摘要
we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 < c 5d/6 < ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) <F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.
we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F（-1/2,1/2;1;1- xc） and F（-1/2- δ,1/2 ＋ δ;1;1- xd） on（0,1） for given 0 c 5d/6 ∞ andδ ∈（-1/2,1/2）,and find the largest value δ1 = δ1（c,d） such that inequality F（-1/2,1/2;1;1- xc） F（-1/2- δ,1/2 ＋ δ;1;1- xd） holds for all x ∈（0,1）. Besides,we also consider the Gaussian hypergeometric functions F（a- 1- δ,1- a ＋ δ;1;1- x3） and F（a- 1,1- a;1;1- x2） for given a ∈ [1/29,1） and δ ∈（a- 1,a）,and obtain the analogous results.
基金
supported by National Natural Science Foundation of China(Grant Nos.11371125,11171307 and 61374086)
Natural Science Foundation of the Hunan Province(Grant No.14JJ2127)
Natural Science Foundation of the Zhejiang Province(Grant No.LY13A010004)
