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Inequalities for the Gaussian hypergeometric function

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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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Authors and Affiliations

  1. School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China

    YingQing Song & YuMing Chu

  2. College of Science and Art, Zhejiang Sci-Tech University, Hangzhou, 320021, China

    PeiGui Zhou

Authors
  1. YingQing Song
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  2. PeiGui Zhou
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  3. YuMing Chu
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Correspondence to YuMing Chu.

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Song, Y., Zhou, P. & Chu, Y. Inequalities for the Gaussian hypergeometric function. Sci. China Math. 57, 2369–2380 (2014). https://doi.org/10.1007/s11425-014-4858-3

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  • DOI: https://doi.org/10.1007/s11425-014-4858-3

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