Let P(z):=∑j=0najzj be a polynomial of degree n and P′(z) its derivative, then it is known that

The above result, which is an immediate consequence of Bernstein’s inequality on the derivative of a trigonometric polynomial is best possible with equality holding for the polynomial P(z) = λzⁿ, where λ is a complex number.

For the class of polynomials having all their zeros in |z| ≥ 1, inequality (1.1) can be sharpened. In fact, Erdös conjectured and later Lax [7] proved that if P(z) ≠ 0 in |z| < 1, then

On the other hand, if the polynomial P(z) of degree n has all its zeros in |z| ≤ 1, then it was proved by Turán [12], that

Inequalities (1.2) and (1.3) are best possible and become equality for polynomials which have all zeros on |z| = 1.

Aziz and Dawood [1] improved inequality (1.2) and under the same hypothesis proved:

Equality in (1.4) holds for P(z) = α + βzⁿ, |α| ≥ |β|.

For the class of polynomials P(z) of degree n having all their zeros in |z| ≥ k, k ≥ 1, Malik [8] proved:

Inequality (1.5) was further improved by Govil [6] who under the same hypothesis proved:

Chan and Malik [2] obtained a generalization of (1.5) by considering the lacunary type of polynomials and obtained the following:

Theorem A

LetP(z):=a0+∑j=μnajzj, 1 ≤ μ ≤ n be a polynomial of degree n having all its zeros in |z| ≥ k, k ≥ 1, then

The result is best possible and extremal polynomial isP(z)=(zμ+kμ)nμ, where n is a multiple of μ.

The next result was proved by Pukhta [10], who infact proved:

Theorem B

LetP(z):=a0+∑j=μnajzj, 1 ≤ μ ≤ n be a polynomial of degree n having all its zeros in |z| ≥ k, k ≥ 1, then

The result is best possible and extremal polynomial isP(z)=(zμ+kμ)nμ, where n is a multiple of μ.

Theorem C

Let P(z) be a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then

The above result was independently given by Govil [5]. For the polynomials of type P(z):=anzn+∑j=μnan-jzn-j, 1 ≤ μ < n, Theorem C was further generalized by Dewan and Hans [3] and proved the following:

Theorem D

LetP(z):=anzn+∑j=μnan-jzn-j, 1 ≤ μ < n, be a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then

For the proofs of above theorems, we need the following lemmas. The first result is due to Qazi [11, Lemma 1].

Lemma 2.1

IfP(z):=a0+∑j=μnajzj, 1 ≤ μ ≤ n is a polynomial of degree n having all its zeros in |z| ≥ k, k ≥ 1, then

The next lemma which we need is due to Dewan and Hans [4].

Lemma 2.2

LetP(z):=anzn+∑j=μnan-jzn-j, 1 ≤ μ < n, be a polynomial of degree n, having all its zeros on |z| = k, k ≤ 1, then

Theorem 2.3

LetP(z):=zs(a0+∑j=μn-sajzj), 1 ≤ μ ≤ n − s, 0 ≤ s ≤ n − 1 be a polynomial of degree n having s-fold zeros at the origin and remaining n − s zeros in |z| > k, k ≥ 1, then

If we take μ = 1 in Theorem 2.3, we have the following result which is an improvement of a result of Mir [9, Theorem 1.6]

Theorem 2.4

LetP(z):=zs(an-szn-s+∑j=μn-san-s-jzn-s-j), 1 ≤ μ < n − s, 0 ≤ s ≤ n − 1, be a polynomial of degree n, having s-fold zeros at the origin and remaining n − s zeros on |z| = k, k ≤ 1, then

If we put μ = 1 in Theorem 2.4, we get the following :

Corollary 2.5

LetP(z):=zs(∑j=0n-sajzj), 0 ≤ s ≤ n − 1 be a polynomial of degree n having s-fold zeros at the origin and remaining n − s zeros in |z| > k, k ≥ 1, then

On taking k = 1 in Theorem 2.3, we get the following:

Corollary 2.6

LetP(z):=zs(a0+∑j=μn-sajzj), 1 ≤ μ ≤ n − s, 0 ≤ s ≤ n − 1 be a polynomial of degree n having s-fold zeros at the origin and remaining n − s zeros in |z| > 1, then

Remark 2.7

Inequality (1.4) that is the result of Aziz and Dawood is a special case of Theorem 2.3 when μ = k = 1 and s = 0.

If we put μ = 1 in Theorem 2.4, we get the following :

Corollary 2.8

LetP(z):=zs(an-szn-s+∑j=1n-san-s-jzn-s-j), 0 ≤ s ≤ n − 1, be a polynomial of degree n, having s-fold zeros at the origin and remaining n − s zeros on |z| = k, k ≤ 1, then

On taking s=0 and μ = 1 in Theorem 2, we get the following result:

Corollary 2.9

LetP(z):=∑j=0nan-jzn-j, be a polynomial of degree n, having all its zeros on |z| = k, k ≤ 1, then