On quotients and differences of hypergeometric functions

For Gaussian hypergeometric functions $F(x)= F(a,b;c;x),$ $a,b,c>0,$ we consider the quotient $ Q_F(x,y)= (F(x)+F(y))/F(z)$ and the difference $ D_F(x,y)= F(x)+F(y)-F(z)$ for $0<x,y<1$ with $z=x+y-xy \,.$ We give best possible bounds for both expressions under various hypotheses about the parameter triple $(a,b;c)\,.$


Introduction
Among special functions, the hypergeometric function has perhaps the widest range of applications. For instance, several well-known classes of special functions such as complete elliptic integrals, Legendre functions, Chebyshev and Jacobi polynomials, and some elementary functions, such as the logarithm, are particular cases of it, cf. [AS]. In a recent paper [KMSV] the authors studied various extensions of the Bernoulli inequality for functions of logarithmic type. In particular, the zero-balanced hypergeometric function F (a, b; a + b; x), a, b > 0, occurs in these studies, because it has a logarithmic singularity at x = 1 , see (2.8) below. We now continue the discussion of some of the questions for quotients and differences of hypergeometric functions that were left open in [KMSV].
Our task in this paper is to give tight bounds for these quotients and differences assuming various relationships between the parameters a, b, c.
For the general case we can formulate the next Theorem 1.1. For a, b, c > 0 and 0 < x, y < 1 let (1.2) Q F (x, y) = F (a, b; c; x) + F (a, b; c; y) F (a, b; c; x + y − xy) .
The bounds in (1.3) are best possible as can be seen by taking [AS,15.1.8] Then Theorem 1.4. For a, b > 0, c > a + b and 0 < x, y < 1, we have ; c; 1) as the best possible constant.
Most intriguing is the zero-balanced case. For example, Theorem 1.6. For a, b > 0 and 0 < x, y < 1 let Then The constant 1 on the right-hand side is the best possible upper bound, but the best constant on the left-hand side is not known to us.
The proofs of these results are based on the monotonicity of some functions and thus they hold under much weaker hypotheses than stated above, see Remark 4.7.
In the sequel we shall give a complete answer to an open question posed in [KMSV].

Preliminary results
In this section we recall some well-known properties of the Gaussian hypergeometric function F (a, b; c; x) and certain of its combinations with other functions, for further applications. For basic information, the handbooks [AS, OLBC] may be recommended.
It is well known that hypergeometric functions are closely related to the classical gamma function Γ(x), the psi function ψ(x), and the beta function B(x, y). For Re x > 0, Re y > 0, these functions are defined by respectively (cf. [AS,Chap. 6] We shall also need the function where γ is the Euler-Mascheroni constant given by Given complex numbers a, b, and c with c = 0, −1, −2, . . . , the Gaussian hypergeometric function is the analytic continuation to the slit plane C \ [1, ∞) of the series Here (a, 0) = 1 for a = 0, and (a, n) is the shifted factorial function or the Appell symbol (a, n) = a(a + 1)(a + 2) · · · (a + n − 1) The hypergeometric function has the following simple differentiation formula ( [AS,15.2 An important tool for our work is the following classification of the behavior of the hypergeometric function near x = 1 in the three cases a + b < c, a + b = c, and a + b > c : Some basic properties of this series may be found in standard handbooks, see for example [AS]. For some rational triples (a, b, c), the function F (a, b; c; x) can be expressed in terms of well-known elementary function. A particular case that is often used in this paper is [AS,15.1 It is clear that for a, b, c > 0 the function F (a, b; c; x) is a strictly increasing map from [0, 1) into [1, ∞) . For a, b > 0 we see by (2.8) that xF (a, b; a + b; x) defines an increasing homeomorphism from [0, 1) onto [0, ∞) .

Main results
By (2.8) the zero-balanced hypergeometric function F (a, b; a + b; x) has a logarithmic singularity at x = 1 . We shall now demonstrate that its behavior is nearly logarithmic also in the sense that some basic identities of the logarithm yield functional inequalities for the zero-balanced function.
(1) For which values of c and d , this function is bounded from below and above?
We shall give a complete answer to this question in the sequel. Note firstly that the quotient Q g is always bounded. Namely, Theorem 3.3. For all c, d > 0 and all x, y ∈ (0, 1) we have that 0 < Q g (x, y) < 2.
A refinement of these bounds for some particular (c, d) pairs is given by the following two assertions. (2) For c, d > 0, 1/c + 1/d ≤ 2 and x, y ∈ (0, 1) we have .
Note that parts (1) and (3)  Theorem 3.6. For cd ≤ 1 and x, y ∈ (0, 1) we have We shall prove now the hypothesis from the second part of Question 3.2 under the condition 1/c + 1/d ≤ 2 in part b) which, in particular, includes the case c > 1, d > 1 .
Theorem 3.7. Fix c, d > 0 and let Q and g be as in Question 3.2.
Counterparts of Theorem 1.6 for the difference D g are given in the next assertion.
Theorem 3.8. Fix c, d > 0 and let D be as in Question 3.2.
Combining results above, we obtain the following two-sided bounds for the quotient Q g .
Corollary 3.9. Fix c, d > 0 and let Q be as in Question 3.2.
The assertions above represent a valuable tool for estimating quotients and differences of a hypergeometric function with different arguments. To illustrate this point, we give an example.
In the paper [KMSV], motivated by the relation g(x) = 2g(1 − √ 1 − x) with g as in (2.9), the authors asked the question about the bounds for the function S(t) defined by where g(t) := tF (a, b; a + b; t), a, b > 0.
An answer follows instantly applying Corollary 3.9.

Proofs
4.1. Proof of Theorem 1.1. The proof is based solely on the monotonicity property of the function F (x) = F (a, b; c; x). Namely, for x, y ∈ (0, 1), put z = x + y − xy, z ∈ (0, 1). Since x ≤ max{x, y}, y ≤ max{x, y}; z ≥ max{x, y}, and F (u) is monotone increasing in u, we obtain The left-hand bound is trivial.

4.2.
Proof of Theorem 1.4. The assertion of this theorem is a consequence of the previous one and (2.8). Indeed, from (1.3) we get −F (a, b; c; z) < D F (x, y) ≤ F (a, b; c; z), that is, 4.3. Proof of Theorem 1.6. By Theorem 2.10(1) we have that is strictly increasing on (0, 1) . Putting x = 1 − e −t , t > 0, we see that and the first part of Theorem 1.6 is proved. The second part follows easily from Theorem 2.10(2).
Remark 4.4. Since lim x,y→0+ D F (x, y) = 1 , the constant on the right hand side of Theorem 1.6 is best possible.
Question 4.5. What is the best possible constant on the left hand side of Theorem 1.6? 4.6. Proof of Theorem 3.3. Analogously to the proof of Theorem 1.1, we have Remark 4.7. As it is seen from the proofs, above results are valid for much more general class of positive and monotone increasing (not necessary differentiable) functions. In this sense, as the referee notes, a direct proof of the assertion from Theorem 1.6 is possible.
This means that the function G(t)/t is monotone decreasing, where By Lemma 2.12, it follows that G is super-additive, that is which is equivalent to g(x) + g(y) ≥ g(x + y − xy), and the proof of the first part of Theorem 3.7 is complete.
The proof of the second part is similar. Note that the condition c ∈ (1/2, ∞), d ≥ c/(2c − 1) of Lemma 2.11 is equivalent to the condition 1/c + 1/d ≤ 2 of Theorem 3.7.
4.13. Proof of Theorem 3.8. (1) The left-hand side of this inequality is a direct consequence of part (1) of Theorem 3.7.
The rest is an application of Corollary 3.9.