Inequalities describing the growth of polynomials not vanishing in a disk of prescribed radius

Abstract

Abstract. In this paper we study the growth of polynomials of degree n having no zeros in |z| < κ , where κ is an arbitrary positive number. Using the notation M(p; t) = max |z|=t |p(z)| we measure the growth of p by estimating M(p; t)/M(p; 1) from above for any t > 1 , and from below for any t < 1 . (2000): 30A10, 30C10, 30E10; 30C15. Mathematics subject classification

Growth of a polynomial not vanishing in a disk

“…For the proof of Theorem 1, we need the following lemmas. The first lemma is due to Govil, Qazi and Rahman [3].…”

Comparison inequalities of Bernstein-type between polynomials with restricted zeros

“…The inequalities (1) and ( 2) are related with each other and it was observed by Bernstein [4] that (1) can be deduced from (2) by making use of Gauss-Lucas theorem and the proof of this fact was given by Govil et al [6]. If we restrict ourselves to the class of polynomials P ∈ P n with P (z) ̸ = 0 in |z| < 1, then (1) and ( 2) can be respectively replaced by…”

A note on sharpening of a theorem of Ankeny and Rivlin

“…It was proved by Bernstein himself that Theorem 3 can be obtained from Theorem 4. However, it was not known if Theorem 4 can also be obtained from Theorem 3, and this has been shown by Govil, Qazi and Rahman [11]. Thus both the Theorems 3 and 4 are equivalent in the sense that anyone can be obtained from the other.…”

“…Several papers, research monographs and books have been published in this area (see Boas [2], Borwein and Erdelyi [3], Frappier, Rahman and Ruscheweyh [7], Gardner, Govil and Weems [8], Govil [9, 10], Govil, Qazi and Rahman [11], Lorentz, Golitshek and Makovoz [12], Milovanović, Mitrinović and Rassias [15], Mitrinović, Pecaric and Fink [17], Milovanović and Rassia [16], Rahman and Schmeisser [19,20], Rassias, Srivastava, and Yanushauskas [21], Sharma and Singh [23] and Telyakovskii [24]).…”