### Weighted Turán type inequality for the derivative of real algebraic polynomials which have complex zeros

#### Abstract

In this paper, we establish a weighted Turán type inequality with N-doubling weights for the polynomials of degree at most *n* with real coefficients that have at most *r* complex zeros, which *r* is a fixed positive integer and n is sufficiently large. This result gives a partial answer to an open problem in [11].

#### Keywords

#### Full Text:

PDF#### References

Bakenko, V.F., Pichugov, S.A., 1986. “ Accurate inequality for the derivative of trigonometric polynomials which have real zeros” (in Russian), Mat. Zameki, 39(1986), 330-336.

Bakenko, V.F., Pichugov, S.A., 1986. “Inequality for the derivative of polynomials with real zeros” (in Russian), Ukrain. Math. Zh., 38(1986), 411-416.

Erǒd, J., 1939. “Bizonyos polinomok maximumairol”, Math. Fiz. Lapok, 46(1939), 58-82.

Milovanovic, G.V., Rassias, T.M., 1998. “New developments on Turán's extremal problems for polynomials”, In Approximation Theory : In Memory of A. K. Varma, ( Marcel Dekker, Inc., New York, 1998), 433-447.

Turán, P., 1939. “Uber dir ableitung von polynomen”, it Compositio Math., 7(1939), 89-95.

Varma, A.K., 1976. “An analogue of some inequality of P. Turán concerning algebraic polynomials satisfying certain conditions”, Proc. Amer. Math. Soc., 55(1976), 305-309.

Varma, A.K., 1978. “ An analogue of some inequality of P. Turán concerning algebraic polynomials having all zeros inside [-1, 1]”, Proc. Amer. Math. Soc., 69(1978), 25-33.

Varma, A.K., 1983. “Some inequalities of algebraic polynomials having all zeros inside [-1, 1]”, Proc. Amer. Math. Soc., 88(1983), 227-233.

Wang, J.L., Zhou, S.P., 2002. “The weighted Turán type inequality for generalized Jacobi weights”, Bull. Austral. Math. Soc. 66(2002), 259-265.

Wei, B.R., Yu, D.S., 2005. “On Turán type inequality fwith doubling weights and A* weights”. J. Zhejiang Univ. Sci 6A, 7(2005), 764-768.

Xie, T.F., Zhou, S. P, 1998. “Approximation Theory of Real Functions” (in Chinese), Hangzhou Univ. Press, Hangzhou, 1998.

Yu, D.S., Zhou, S.P., 2005. “Turán’s type for rational functions with prescribed poles”, Acta Math. Hungar., 109(2005), 281-287.

Zhou, S. P., 1992. “Some remark On Turán type inequality” , J Approx. Theory, 68(1992), 45-48.

Zhou, S. P., 1993. “Some remark On Turán type inequality II”, J. Math. Anal. Appl., 180(1993), 138-143.

Zhou, S. P., 1995. “Some remark On Turán type inequality III: the completion”, Anal. Math., 21(1995), 313-318.

Zhao, D.J., Zhou, S.P., Yu, D.S., Wang, J.L., 2014. “Weighted Turán type inequality for rational functions with prescribed poles”. J. math. Ineq. 2(2014), 251-265.

### Refbacks

- There are currently no refbacks.

Copyright (c) 2017 Global Journal of Mathematics

Copyright © 2016 by** Global Publishing Corporation**

**ISSN: 2395-4760**

For any Technical Support contact us at **editor@gpcpublishing.org**, **editorglobalpublishing@gmail.com**.