Retracted: Growth of polynomials with prescribed zeros. II. (English) Zbl 1318.30004

Dewan, K. K.; Ahuja, Arty

Tbil. Math. J. 8, No. 2, 69-74 (2015); retraction note ibid. 8, No. 2, 194 (2015).

Summary: In this paper we consider a class of polynomials \(p(z) = c_nz^n+\sum^n_{j=\mu}c_{n-j}z^{n-j}\), \(1 \leq \mu \leq n\), having all its zeros on \(|z| = k\), \(k \leq 1\). Using the notation \(M(p, t) = \underset{|z|=t}\max |p(z)|\), we measure the growth of \(p\) by estimating \(\Big\{\frac{M(p,t)}{M(p,1)}\Big\}^s\) from above for any \(t \geq 1\), \(s\) being an arbitrary positive integer.
Editorial remark: This paper has been retracted by the authors, see [Zbl 1330.30013] for the retraction note.

MSC:

30A10	Inequalities in the complex plane
30C10	Polynomials and rational functions of one complex variable
30C15	Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Keywords:

polynomials; zeros; growth

Citations:

Zbl 1330.30013

References:

[1]	N. C. Ankeny and T.J. Rivlin, On a Theorem of S. Bernstein, Pacific J. Math., 5(1955), 849-852. · Zbl 0067.01001
[2]	K.K. Dewan and Arty Ahuja, Growth of polynomials with prescribed zeros, J. Math. Inequalities, to appear.
[3]	K.K.Dewan and Sunil Hans, On extremal properties for the derivative of polynomials, Mathematica Balkanica, 23 (2009), Fasc. 1-2, 27-36. · Zbl 1176.30002
[4]	K.K.Dewan and Sunil Hans, On maximum modulus for the derivative of a polynomial, Annales Univ. Mariae Curie-Sklodowska Lublin, LXIII (2009), 55-62. · Zbl 1190.30003
[5]	G. Pólya and G. Szegö, Aufgaben and Lehrsatze aus der Analysis, Springer-Verlag, Berlin, 1925.
[6]	M.A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115(1992), 337-343. · Zbl 0772.30006