Application of the Euler's gamma function to a problem related to F. Carlson's uniqueness theorem

M. A. Qazi


In his work on F. Carlson's uniqueness theorem for entire functions of exponential type, Q. I. Rahman [5] was led to consider an infinite integral and needed to determine the rate at which the integrand had to go to zero for the integral to converge. He had an estimate for it which he was content with, although it was not the best that could be done. In the present paper we find a result about the behaviour of the integrand at infinity, which is essentially best possible. Stirling's formula for the Euler's Gamma function plays an important role in its proof.


Entire functions; Hadamard's three circles theorem; Euler's Gamma function

Full Text:



Boas, Jr., R. P., Entire Functions, Academic Press, New York, 1954.

Boas, Jr., R. P., A Primer of Real Functions, The Carus mathematical monographs, No. 13, The Mathematical Association of America, 1960.

Hardy, G. H., The mean value of the modulus of an analytic function, Proc. London Math. Soc. 14 (1915), 269-277.

Henrici, P., Applied and Computational Complex Analysis, Vol. 2, (A Wiley-

Interscience publication), John Wiley & Sons, New York, 1977.

Rahman, Q. I., Interpolation of Entire functions, Amer. J. Math. 87 (1965), 1029-1076.

Titchmarsh, E. C., The Theory of Functions, 2nd ed. Oxford University Press, 1939.

Valiron, G., Lectures on the General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949.

Data publikacji: 2016-07-04 15:44:01
Data złożenia artykułu: 2016-07-04 12:22:34


  • There are currently no refbacks.

Copyright (c) 2016 M. A. Qazi