Monotonicity, convexity properties and inequalities involving Gaussian hypergeometric functions with applications

In this paper, we mainly prove monotonicity and convexity properties of certain functions involving zero-balanced Gaussian hypergeometric function $ F(a, b; a+b; x) $. We generalize conclusions of elliptic integral to Gaussian hypergeometric function, and get some accurate inequalities about Gaussian hypergeometric function.

For r ∈ (0, 1), one kind of known elegant functional inequalities for K a (r) are of the following form sin(πa) , with constants c 1 , c 2 ∈ (0, 1). For example, Wang and Chu [19] proved that sin(πa) In present paper, we try to generalize the above inequality (1.9) to zero-balanced hypergeometric function.
It is natural to think about the monotonicity of the reciprocal of F(x), that is 11) and the relationship between the monotonicity of the function f (x) and the value of R(a, b). Thus we consider the following questions: is the function f (x) stictly increasing or decreasing on (0, 1)? What's the relationship between the monotonicity of the function f (x) and the value of R(a, b)?
Let f (x) be in (1.11) and (1.14) In [7], Huang, Qiu and Ma considered the above functions for the particular case of a + b = 1 and obtained the following theorems: (1) f (x) is convex on (0, 1).
For the particular case of a + b = 1 , Theorem 1.3 and 1.4 actually obtain the conclusions of generalized elliptical integral K a (r). In light of the above results, we are trying to extend the above theorem to the zero-balanced hypergeometric function and it is natural to consider the following questions: The purpose of this paper is to give complete answers to Question 1.2 and 1.5. This paper is organized as follows. The preliminaries we needed are listed in Section 2, and the main results and their complete proofs of this paper are listed in Section 3. As applications, inequalities of hypergeometric function are displayed in Section 4.

Preliminaries
Before proving our main results, we firstly introduce the following important lemmas, which will be used in the proofs of main results. .
is strictly monotone, then the monotonicity in the conclusion is also strict.
Next by the monotonicity of φ(x) = xψ (x) and The proof of this lemma is completed.

Main results and proofs
In the following statement, we always let R = R(a, b), B = B(a, b) for (a, b) ∈ (0, ∞).
Remark 3.10. Theorem 3.8 is a general conclusion of Theorem 1.4 (1) in region D 1 ∩ D 2 and Theorem 3.9 is a general conclusion of Theorem 1.4 (2) in region D 3 .

Applications for inequalities
From our main Theorems, we can easily obtain several asymptotically sharp inequalities for F(a, b; a + b; x). Let x = r 2 , we can get the inequalities of generalized elliptic integral K a (r).   /(a + b)).
Remark 4.6. Acorrding to the above Corollaries, let a + b = 1 and x = r 2 , we can obtain the conculsions of genelized ellptic integral K a (r) in [7].