# A Note on the Hypergeometric Mean Value

A Note on the Hypergeometric Mean Value Recent efforts to obtain bounds for the complete elliptic integral % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $${\pi \over 2} \cdot {_2F_1} \bigg(-{1 \over 2},{1 \over 2}; 1;r^{2}\bigg)$$ in terms of power means and other related means have precipitated the search for similar bounds for the more general 2 F 1(α,β;γ;r). In an early paper, B. C. Carlson considered the approximation of the hypergeometric mean values (2 F 1(−a,b;b + c;r))1/a in terms of means of order t, given by M t(s,r):= (1 − s) + s(1 − r)t 1/t . In this note, a refinement of one such approximation is established by first proving a general positivity result involving 3 F 2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# A Note on the Hypergeometric Mean Value

, Volume 1 (1) – Mar 7, 2013
8 pages

/lp/springer-journals/a-note-on-the-hypergeometric-mean-value-F0paxkJKEJ
Publisher
Springer Journals
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03320978
Publisher site
See Article on Publisher Site

### Abstract

Recent efforts to obtain bounds for the complete elliptic integral % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $${\pi \over 2} \cdot {_2F_1} \bigg(-{1 \over 2},{1 \over 2}; 1;r^{2}\bigg)$$ in terms of power means and other related means have precipitated the search for similar bounds for the more general 2 F 1(α,β;γ;r). In an early paper, B. C. Carlson considered the approximation of the hypergeometric mean values (2 F 1(−a,b;b + c;r))1/a in terms of means of order t, given by M t(s,r):= (1 − s) + s(1 − r)t 1/t . In this note, a refinement of one such approximation is established by first proving a general positivity result involving 3 F 2.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: Mar 7, 2013

### References

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