New bounds for the function involving incomplete gamma function

Original Paper
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Abstract

This paper reveals a new relation between the two functions \( A_{p}\left( x\right) =\) \(\left( \int _{0}^{x}e^{-t^{p}}dt\right) /x\) and \(B_{q}\left( x\right) =\) \(1-\left( 1-e^{-qx^{p}}\right) /\left( q\left( p+1\right) \right) \) . At the same time, we study the upper and lower bounds for the function \(A_{p}\left( x\right) /B_{\alpha }\left( x\right) \) in another sense.

Keywords

Bounds Incomplete gamma function Gamma function Exponential integral Integral inequality 

Mathematics Subject Classification

Primary 33B20 Secondary 33A23 

Notes

Acknowledgements

This paper is supported by the Natural Science Foundation of China Grants no. 11471285 and the Natural Science Foundation of China Grants no. 61772025.

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Gongshang UniversityHangzhouChina

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