On the convergence of certain integrals

Mohamed Amine Hachani

Abstract


Let \(M(r) := \max_{|z|=r} |f(z)|\), where \(f(z)\) is an entire function. Also let \(\alpha> 0\) and \(\beta>1\). We discuss the behavior of the integrand \(M(r)e^{-\alpha(log r)^\beta}\) as \(r \to \infty\) if \(\int_1^\infty M(r)e^{-\alpha(log r)^\beta}dr\) is convergent.

Keywords


Entire functions; Hadamard’s three-circles theorem; infinite integrals

Full Text:

PDF

References


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

Hardy, G. H., A Course of Pure Mathematics, Cambridge University Press, London, 1921.

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

Hardy, G. H., Rogosinski, W. W., Notes on Fourier series (III), Q. J. Math. 16 (1) (1945), 49–58.

Qazi, M. A., Application of the Euler’s gamma function to a problem related to F. Carlson’s uniqueness theorem, Ann. Univ. Mariae Curie-Skłodowska Sect. A 70 (1) (2016), 75–80.

Rahman, Q. I., On means of entire functions, Q. J. Math. 7 (1) (1956), 192–195.

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

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

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




DOI: http://dx.doi.org/10.17951/a.2019.73.1.19-25
Date of publication: 2019-12-19 10:33:45
Date of submission: 2019-12-17 09:41:01


Statistics


Total abstract view - 811
Downloads (from 2020-06-17) - PDF - 689

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2019 Mohamed Amine Hachani