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.

Introduction

The Gaussian hypergeometric function with real parameters , and c is defined by [1, 4, 24, 41] for , where for , and for . The function is called zero-balanced if . The asymptotical behavior for as is as follows (see [4, Theorems 1.19 and 1.48]) where [10, 25, 43, 44, 47] and are the classical gamma and beta functions, respectively, and ), and is the Euler–Mascheroni constant [21, 50].

As is well known, making use of the hypergeometric function, Branges proved the famous Bieberbach conjecture in 1984. Since then, and its special cases and generalizations have attracted attention of many researchers, and was studied deeply in various fields [2, 5, 9, 11–18, 20, 22, 23, 26, 30, 31, 35–37, 40, 45, 46, 48]. A lot of geometrical and analytic properties, and inequalities of the Gaussian hypergeometric function have been obtained [3, 6–8, 19, 29, 32, 34, 38, 49].

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), (15.8.21)] Wang and Chu [32] found the maximal regions of the -plane in the first quadrant such that inequality or its reversed inequality holds for each . Moreover, very recently in [33], some Landen-type inequalities for a class of Gaussian hypergeometric function , 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] have been derived. In fact, the authors have proved

Theorem 1.1

For , let , then the Landen-type inequality holds for all .

Theorem 1.2

For , define the function g on by Then g is strictly increasing from onto . In particular, the inequality holds for each with .

The purpose of this paper is to establish several quadratic transformation inequalities for Gaussian hypergeometric function , such as inequalities (1.4), (1.5) and (1.7), and thereby prove the analogs of Theorem 1.2.

We recall some basic facts about (see [33]). The limiting values of at 0 and 1 are and the derivative formula of is 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 for with and , and For the convenience of readers, we introduce some regions in and refer to Fig. 1 for illustration:

Figure 1

The regions for , and their boundary curves

Obviously, and for except that . Moreover, and .

Lemma 2.1

([42, Theorem 2.1])

Suppose that the power series and have the radius of convergence with for all . Let and , then the following statements hold true:

If the non-constant sequence is increasing (decreasing) for all , then is strictly increasing (decreasing) on ;

If the non-constant sequence is increasing (decreasing) for and decreasing (increasing) for , then is strictly increasing (decreasing) on if and only if . Moreover, if , then there exists an such that is strictly increasing (decreasing) on and strictly decreasing (increasing) on .

Lemma 2.2

The function is strictly decreasing on if and strictly increasing on if . Moreover, if , then there exists such that is strictly increasing (decreasing) on and strictly decreasing (increasing) on .

The function is strictly decreasing on if and strictly increasing on if . In the remaining case, namely for , is piecewise monotone on .

Proof

Suppose that then we have It suffices to take into account the monotonicity of . By simple calculations, one has where

We divide the proof into four cases.

Case 1 . Then it follows easily that , and . This, in conjunction with (2.4) and (2.5), implies that is strictly decreasing for all . Therefore, (2.3) and Lemma 2.1(1) lead to the conclusion that is strictly decreasing on .

Case 2 . Then a similar argument as in Case 1 yields and this implies that is strictly increasing on from (2.3), (2.4) and Lemma 2.1(1).

Case 3 . It follows from (2.4) and (2.5) that the sequence is increasing for and decreasing for for some integer . Furthermore, making use of the derivative formula for Gaussian hypergeometric function and in conjunction with (1.1) and , we obtain as . Combing with (2.3), (2.6) and Lemma 2.1(2), we conclude that there exists an such that is strictly increasing on and strictly decreasing on .

Case 4 . In this case, we follow a similar argument as in Case 3 and use the fact that as since . Therefore, (2.3), (2.7) and Lemma 2.1(2) lead to the conclusion that there exists an such that is strictly decreasing on and strictly increasing on .

Let then we can write Easy calculations lead to the conclusion that the monotonicity of depends on the sign of

Notice that where 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). □

Lemma 2.3

Let and for , then the function is strictly increasing on if .

Taking the derivative of yields where

We clearly see from (1.1) that for . This implies, in conjunction with (2.15), that where

It follows from the definition of hypergeometric function that where

If , namely, and , we can verify This, in conjunction with (2.17) and (2.18), implies that for . Therefore, is strictly increasing on , which follows from (2.14) and (2.16) if . □

Remark 2.4

The function defined in Lemma 2.3 is not monotone on if two positive numbers satisfy , since and Lemma 2.1(1) shows the monotonicity of on if . In the remaining case , it follows from (2.15) that . This, in conjunction with (2.14), implies that is strictly increasing on for a sufficiently small . This enables us to find a sufficient condition for with such that is strictly increasing on in Lemma 2.3.

The following corollary can be derived immediately from the monotonicity of in Lemma 2.3 and the quadratic transformation equality (1.3).

Corollary 2.5

Let , if , then the inequality holds for all .

Main results

Theorem 3.1

The quadratic transformation inequality holds for all with if and only if and the reversed inequality takes place for all if and only if , with equality only for .

In the remaining case , neither of the above inequalities holds for all .

Suppose that , then we clearly see that for . It follows from Lemma 2.1(1) that for and for . This, in conjunction with the quadratic transformation formula (1.3), implies for , and it degenerates to the quadratic transformation equality for . This completes the proof of (3.1).

Inequality (3.2) can be derived analogously, and the remaining case follows easily from Lemma 2.2(1). □

Theorem 3.2

We define the function for with and . Let and . Then the following statements hold true: As a consequence, the inequality holds for all if , and the following inequality is valid for all : if .

If , then is strictly increasing (resp., decreasing) from onto (resp., );

If , then is strictly increasing from onto ;

If , then is strictly decreasing from onto .

Let , then we clearly see that

Taking the derivative of with respect to r and using (3.5) yields

We substitute for r in the quadratic transformation equality (1.3), then differentiate it with respect to r to obtain in other words,

If , then it follows from Lemma 2.2(2) that is strictly decreasing on . This, in conjunction with , implies that , that is,

Combing (3.6), (3.7) with the inequality (3.8), we clearly see that

It follows from Lemma 2.2(1) that is strictly increasing on if . This, in conjunction with (3.9), implies that is strictly increasing on if .

Analogously, if , then we obtain the following inequality: By using a similar argument as above, we have since is strictly increasing on (0,1) if by Lemma 2.2(1). Hence, is strictly decreasing on if .

Notice that and

Therefore, we obtain the desired assertion from (3.10). □

Theorem 3.3

If we define the function for , then is strictly increasing from onto . As a consequence, the inequality holds for all if .

Remark 2.4 enables us to consider the case for . Note that and

Let and . Then

Taking the derivative of and using (3.12) leads to

Therefore, the monotonicity of 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 . As applications, we provide the analogs of duplication inequalities for the generalized Grötzsch ring function introduced in [33].

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.