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 AM  Vol.2 No.9 , September 2011
On the Growth and Polynomial Coefficients of Entire Series
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)|=Σpk(z) [q(z)]k-1 where q(z) is a polynomial of degree m and pk(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.
Cite this paper: nullH. Khan and R. Ali, "On the Growth and Polynomial Coefficients of Entire Series," Applied Mathematics, Vol. 2 No. 9, 2011, pp. 1124-1128. doi: 10.4236/am.2011.29155.
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.

 
 
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