Keywords: derivative of a polynomial, zeros, exterior of circle, lacunary, inequalities
American Journal of Mathematical Analysis, 2015 3 (1),
pp 14.
DOI: 10.12691/ajma311
Received December 29, 2014; Revised January 17, 2015; Accepted January 23, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction and Statement of Results
Let be a polynomial of degree n and P’(z) its derivative, then it is known that
 (1) 
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 , where is a complex number.
If we restrict ourselves to the class of polynomials having all their zeros in , inequality (1) can be sharpened. In fact, Erdös conjectured and later Lax ^{[6]} proved that if P(z)≠ 0 in , then
 (2) 
On the other hand, if the polynomial P(z) of degree n has all its zeros in , then it was proved by Turán ^{[10]}, that max
 (3) 
Inequalities (2) and (3) are best possible and become equality for polynomials which have all zeros on .
Inequality (2) was refined by Aziz and Dawood ^{[1]} by showing that under the same hypothesis that
 (4) 
Equality in (4) holds for .
For the class of polynomials P(z) of degree n having all their zeros in , Malik ^{[7]} proved:
 (5) 
Inequality (5) was further improved by Govil ^{[5]} who under the same hypothesis proved:
 (6) 
Chan and Malik ^{[3]} obtained a generalization of (5) by considering the lacunary type of polynomials and obtained the following:
Theorem A: Let , be a polynomial of degree n having all its zeros in , then
 (7) 
The result is best possible and extremal polynomial is ; where n is a multiple of .
The next result was proved by Pukhta ^{[8]}, who infact proved:
Theorem B: Let , be a polynomial of degree n having all its zeros in , then
 (8) 
The result is best possible and extremal polynomial is ; where n is a multiple of .
Bidkham and Dewan ^{[2]} obtained a generalization of (5) by proving the following result:
Theorem C: Let P(z) be a polynomial of degree n having all its zeros in , then for
 (9) 
The result is best possible and equality holds for P(z) = (z + k)^{n}.
In this paper, we prove the following generalization as well as an improvement of Theorem C by considering the s^{th} derivative of P(z).
Theorem 1: If , n>3, is a polynomial of degree n having all its zeros in , then for , and
2. Lemmas
For the proof of above theorem, we need the following lemmas. The first result is due to Qazi [9, Lemma 1].
Lemma 1: If , is a polynomial of degree n having all its zeros in , then
The next lemma is due to Dewan, Kour and Mir ^{[4]}.
Lemma 2: Let be a polynomial of degree n, then for R 1;
 (10) 
and
 (11) 
Lemma 3: If , n > 3, is a polynomial of degree n having all its zeros in , then for , we have
Proof of Lemma 3: Since P(z) is a polynomial of degree n > 3, the polynomial P′(z) is of degree n 3, hence on applying inequality (10) of Lemma 2 to the polynomial P′(z), we obtain
This proves Lemma 3.
Lemma 4: If , n > 3, is a polynomial of degree n having all its zeros in , then for R 1,
Proof of Lemma 4: For each and for we have
Hence
 (12) 
which when combined with Lemma 3, gives
which gives
Hence the proof.
Lemma 5: If P(z) is a polynomial of degree n having all its zeros in , then for .
This Lemma is due to Govil ^{[5]}.
Proof of Theorem 1: Since P(z) has all its zeros in and if , then G(z) = P(Rz) has all its zeros in , therefore by applying Lemma 5 to G(z), we obtain
which implies
which is equivalent to
 (13) 
Inequality (13) in conjunction with Lemma 4 yields
The proof of Theorem 1 is completed.
References
[1]  A. Aziz and Q.M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory, 54 (1988), No. 3, 306313. 
 In article  

[2]  M. Bidkham and K. K. Dewan, Inequalities for a polynomial and its derivative, J. Math. Anal. Appl., Vol. 166 (1992), 319324. 
 In article  

[3]  T. N. Chan and M. A. Malik, On ErdosLax Theorem, Proc. Indian Acad. Sci. (Math. Sci.), 92 (3) (1983), 191193. 
 In article  

[4]  K. K. Dewan, Jagjeet Kaur and Abdullah Mir, Inequalities for the derivative of a polynomial, J. Math. Anal. Appl., 269 (2002), 489499. 
 In article  CrossRef 

[5]  N. K. Govil, Some inequalities for derivatives of polynomials, J. Approx. Theory, 66 (1) (1991), 2935. 
 In article  

[6]  P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Amer. Math. Soc., Bulletin, 50 (1944), 509513. 
 In article  

[7]  M. A. Malik, On the derivative of a polynomial, J. London Math. Soc., 2 (1) (1969), 5760. 
 In article  

[8]  M. S. Pukhta Extremal Problems for Polynomials and on Location of Zeros of Polynomials, Ph. D Thesis, Jamia Millia Islamia, New Delhi (1995). 
 In article  

[9]  M. A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115(1992), 337343. 
 In article  CrossRef 

[10]  P. Turán, Uber die Ableitung von Polynomen, Compositio Mathematica, 7 (1939), 8995 (German). 
 In article  
