Inequalities for the Gaussian hypergeometric function |
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Authors: | YingQing Song PeiGui Zhou YuMing Chu |
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Affiliation: | 1. School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China 2. College of Science and Art, Zhejiang Sci-Tech University, Hangzhou, 320021, China
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Abstract: | we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(?1/2, 1/2; 1; 1 ? x c ) and F(?1/2 ? δ, 1/2 + δ; 1;1 ? x d ) on (0, 1) for given 0 < c ? 5d/6 < ∞ and δ ∈ (?1/2, 1/2), and find the largest value δ 1 = δ 1(c, d) such that inequality F(?1/2, 1/2; 1; 1 ? x c ) < F(?1/2 ? δ, 1/2 + δ; 1; 1 ? x d ) holds for all x ∈ (0, 1). Besides, we also consider the Gaussian hypergeometric functions F(a?1 ?δ, 1-a+δ; 1;1 ?x 3) and F(a?1, 1 ?a; 1; 1?x 2) for given a ∈ [1/29, 1) and δ ∈ (a?1, a), and obtain the analogous results. |
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Keywords: | Gaussian hypergeometric function monotonicity inequality |
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