Generalization of certain well known polynomial inequalities

Document Type : Original Article

Authors

Department of Mathematics, Semnan University, Semnan, Iran

Abstract

In this paper, we investigate a generalized class of complex polynomials of the form G(z) = a_n z^n +\sum_{v=t}^{n} a_{n-v}z^{n-v}, 1\leq t \leq n, whose zeros are constrained to lie within or on the boundary of the disk  |z|\leq k, k\leq 1.  We establish new  Bernstein-type inequalities for such polynomials, extending and improving earlier results due to Jain [6], Aziz and Rather [4]. Several known inequalities follow as special cases, and the results have implications for the growth estimates and derivative bounds of analytic functions within specified domains.

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References

[1] Ankeny, N.C., & Rivlin, T.J. (1955). On a theorem of S.Bernstein. Pacific J. Math, 5, 849–851.
[2] Aziz, A. (1987). Growth of polynomials whose zeros are within or outside a circle. Bulletin of the Australian Mathematical Society, 35(2), 247–256.
[3] Aziz, A., & Dawood, Q.M. (1988). Inequalities for a polynomial and its derivative. Journal of Approximation Theory, 54(3), 306–313.
[4] Aziz, A., & Rather, N.A. (2004). Some compact generalizations of Bernstein-type inequalities for polynomials. Mathematical Inequalities and Applications, 7(3), 393–403.
[5] Bernstein, S. (1930). Sur la limitation des dérivées des polynômes. C. R. Acad. Sc. Paris, 190, 338–340.
[6] Jain, V.K. (2000). Inequalities for a polynomial and its derivative. Proc. Indian Acad. Sci, 110, 137–146.
[7] Lax, P.D. (1944). Proof of a conjecture of P.Erdos on the derivative of a polynomial. Bull. Amer. Math. Soc, 50, 509–513.
[8] Malik, M.A. (1969). On the Derivative of a Polynomial. Journal of the London Mathematical Society, 1, 57–60.
[9] Polya, G., & Szego, G. (1976). Problems and Theorems in Analysis. Springer-Verlag.
[10] Soleiman Mezerji, H.A., Bidkham, M., & Zireh, A. (2012). Bernstien type inequalities for polynomial and its derivative. Journal of Advanced Research in Pure Mathematics, 4, 26–33.
[11] Zargar, B.A. (2014). On an inequality of Paul Turan. Matematički Vesnik, 66(2), 148–154. 

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