Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function

In the article, we present several new monotonicity properties and bounds involving the generalized Grötzsch ring functions $\mu _{a,b}$μa,b in the theory of Ramanujan’s generalized modular equation for $0< a, b<1$0<a,b<1. Our results are the variants and extensions of some previously known results.

Let 0 < a, b < 1 with a + b ≥ 1 and r ∈ (0, 1). Then the two-parameter generalized elliptic integrals of first and second kinds [34,  where and in what follows r = √ 1r 2 . Moreover, it follows from (1.2) that In this paper, we study the two-parameter generalized Grötzsch ring function μ a,b (r) for a, b ∈ (0, 1), as well as the related functions K a,b , E a,b , and The so-called Legendre M-function introduced in [35] can be used to study the derivative of m a,b (r) and satisfies the formula for r ∈ (0, 1). Furthermore, M(r) can be rewritten as (1.10) and M(r) becomes a constant if and only if a + b = 1, in which case M(r 2 ) degenerates to be the generalized Legendre relation.
The main purpose of the article is to find the sub-regions of {(a, b) ∈ R 2 |0 < a, b < 1, a + b > 1} such that certain quotient functions involving μ a,b (r), K a,b (r), E a,b (r), and m a,b (r) are monotonic on their corresponding sub-regions. As a consequence, several new bounds for μ a,b (r) and m a,b (r) are discovered, which are the variants and extensions of the results given in [42, Theorems 1.1 and 1.2] for the case of zero-balanced.

Notations, formulas, and lemmas
In order to prove our main results, we need several derivative formulas and lemmas, which we present in this section.

Notations
Throughout the article, we denote B(a, b) by B if no risk for confusion. Let For the convenience of readers, we also introduce three sub-regions Ω 1 , Ω 2 , and Ω 3 of {(a, b) ∈ R 2 |0 < a, b < 1}, which are illustrated in Fig. 1.
is strictly monotone, then the monotonicity in the conclusion is also strict.
Then the following assertions are valid: The function r a+b-1 K has positive Maclaurin coefficients and maps (0, 1) onto Proof Items (i) and (ii) follow directly from [34,Lemma 4.22]. We only need to prove item (iii).
It follows from (2.1) that Lemma 2.3(i) and (2.7) enable us to know that r p K is strictly decreasing Note that This in conjunction with (1.2) and (2.8) gives lim r→1 -r p K = 0.
In the following Lemma 2.4 we provide an asymptotic formula for K as r → 1 in the case of a + b > 1, which is the analog for the zero-balanced hypergeometric function (1.2).
Theorem 1.19(5)] as r → 1 for a + b > 1 and the derivative formula given in [19, (1.16)] for the hypergeometric function together with (1.2), and L'Hôpital's rule that This completes the proof. Lemma 2.4 leads to Corollary 2.5 immediately.
Proof By replacing r with 1r 2 in Lemma 2.4, we clearly see that (2.10) By definition, it is easy to know that (K -B/2)/r → 0 as r → 0. This in conjunction with (2.10) and a + b < 2 yields as r → 0. The second asymptotic formula can be proved by similar arguments.
On the other hand, it follows from L'Hôpital's rule and (2.12) that (a, b) ∈ Ω 1 and f (r) be defined by
• Lemma 2.1 and (2.22) enable us to know that the monotonicity of g 1 (r) depends on the monotonicity of the following sequence: .

Then F(r) is strictly decreasing from
.
Since F 1 (r)/ F 2 (r) = F 1 (r)/F 2 (r), Lemma 2.2 enables us to know the monotonicity of F(r) depends on that of F 1 (r)/F 2 (r), which follows from Theorem 3.1. It only remains to compute two limiting values F(0 + ) and F(1 -).
Proof Since H(r) is symmetric with respect to a, b, we may assume that 0 < a ≤ b < 1.

Consequences and discussion
In the article, we study the monotonicity of the functions F(r), G(r), and H(r) related to generalized Grötzsch ring function and generalized elliptic integrals, where F(r), G(r), and H(r) are explicitly given by

Conclusion
In the article, we have found the sub-regions of {(a, b) ∈ R 2 |0 < a, b < 1, a + b > 1} such that several quotient functions involving μ a,b (r), K a,b (r), E a,b (r), and m a,b (r) are monotonic on their corresponding sub-regions, and established several inequalities for μ a,b (r) and m a,b (r). Our results are the variants and extensions of the previous results of [42, Theorems 1.1 and 1.2] in the case of zero-balanced.