Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A Circle

GULSHAN SINGH

Govt. Department of Education Jammu and Kashmir, India

Abstract: Let P(z) be a polynomial of degree n having all its zeros in , then for , Bidkham and Dewan [J. Math. Anal. Appl. 166(1992), 191-193] proved max In this paper, we prove an interesting generalization as well as an improvement of this result by considering the sth derivative of lacunary type of polynomials P(z) of degree n > 3.

Keywords: derivative of a polynomial, zeros, exterior of circle, lacunary, inequalities

Received December 29, 2014; Revised January 17, 2015; Accepted January 23, 2015

Citation: GULSHAN SINGH, Inequalities for the Sth Derivative of Polynomials Not Vanishing inside A Circle. American Journal of Mathematical Analysis, 3 (1). 1-4. January 2015. doi:10.12691/ajma-3-1-1

1. Introduction and Statement of Results

Let be a polynomial of degree n and P’(z) its derivative, then it is known that

[picture] | (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

[picture] | (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

[picture] | (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

[picture] | (4)

Equality in (4) holds for .

For the class of polynomials P(z) of degree n having all their zeros in , Malik [7] proved:

[picture] | (5)

Inequality (5) was further improved by Govil [5] who under the same hypothesis proved:

[picture] | (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

[picture] | (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

[picture] | (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

[picture] | (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 sth 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;

[picture] | (10)

and

[picture] | (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

[picture] | (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

[picture] | (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, 306-313.
[2]   | M. Bidkham and K. K. Dewan, Inequalities for a polynomial and its derivative, J. Math. Anal. Appl., Vol. 166 (1992), 319-324.
[3]   | T. N. Chan and M. A. Malik, On Erdos-Lax Theorem, Proc. Indian Acad. Sci. (Math. Sci.), 92 (3) (1983), 191-193.
[4]   | K. K. Dewan, Jagjeet Kaur and Abdullah Mir, Inequalities for the derivative of a polynomial, J. Math. Anal. Appl., 269 (2002), 489-499.
[5]   | N. K. Govil, Some inequalities for derivatives of polynomials, J. Approx. Theory, 66 (1) (1991), 29-35.
[6]   | P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Amer. Math. Soc., Bulletin, 50 (1944), 509-513.
[7]   | M. A. Malik, On the derivative of a polynomial, J. London Math. Soc., 2 (1) (1969), 57-60.
[8]   | M. S. Pukhta Extremal Problems for Polynomials and on Location of Zeros of Polynomials, Ph. D Thesis, Jamia Millia Islamia, New Delhi (1995).
[9]   | M. A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115(1992), 337-343.
[10]   | P. Turán, Uber die Ableitung von Polynomen, Compositio Mathematica, 7 (1939), 89-95 (German).