On the order of convexity for the shifted hypergeometric functions

Theorem 1.1. For real parameters a, b and c none of which are negative integers satis- fying c −a ̸∈−N and c −b ̸∈−N, if one of the following conditions holds: (1) 0 < ab < 1 and a + b ≤c < 1 + a + b −ab; (2) ab < 0 and a + b ≤c < 1 + a + b; then the order of convexity of the function z2F1(a, b; c; z) is −∞. Note that Theorem 1.1 generalizes the case (d) in Theorem A and a result due to K¨ustner in (6, Corollary 9, case (e)). Letting c = a+b in Theorem 1.1, we obtain a result on the nonconvexity

Corollary 1.2. If a and b are real constants satisfying a ̸= 0, −1, −2, · · ·, b ̸= 0, −1, −2, · · · and ab < 1, then the shifted zero-balanced hypergeometric function z2F1(a, b; a + b; z) is not convex. By applying the continued fraction representations of the ratio of two hypergeometric functions, we get the following results on the order of convexity of the shifted hypergeo- metric functions.

Theorem 1.3. Suppose b and c are real parameters with 0 < b ≤c. (1) If c ≥2, then κ(z2F1(1, b; c; z)) = 4 −b −c 2 + c −2 2 2F1(1, b; c; −1) 2F1(2, b; c; −1). (2) If 1 ≤c < min{2, 1 + b}, then κ(z2F1(1, b; c; z)) = (c −b)(c + b −3) 2(1 + b −c) . It is worth to point that the order of convexity of z2F1(1, b; c; z) with real parameters b and c satisfying 0 ≤b ≤c and 1 + b ≤c < 2 is already shown in Theorem 1.1.

Theorem 1.5. For real parameters a, b and c satisfying 0 < a < 1, a ≤c and 0 ≤b ≤c, the following hold for the order of convexity of the function z2F1(a, b; c; z): (1) If 1 −b > 0 and c −2 + (1 −a)(1 −b) ≥0, then κ(z2F1(a, b; c; z)) = 5 −c −a −b 2 + M(−1). (2) If 1 −b < 0 and c −2 + (1 −a)(1 −b) ≤0, then κ(z2F1(a, b; c; z)) = 5 −c −a −b 2 + M(1).

Corollary 1.7. For 0 < a < 1, a ≤c and 0 < b ≤min{1, c}, then the order of convexity for the shifted hypergeometric function z2F1(a, b; c; z) satisfies κ ≥        (4 −ab)c −ab(5 −a −b) 2(2c −ab) , c ≥3 −a −b + ab 2c + (a2 −5a + 2)b 2(b + c −ab)

Corollary 1.8. For 0 < a < 1 < b ≤c < min{a + b, 1 + a + b −ab}, then the order of convexity for the shifted hypergeometric function z2F1(a, b; c; z) is c2 −a2 −b2 + 3(a + b −c) −2 2(a + b −c) .

Corollary 1.10. (1) The shifted hypergeometric function z2F1(a, b; c; z) is convex in D if 0 < a < 1, 0 < b < 1 and c ≥1 + a + b −ab. (2) For real parameters a, b and c with 0 < a < 1 < b ≤c, the shifted hypergeometric function z2F1(a, b; c; z) is not convex in D if one of the following conditions holds: (a) ab < 1 and c < a + b; (b) ab > 1 and c < min{1 + a + b −ab, 3+√ 9+4(a2+b3−3a−3b+2) 2 }.

Lemma 2.1 ((5), Thm. 1.5, (17), p.337-339 and Thm. 69.2). If −1 ≤a ≤c and 0 ≤b ≤c ̸= 0, the ratio of two hypergeometric functions can be written in continued fraction and integral as 2F1(a + 1, b; c; z) 2F1(a, b; c; z) = 1 1 − (1−g0)g1z 1−(1−g1)g2z 1−... = Z 1

Theorem 1.5, the next two lemmas deal with the estimations at z = −1 and the behaviors around z = 1 respectively.

Lemma 2.2. If −1 ≤a ≤c and 0 ≤b ≤c ̸= 0, then c b + c ≤2F1(a + 1, b; c; −1) 2F1(a, b; c; −1) ≤2c −b 2c .

Lemma 2.3. Let a, b and c be real constants with a, b, c ̸∈−N, c−a ̸∈−N and c−b ̸∈−N. (1) If a + b < c < a + b + 1, then (2.2) 2F1(a + 1, b; c; z) 2F1(a, b; c; z) = A (1 −z)1−α + O(|1 −z|ε−1) where A = Γ(a + b + 1 −c)Γ(c −a)Γ(c −b) Γ(a + 1)Γ(b)Γ(c −a −b) , α = c −a −b ∈(0, 1) and ε = min{2α, 1}. (2) If a + b = c, then (2.3)

Lemma 2.4 ((13)). Let F(z) be analytic in the slit domain C \ [1, +∞). Then F(z) = Z 1 0 dµ(t) 1 −tz for some probability measure µ on [0, 1], if and only if the following conditions are fulfilled: (1) F(0) = 1; (2) F(x) ∈R for x ∈(−∞, 1); (3) Im F(z) ≥0 for Im z > 0; (4) lim n→∞F(zn)/zn = 0 for some sequence zn ∈C with Im zn →+∞, and Im zn ≥ δ Re zn for some positive constant δ; (5) lim sup x→+∞F(−x) ≥0.

Lemma 2.5. If f(z) = Z 1 0 dµ(t) 1 −tz

Corollary 4.3. Assume b and c are real constants satisfying 0 < b ≤c, the function z2F1(1, b; c; z) is convex if one of the following conditions holds: (1) 0 ≤b ≤1 and c ≥2; (2) 1 < b < 2 ≤c < 4b−b2 2−2b ; (3) 1 ≤c < min{2, 1 + b} and c ≥3 −b.