Quadratic transformation inequalities for Gaussian hypergeometric function

In the article, we present several quadratic transformation inequalities for Gaussian hypergeometric function and find the analogs of duplication inequalities for the generalized Grötzsch ring function.

Recently, in order to investigate the Ramanujan's generalized modular equation in number theory, Landen inequalities, Ramanujan cubic transformation inequalities, and several other quadratic transformation inequalities for zero-balanced hypergeometric function have been proved in [27,28,32,39,42]. For instance, using the quadratic transformation formula [24, (15.8.15) Wang and Chu [32] found the maximal regions of the (a, b)-plane in the first quadrant such that inequality holds for each r ∈ (0, 1). Moreover, very recently in [33], some Landen-type inequalities for a class of Gaussian hypergeometric function 2 F 1 (a, b; (a + b + 1)/2; x) (a, b > 0), which can be viewed as a generalization of Landen identities of the complete elliptic integrals of the first kind have also been proved. As an application, the analogs of duplication inequalities for the generalized Grötzsch ring function with two parameters [33]  holds for all r ∈ (0, 1).
We recall some basic facts about μ a,b (r) (see [33]). The limiting values of μ a,b (r) at 0 and 1 are 9) and the derivative formula of μ a,b (r) is 1 r a+b r a+b+1 F(a, b; (a + b + 1)/2; r 2 ) 2 . (1.10) Here and in what follows, .

Lemmas
In order to prove our main results, we need several lemmas, which we present in this section. Throughout this section, we denote and H f ,g = (f /g )gf , then the following statements hold true: n=0 is increasing (decreasing) for 0 < n ≤ n 0 and decreasing (increasing) for n > n 0 , then h(x) is strictly increasing (decreasing) on (0, r) if and only if H f ,g (r -) ≥ (≤)0. Moreover, if H f ,g (r -) < (>)0, then there exists an x 0 ∈ (0, r) such that h(x) is strictly increasing (decreasing) on (0, x 0 ) and strictly decreasing (increasing) on (x 0 , r).
In the remaining case, namely for
Case 3 (a, b) ∈ D 3 . It follows from (2.4) and (2.5) that the sequence {A n /A * n } is increasing for 0 ≤ n ≤ n 0 and decreasing for n ≥ n 0 for some integer n 0 . Furthermore, making use of the derivative formula for Gaussian hypergeometric function and in conjunction with (1.1) and a + b < 1, we obtain 3), (2.6) and Lemma 2.1(2), we conclude that there exists an x 1 ∈ (0, 1) such that η(x) is strictly increasing on (0, x 1 ) and strictly decreasing on (x 1 , 1). Case 4 (a, b) ∈ D 4 . In this case, we follow a similar argument as in Case 3 and use the fact that as x → 1since a + b > 1. Therefore, (2.3), (2.7) and Lemma 2.1(2) lead to the conclusion that there exists an x 2 ∈ (0, 1) such that η(x) is strictly decreasing on (0, x 2 ) and strictly increasing on (x 2 , 1). Let then we can write Notice that where ω(a, b; x) = 64(a + 1)(b + 1) It follows easily from (1.1) and (2.11) that Employing similar arguments mentioned in part (1), we obtain the desired assertions easily from (2.8)-(2.12).
Proof Remark 2.4 enables us to consider the case for a + b > 1. Note that φ(1 -) = 0 and Therefore, the monotonicity of φ(r) follows immediately from (2.19) and (3.13). This, in conjunction with (3.11), gives rise to the desired result.

Results and discussion
In the article, we establish several quadratic transformation inequalities for Gaussian hypergeometric function 2 F 1 (a, b; (a + b + 1)/2; x) (0 < x < 1). As applications, we provide the analogs of duplication inequalities for the generalized Grötzsch ring function

Conclusion
We find several quadratic transformation inequalities for the Gaussian hypergeometric function and Grötzsch ring function. Our approach may have further applications in the theory of special functions.