2 −

𝛼 𝑅2 𝑀4 + 𝑅2 𝑀4 − 𝑎 𝑛 𝑎 𝑛 𝑡1− 𝑡2 + 𝛼 − 𝑎 𝑛 −1

and

1 2[𝑅 2 𝛽 2
𝑎 1 𝑡 1 𝑡 2 +𝑎 0 𝑡 1− 𝑡 2 +𝛽 −𝑀5 ]
(15) =
𝑟2 𝑅2 𝑎 1 𝑡 1 𝑡 2 +𝑎 0 𝑡 1− 𝑡 2 +𝛽 ( 𝑎 0 𝑡 1 𝑡 2 −𝑀5 )− 𝛽 𝑀5 +

𝑅4 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1− 𝑡2 + 𝛽 𝑎0 𝑡1 𝑡2 − 𝑀5 2

2
+4 𝑎0 𝑀5 𝑡1 𝑡2 (𝑅 2 𝛽 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1− 𝑡2 ) + 𝛽 + 𝑀5 1/2

Taking 𝛼 = 0, 𝛽 = 0, in Theorem 3, we get the following.

Corollary 5. Let P(z) = ∑ 𝑗𝑛=0 𝑎𝑗 𝑧 𝑗 be a polynomial of degree n, If ,𝑡1 ≠ 0 𝑎𝑛𝑑 𝑡2 are real numbers with 𝑡1 ≥ 𝑡2 ≥
0,
𝑀𝑎𝑥 𝑧 =𝑅 ∑ 𝑗𝑛+1 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1− 𝑡2 − 𝑎𝑗 −2 𝑧 𝑛−𝑗 +2 ≤ 𝑀4 ,
=1

𝐧+𝟐
∑ 𝐣=𝟏 𝐚 𝐣 𝐭 𝟏 𝐭 𝟐 + 𝐚 𝐣−𝟏 𝐭 𝟏− 𝐭 𝟐 − 𝐚 𝐣−𝟐 𝐳 𝐣 ≤ 𝐌 𝟓 ,
𝐌𝐚𝐱 𝐳 =𝐑

Where R is any positive real number . then all the zeros of P(z) lie in the ring shaped region
1
min (r2,R) ≤ 𝑧 ≤ max 𝑟1 ,
𝑅

Where
2𝑀42
𝑟1 = 3
𝑅4 𝑡 1− 𝑡 2 ) 𝑎 𝑛 −𝑎 𝑛 −1 2 𝑀4 − 𝑎 𝑛 2 +4 𝑎
𝑛 𝑅 2 𝑀4 1/2 − 𝑡 1 −𝑡 2 )𝑎 𝑛 −𝑎 𝑛 −1 𝑀4 − 𝑎 𝑛 𝑅2

𝑟2 =


1
1 3
2
2𝑀5
𝑅4 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1 − 𝑡2 2
𝑀5 − 𝑎0 𝑡1 𝑡2 2
+ 4𝑀5 𝑅 2 𝑎0 𝑡1 𝑡2 2 −
𝑅 2 𝑀5 − 𝑎0 𝑡1 𝑡2 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1 − 𝑡2

The result was also proved by Shah and Liman [12].

II. LEMMAS
For the proof of these Theorems, we need the following Lemmas. The first Lemma is due to Govil, Rahman and
Schmesser [7].
LEMMA. 1. If 𝑓(z) is analytic in 𝑧 ≤ 1, 𝑓 𝑜 = 𝑎 𝑤ℎ𝑒𝑟𝑒 𝑎 < 1, 𝑓′ 𝑜 = 𝑏, 𝑓(𝑧) ≤ 1, 𝑜𝑛 𝑧 = 1, 𝑡ℎ𝑒𝑛 𝑓𝑜𝑟 𝑧 ≤
1.
1− 𝑎 𝑧 2 + 𝑏 𝑧 + 𝑎 1− 𝑎
𝑓(𝑧) ≤ 𝑎 1− 𝑎 𝑧 2 + 𝑏 𝑧 + 1− 𝑎

The example
𝑏
𝑎+ 𝑧−𝑧 2
1+𝑎
𝑓 (z)= 𝑏
1− 𝑧−𝑎𝑧 2
1+𝑎

Shows that the estimate is sharp.

form Lemma 1, one can easily deduce the following:

LEMMA. 2. If 𝑓 (z) is analytic in 𝑧 ≤ 𝑅, 𝑓 0 = 0, 𝑓 ′ 0
= 𝑏 𝑎𝑛𝑑 𝑓(𝑧) ≤ 𝑀,

𝑓𝑜𝑟 𝑧 = 𝑅, 𝑡ℎ𝑒𝑛
𝑀 𝑧 𝑀 𝑧 +𝑅 2 𝑏
𝑓(𝑧) ≤ for 𝑧 ≤ 𝑅
𝑅2 𝑀+ 𝑏 𝑧

III. PROOFS OF THE THEOREMS.
Proof of Theorem . 1 Consider
(16) F(z) = 𝑡2 + 𝑧 𝑡1 − 𝑧 𝑃(𝑧)

= −𝑎 𝑛 𝑧 𝑛+2 + 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛−1 𝑧 𝑛+1

+ 𝑎 𝑛 𝑡1 𝑡2 + 𝑎 𝑛 −1 𝑡1 − 𝑡2 − 𝑎 𝑛−2 𝑧𝑛

+….+ 𝑎 𝑛 𝑡1 𝑡2+ 𝑎0 𝑡1 − 𝑡2 𝑧 + 𝑎0 𝑡1 𝑡2

Let
1
G(z) + 𝑧 𝑛+2 𝐹
𝑧

= −𝑎 𝑛 + 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛−1 𝑧 + 𝑎 𝑛 𝑡1 𝑡2 + 𝑎 𝑛 −1 𝑡1 − 𝑡2 − 𝑎 𝑛 −2 𝑧 2 + ⋯

…+ 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1 − 𝑡2 𝑧 𝑛 +1 + 𝑎0 𝑡1 𝑡2 𝑧 𝑛+2

= −𝑎 𝑛 − 𝛼𝑧 + 𝑎 𝑛 𝑡1 − 𝑡2 + 𝛼 − 𝑎 𝑛−1 𝑧

+ 𝑎 𝑛 𝑡1 𝑡2 + 𝑎 𝑛−1 𝑡1 − 𝑡2 − 𝑎 𝑛 −2 𝑧 2 + ⋯

… + (𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1 − 𝑡2 𝑧 𝑛+1 + 𝑎0 𝑡1 𝑡2 ) 𝑧 𝑛+2

= − 𝑎 𝑛 + 𝛼𝑧 + 𝐻(𝑧)

Where,

H(z)= 𝑧 𝑎 𝑛 𝑡1 − 𝑡2 + 𝛼 − 𝑎 𝑛 −1 + ∑ 𝑗𝑛=0 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1 − 𝑡2 − 𝑎𝑗 −2 𝑧 𝑛−𝑗


Clearly H(o)=0 and 𝑀𝑎𝑥 𝑧 =𝑅 𝐻(𝑧) ≤ 𝑅𝑀1,

We first assume that 𝑎 𝑛 ≤ 𝑅 𝑀1 + 𝛼

Now for 𝑧 ≤ 𝑅, by using Schwarz Lemma, we have

(17) G(z) = − a n + αz + H(z)

≥ a n − 𝛼 𝑧 − 𝐻(𝑧)

≥ a n − 𝛼 𝑧 − |𝑀1 𝑧|

= an − 𝑀1 + 𝛼 𝑧|

> 𝑜, 𝑖𝑓
𝑎𝑛
𝑧 < 𝑀1 + 𝛼
≤ 𝑅

𝑎𝑛
This shows that all the zeros of G(z) lie in 𝑧 ≥
𝑀1 + 𝛼

1 1
Replacing z by 𝑧 and noting that F(z) = 𝑧 𝑛+2 𝐺 𝑧
, it follows that all the zeros of F(z) lie in

𝑀1 + 𝛼
𝑧 ≤ 𝑎𝑛
if , 𝑎 𝑛 ≤ 𝑅 (𝑀1 + 𝛼 )

Since all the zeros of P(z) are also the zeros of F(z), we conclude that all the zeros of P(z) lie in
𝑎𝑛
(18) 𝑧 ≤
𝑀1 + 𝛼

Now assume 𝑎 𝑛 > 𝑅 (𝑀1 + 𝛼 ), then for 𝑧 ≤ 𝑅, we have from (17).

G z ≥ a n − 𝛼 𝑧 − 𝐻(𝑍)

≥ 𝑎 𝑛 − 𝛼 𝑅 − 𝑀1 𝑅

= 𝑎𝑛 − 𝛼 + 𝑀 𝑅 > 0,

Thus G(z) ≠ 0 𝑓𝑜𝑟 𝑧 < 𝑅, 𝑓𝑟𝑜𝑚 𝑤ℎ𝑖𝑐ℎ it follows as before that all the zeros of F(z) and hence all the zeros of P(z) lie
1,
in 𝑧 ≤ 𝑅 Combining this with (18), we infer that all the zeros of P(z) lie in

𝐚𝐧 𝟏,
19. 𝐳 ≤ 𝐦𝐚𝐱 𝐌 𝟏+ 𝛂
, 𝐑

Now to prove the second part of the Theorem, from (16) we can write F(z) as

F(z)= t 2 + z t1 − z P(z)

=

a 0 t1 t 2 + a1 t1 t 2 + a 0 t1 − t 2 z + ⋯ + a n t1 − t 2 − a n−1 z n+1 − a n z n+2

=a 0 t1 t2 − βz + a1 t1 t 2 + a 0 t1 − t 2 + β z + ⋯ + a n t1 − t 2 − a n−1 z n+1 − a n z n+2

= a 0 t1 t 2 − βz + T(z)

Where

T(z) = z 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1 −𝑡2 + 𝛽 + ∑n+2 𝑎𝑗 𝑡1 𝑡2 + 𝑎𝑗 −1 𝑡1 −𝑡2 𝑎𝑗 −2 𝑧 𝑗 −1
j=2

Clearly T(o)=0, and 𝑀𝑎𝑥 𝑧 =𝑅 𝑇(𝑧) ≤ 𝑅𝑀2

We first assume that 𝑎0 𝑡1 𝑡2 ≤ β + M2


Now for z ≤ R, by using Schwasr lemma, we have

F(z) ≥ 𝑎0 𝑡1 𝑡2 − β z − T(z)

≥ 𝑎0 𝑡1 𝑡2 − β z − M2 Z

>0, if,
𝑎0 𝑡1 𝑡2
z < ≤R
β +M 2

𝑎0 𝑡1 𝑡2
This shows that F(z) ≠0, for z < , hence all the zeros of F(z) lie in
β +M 2

𝑎0 𝑡1 𝑡2
z ≥ β +M 2

But all the zeros of P(z) are also the zeros of F(z), we conclude that all the zeros of P(z) lie in

𝑎0 𝑡1 𝑡2
20. z ≥ β +M 2

Now we assume 𝑎0 𝑡1 𝑡2 > β + M2 , then for z ≤ R, we have

` F(z) ≥ 𝑎0 𝑡1 𝑡2 − β z − T(z)

≥ 𝑎0 𝑡1 𝑡2 − β R − M2 R

> 𝑡1 𝑡2 𝑎0 − β + M2 R

>0,

This shows that all the zeros of F(z) and hence that of P(z) lie in

z ≥R

Combining this with (20) , it follows that all the zeros of P(z) lie in
𝑎0 𝑡1 𝑡2
21 z ≥ min β +M 2
,R

Combining (19) and (20), the desired result follows.

Proof of Theorem 2:- Consider

22. F(z) = 𝑡2 + 𝑧 𝑡1 − 𝑧 𝑃(𝑧)

= 𝑡2 + 𝑧 𝑡1 − 𝑧 𝑎 𝑛 𝑧 𝑛 + 𝑎 𝑛−1 𝑧 𝑛−1 + ⋯ + 𝑎1 𝑧 + 𝑎0

= −𝑎 𝑛 𝑧 𝑛+2 + 𝑎𝑛 𝑡1− 𝑡2 − 𝑎 𝑛−1 𝑧 𝑛 +1 + ⋯ + 𝑎0 𝑡1 𝑡2

Let
1
23. G(z) = 𝑧 𝑛+2 𝐹( 𝑧 )

= −𝑎 𝑛 + 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛−1 𝑧 +

𝑎 𝑛 𝑡1 𝑡2 − 𝑎 𝑛 −1 𝑡1 − 𝑡2 𝑎 𝑛 −2 𝑧 2 + 𝐻(𝑧)

Where

H(z) = 𝑧 2 𝑎 𝑛 𝑡1 𝑡2 − 𝑎 𝑛−1 𝑡1 − 𝑡2 − 𝑎 𝑛 −2 + ⋯ + 𝑎0 𝑡1 𝑡2 𝑧 𝑛

Clearly H(o)=0 = H(o), since H(z) ≤ M3 R2 for z = R

We first assume,

24 a n ≤ MR2 + R 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1

Then by using Lemma 2. To H(z), it follows that
M3 z M3 z
𝐻z ≤ R2 M3
𝑀𝑎𝑥 𝑧 =𝑅 H(z)

Hence from(23) we have
z 2M3
G(z) ≥ a n − 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1 z − 𝑅2
R2

>0, if

2
M3 z + 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛−1 z − a n < 0

That is if,
1/2
𝑎 𝑛 𝑡 1 −𝑡 2 −𝑎 𝑛 −1 2 +4 a n M − 𝑎 𝑛 𝑡 1 −𝑡 2 −𝑎 𝑛 −1
25. z < 2M 3

1
=r ≤ R

If
2 2
𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1 + 4 a n M3 ≤ 2M3 R + 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛−1

Which implies

a n ≤ M3 R2 + R 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1

Which is true by (24)..Hence all the zeros of G(z) lie in Z ≥ r. since

F z = z n+2 G z ,

It follows that all the zeros of F(z) and hence that of P(z) lie in z ≤ r. We now assume.

26. a n ≤ M3 R2 + R 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1 ,

then for z ≤ R, from (23) it follows that

G(z) ≥ a n − 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1 z − H(z)

≥ 𝑎 𝑛 − 𝑅 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛−1 − R2 M3

>0, by (26).
1
This shows that all the zeros of G(z) lie in Z > 𝑅, and hence all the zeros of F(z) = z n+2 G(z ) lie in
1
z ≤ , but all the zeros of P(z) are also the zeros of F(z) , therefore it follows that all the zeros of P(z) lie in
R

1
Z ≤R

27. From (25) amd (27) , we conclude that all the zeros of P(z) lie in
1
Z ≤ max(𝑟, )
R

Which completes the proof of Theorem 2.

28. PROOF OF THEOREM 3. Consider the polynomial

F(z) = 𝑡2 + 𝑧 𝑡1 − 𝑧 𝑃(𝑧)

= 𝑡1 𝑡2 + 𝑡1 − 𝑡2 𝑧 − 𝑧 2 ) 𝑎 𝑛 𝑧 𝑛 + 𝑎 𝑛−1 𝑧 𝑛 −1 + ⋯ + 𝑎1 𝑧 + 𝑎0

= −𝑎 𝑛 𝑧 𝑛+2 + 𝑎𝑛 𝑡1− 𝑡2 − 𝑎 𝑛−1 𝑧 𝑛 +1 + ⋯ + 𝑎 𝑛 𝑡1 𝑡2

+ 𝑎 𝑛−1 𝑡1− 𝑡2 − 𝑎 𝑛 −1 𝑧 𝑛+1 + ⋯ +

a 2 t1 t 2 + a1 𝑡1− 𝑡2 − 𝑎0 z 2 + a1 t1 t 2 + a 0 𝑡1− 𝑡2 𝑧 + 𝑎0 t1 t 2
We have,
1
G(z) = 𝑧 𝑛+2 𝐹( )
𝑧
= 𝑟𝑎 𝑛 + 𝑎 𝑛 𝑡1 − 𝑡2 − 𝑎 𝑛 −1 𝑧 + 𝑎 𝑛 𝑡1 𝑡2 + 𝑎 𝑛 −1 𝑡1 − 𝑡2 − 𝑎 𝑛 −2 𝑧 2
+…+ 𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1− 𝑡2 𝑧 𝑛 +1 + 𝑎0 𝑡1 𝑡2 𝑧 𝑛+2
= −𝑎 𝑛 − 𝛼𝑧 + 𝑎 𝑛 𝑡1 − 𝑡2 + 𝛼 − 𝑎 𝑛 −1 𝑧 +
𝑎 𝑛 𝑡1 𝑡2 + 𝑎 𝑛 −1 𝑡1− 𝑡2 − 𝑎 𝑛 −2 z 2 +. . +
𝑎1 𝑡1 𝑡2 + 𝑎0 𝑡1− 𝑡2 z n+1 + 𝑎0 𝑡1 𝑡2 𝑧 𝑛+2
29. = −𝑎 𝑛 − 𝛼𝑧 + 𝐻 𝑧
Where
H(z) = 𝑎 𝑛 𝑡1 −𝑡2 + 𝛼 − 𝑎 𝑛 −1 𝑧 + ⋯ + 𝑎0 𝑡1 𝑡2 𝑧 𝑛+2
We first assume that
30. a n ≥ α R + M4
Then for z < 𝑅, we have
31. G(z) ≥ a n − α z − H(z)
Since
H(z) ≤ M4 𝑓𝑜𝑟 z ≤ R
Therefore for 𝑧 <R, from (31) with the help of (30), we have
𝐺(𝑧) > 𝑎 𝑛 − ∝ 𝑅 − 𝑀4
1
therefore , all the zeros of G(z) lie in z ≥ R, in this case. Since F(z) = 𝑧 𝑛 +2 G 𝑧 therefore all the zeros of G(z) lie in
1
Z ≤ R . As all the zeros of P(z) are the zeros of F(z), it follows that all the zeros of P(z) lie in
1
32. z ≤R
Now we assume a n < α R + M4 , clearly H(o) = 0 and H(o) = (𝑎 𝑛 𝑡1 −𝑡2 −
𝛼 − 𝑎 𝑛 −1 , Since by (11), H(z) ≤ M4 , for z = R,
therefore it follows by Lemma2. that
M4 z M4 z + R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼
H(z) ≤
R2 M4 + a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 z
for z ≤ R
Using this in (31), we get,
M4 z M4 z + R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛−1 + 𝛼
G(z) ≥ 𝑎 𝑛 − 𝛼 𝑧 − 2
R M4 + a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 z
>0, if

𝑎 𝑛 R2 M4 + a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 z − z α R2
M4 + a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 z + M4 M4 z + R2
a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 > 0,
this implies,
α R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛−1 + 𝛼 + M4 z 2 + α R2 M4 +
2
2 2
R M4 a n t1 − t 2 − a n−1 +∝ … R M4 an 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 }
z − a n R2 M4 < 0,
This gives,
G(z) > 0, 𝑖𝑓
z < [ α R2 M4 + R2 M4 − a n a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 2 +..
….4 α R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 + M4 2 𝑎 𝑛 R2 M4 ]

𝛼 R 2 M 4 +R 2 M 4 − 𝑎 𝑛 a n 𝑡 1 −𝑡 2 −𝑎 𝑛 −1 +𝛼
−
2 α R 2 a n 𝑡 1 −𝑡 2 −𝑎 𝑛 −1 +𝛼 +M 4 2
1
=r ,
1
So, z < 𝑅, 𝑖𝑓
α R2 M4 + M4 − a n R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 2
+

4 α R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛−1 + 𝛼 + M4 2 𝑎 𝑛 R2 M4

< 2 α a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 R + α R2 M4 + 2RM4 2

+ M4 − a n R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 2

That is if,

a n R2 M4 ≤ R2 α 𝑅2 an 𝑡1 −𝑡2 − 𝑎 𝑛−1 + 𝛼 + M4 2 +

α R2 M4 + M4 − a n R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 R

Which gives

a n M4 ≤ M4 α R + M4 − 𝑎 𝑛 + 𝑅 α R2 a n 𝑡1 −𝑡2 − 𝑎 𝑛 −1 + 𝛼 + M4 2

α R2 + M4 − 𝑎 𝑛

Which is true because 𝑎 𝑛 < α R + M4 ,
1 1
Consequently , all the zeros of G(z) lie in z ≥ r , as F(z) = z n+1 G( z ), we conclude that all the zeros of
1
F(z) lie z ≤ 𝑟1 , since every zero of P(z) is also a zero of F(z), it follows that all the zeros of P(z) lie in

33. z ≤ 𝑟1

Combining this with (32) it follows that all the zeros of P(z) lie in
1
34. z ≤ 𝑚𝑎𝑥 𝑟1 , 𝑅

Again from (28) it follows that

𝐹 𝑧 = 𝑎0 t1 t 2 − βz + a1 t1 t2 𝑡1− 𝑡2 + 𝛽 z + ⋯ + a n ( 𝑡1 −𝑡2

−. 𝑎 𝑛−1 )z n+2 − a n 𝑧 𝑛+2

35. ≥ 𝑎0 t1 t 2 − 𝛽 𝑧 − 𝑇(𝑧)

Where,

T(z) = a n 𝑧 𝑛+2 + a n 𝑡1 −𝑡2 −. 𝑎 𝑛 −1 𝑧 𝑛 +1 +. . +
𝑎0 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 z
Clearly T(o)=0, and T(o) 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽
Since by (12) , T(z) ≤ M5 for z = R
using Lemma 2. To T(z), we have

M5 z
F(z) ≥ 𝑎0 t1 t 2 − β z − R2
M5 z + R2 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 z

= - R2 β 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 + 𝑀5 2 z 2 +

𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 R2 𝑎0 t1 t 2 − R2 M5 − β R2 M5 z + M5 R2 𝑎1 t1 t2

R2 M5 + 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 z

>0, if,

R2 β 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 + 𝑀5 2 z 2 − 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽

R2 𝑎0 t1 t2 − R2 M5 − β R2 M5 z − R2 M5 𝑎0 t1 t 2 < 0,

Thus, F(z) > 0, 𝑖𝑓

z < −R2 𝑎1 t1 t2 + a 0 𝑡1− 𝑡2 + 𝛽 𝑎0 t1 t 2 − M5 β M5


− R4 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 𝑎0 t1 t 2 − M5 β M5 2

+4 R2 β 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 + M5 2 R2 M5 𝑎0 t1 t2 1/2
𝑟𝑧
2 R2 β 𝑎1 t1 t 2 + a 0 𝑡1− 𝑡2 + 𝛽 + M5 2

Thus it follows by the same reasoning as in Theorem 1, that all the zeros of F(z) and hence that of P(z) lie in

36. z ≥ min(r2, R)

Combining (34) and (36) , the desired result follows.

REFERENCES
[1] A. Aziz and Q.G Mohammad, On thr zeros of certain class of polynomials and related analytic functions, J. Math.
Anal Appl. 75(1980), 495-502
[2] A.Aziz and W.M Shah, On the location of zeros of polynomials and related analytic functions, Non-linear studies,
6(1999),97-104
[3] A.Aziz and W.M Shah, On the zeros of polynomials and related analytic functions, Glasnik, Matematicke, 33 (1998),
173-184;
[4] A.Aziz and B.A. Zarger, Some extensions of Enestrem-Kakeya Theorem, Glasnik, Matematicke, 31 (1996), 239-244
[5] K.K.Dewan and M. Biakham,, On the Enestrom –Kakeya Theorem, J.Math.Anal.Appl.180 (1993), 29-36
[6] N.K, Govil and Q.I. Rahman, On the Enestrem-Kakeya Theorem, Tohoku Math.J. 20 (1968), 126-136.
[7] N.K, Govil , Q.I. Rahman and G. Schmeisser, on the derivative of polynomials, Illinois Math. Jour. 23 (1979), 319-
329
[8] P.V. Krishnalah, On Kakeya Theorem , J.London. Math. Soc. 20 (1955), 314-319
[9] M. Marden, Geometry of polynomials, Mathematicial surveys No.3. Amer. Math. Soc. Providance , R.I. 1966
[10] G.V. Milovanovic, D.S. Mitrinovic, Th.M. Rassias Topics in polynomials, Extremal problems Inequalitics,
Zeros(Singapore, World Scientific)1994.
[11] Q.I. Rahman and G.Schmessier, Analytic Theory of polynomials, (2002), Clarantone, Press Oxford.
[12] W.M. Shah and A. Liman, On the zeros of a class of polynomials, Mathematical inequalioties and Applications,
10(2007), 793-799


On The Zeros of Certain Class of Polynomials

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to On The Zeros of Certain Class of Polynomials (20)

More from IJMER (20)

On The Zeros of Certain Class of Polynomials