On the Growth and Polynomial Coefficients of Entire Series

Huzoor H. Khan, Rifaqat Ali
DOI: 10.4236/am.2011.29155                

Abstract

In this paper we have generalized some results of Rahman [1] by considering the maximum of |f(z)| over a certain lemniscate instead of considering the maximum of|f(z)|, for |z|=r and obtain the analogous results for the entire function |f(z)|=Σp_k(z) [q(z)]^k-1 where q(z) is a polynomial of degree m and p_k(z)is of degree m-1. Moreover, we have obtained some inequalities on the lover order, type and lower type in terms of polynomial coefficients.

Keywords

Lemniscate, Lower Order, Lower Type, Slowly Changing Function, Polynomial Coefficients and Entire Functions.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Q. I. Rahman, “On the Coefficients of an Entire Series of Finite Order,” Math Student, Vol. 25, 1957, pp. 113-121.
[2]	R. P. Boas, “Entire Functions,” Academic Press, New York, 1954, pp. 9-11.
[3]	J. L. Walsh, “Interpolation and Approximation,” American Mathematical Society, Colloquim Publications, Providence, Vol. 20, 1960, p. 56.
[4]	P. Borwein, “The Arc Length of the Lemniscate {|p(z)| = 1},” Proceedings of the AMS—American Mathematical Society, Vol. 123, 1995, pp. 797-799.
[5]	J. R. Rice, “A Characterization of Entire Functions in Terms of Degree of Convergence,” Bulletin of the American Mathematical Society, Vol. 76, 1970, p. 129. doi:10.1090/S0002-9904-1970-12396-5
[6]	J. R. Rice, “The Degree of Convergence for Entire Functions,” Duke Mathematical Journal, Vol. 38, No. 3, 1971, pp. 429-440. doi:10.1215/S0012-7094-71-03852-X
[7]	O. P. Juneja, “On the Coefficients of Entire Series,” Journal of Mathematical Analysis, Vol. 24, 1971, pp. 395-401.
[8]	O. P. Juneja and G. P. Kapoor, “Polynomial Coefficients of Entire Series,” Yokohama Mathematical Journal, Vol. 22, 1974, pp. 125-133.
[9]	D. Kumar, “Approximation Error and Generalized Orders of an Entire Function,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 25, No. 2, 2009, pp. 225- 235.
[10]	D. Kumar and H. Kaur, “Lp—Approximation Error and Generalized Growth Parameters of Analytic Functions in Coratheodory Domains,” International Journal of Mathematical Analysis, Vol. 3, No. 30, 2009, pp. 1461-1472.