# New Sharp Approximations Involving Incomplete Gamma Functions

New Sharp Approximations Involving Incomplete Gamma Functions In the article, we deal with the monotonicity of the function $$x\mapsto x(c+be^{-ax^{p}}){/}\int _{0}^{x}e^{-t^{p}}dt$$ x ↦ x ( c + b e - a x p ) / ∫ 0 x e - t p d t for $$a,b,c,p>0$$ a , b , c , p > 0 on $$(0,\infty )$$ ( 0 , ∞ ) and prove that the double approximating inequality \begin{aligned} 1-\frac{1-e^{-ax^{p}}}{a(p+1)}<\frac{1}{x}\int _{0}^{x}e^{-t^{p}}dt<1-\frac{ 1-e^{-bx^{p}}}{b(p+1)} \end{aligned} 1 - 1 - e - a x p a ( p + 1 ) < 1 x ∫ 0 x e - t p d t < 1 - 1 - e - b x p b ( p + 1 ) holds for all $$x>0$$ x > 0 if and only if $$0<a\le 1/(p+1)$$ 0 < a ≤ 1 / ( p + 1 ) and $$b\ge (p+1)/(2p+1)$$ b ≥ ( p + 1 ) / ( 2 p + 1 ) . Also, some approximating properties concerning incomplete gamma function and exponential integral are established. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# New Sharp Approximations Involving Incomplete Gamma Functions

, Volume 72 (2) – Jul 10, 2017
14 pages

/lp/springer_journal/new-sharp-approximations-involving-incomplete-gamma-functions-s3g60UgAz5
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-017-0713-5
Publisher site
See Article on Publisher Site

### Abstract

In the article, we deal with the monotonicity of the function $$x\mapsto x(c+be^{-ax^{p}}){/}\int _{0}^{x}e^{-t^{p}}dt$$ x ↦ x ( c + b e - a x p ) / ∫ 0 x e - t p d t for $$a,b,c,p>0$$ a , b , c , p > 0 on $$(0,\infty )$$ ( 0 , ∞ ) and prove that the double approximating inequality \begin{aligned} 1-\frac{1-e^{-ax^{p}}}{a(p+1)}<\frac{1}{x}\int _{0}^{x}e^{-t^{p}}dt<1-\frac{ 1-e^{-bx^{p}}}{b(p+1)} \end{aligned} 1 - 1 - e - a x p a ( p + 1 ) < 1 x ∫ 0 x e - t p d t < 1 - 1 - e - b x p b ( p + 1 ) holds for all $$x>0$$ x > 0 if and only if $$0<a\le 1/(p+1)$$ 0 < a ≤ 1 / ( p + 1 ) and $$b\ge (p+1)/(2p+1)$$ b ≥ ( p + 1 ) / ( 2 p + 1 ) . Also, some approximating properties concerning incomplete gamma function and exponential integral are established.

### Journal

Results in MathematicsSpringer Journals

Published: Jul 10, 2017

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