#### Glasnik Matematicki, Vol. 34, No.2 (1999), 197-202.

### ON WEIGHTED TURAN TYPE INEQUALITY

### Wei Xiao and Songping Zhou

Ningbo University, The Mathematical Institute, Ningbo,
Zhejiang 315211, China
State Key Laboratory, Southwest Institute of Petroleum,
Nanhchong, Sucian 637001, China

*e-mail:* `spz@public.hz.zj.cn`

**Abstract.** Let *H*_{n} be the class
of real algebraic polynomials of degree *n*, whose zeros all
lie in the interval [-1, 1]. Define || *f* || =
max_{-1 ≤ x ≤ 1} | *f*(*x*)|.
In 1939, Turan proved that for *f*
∈ *H*_{n},
|| *f* ' || ≥ C√*n* || *f* ||.
Usually, to establish weighted inequalities for polynomials requires
more complicated techniques, especially in general cases. The object
of this paper is to start the work in uniform norm under general
requirements for weight functions.
**1991 Mathematics Subject Classification.**
26D10.

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