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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.
Computational Methods and Function Theory – Springer Journals
Published: Mar 7, 2013
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