Growth of the Maximum Modulus of Polynomials With Prescribed Zeros
DOI:
https://doi.org/10.5644/SJM.07.1.02Keywords:
Polynomials, maximum modulus, zeros, extremal problemsAbstract
If $p(z)$ be a polynomial of degree $n$ which does not vanish in the disk $\left|z\right|<k$, then for $k=1$, it is
well known that
\begin{align*}
\max_{\left|z\right|=r<1}\left|p(z)\right|&\geq
\left(\frac{r+1}{2}\right)^n\max_{\left|z\right|=1}\left|p(z)\right|,\\
\intertext{and} \max_{\left|z\right|=R>1}\left|p(z)\right|&\leq
\frac{R^n+1}{2}\max_{\left|z\right|=1}\left|p(z)\right|.\end{align*}
In this paper, we consider a class of lacunary polynomials and present certain generalizations as well as improvements of the above inequalities for the two cases $k\geq 1$ and $k<1$.
2000 Mathematics Subject Classification. 30A10, 30C10, 30C15
Downloads
References
N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math., 5 (1955), 849–852.
A. Aziz, Growth of polynomials whose zeros are within or outside a circle, Bull. Austral. Math. Soc., 35 (1987), 247–256.
K. K. Dewan and Sunil Hans, Growth of polynomials whose zeros are outside a circle, Ann. Univ. Mariae Curie-Sklodowska, Lublin-Polonia, LXII (2008), 61–65.
K. K. Dewan and Sunil Hans, On extremal properties for the derivative of polynomials, Math. Balk., 23 (2009), 1–8.
N. K. Govil and Q. I. Rahman, Functions of exponential type not vanishing in a half plane and related polynomials, Trans. Amer. Math. Soc., 137 (1969), 501–517.
G. P´olya and G. Szeg¨o, Aufgaben und Lehrsatze aus der Analysis, Springer–Verlag, Berlin, (1925).
T. J. Rivlin, On the maximum modulus of polynomials, Amer. Math. Monthly, 67 (1960), 251–253.
