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# On quotients and differences of hypergeometric functions

- Slavko Simić
^{1}Email author and - Matti Vuorinen
^{2}

**2011**:141

https://doi.org/10.1186/1029-242X-2011-141

© 2011 Simic and Vuorinen; licensee Springer. 2011

**Received:**15 July 2011**Accepted:**21 December 2011**Published:**21 December 2011

## Abstract

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*).

**2010 Mathematics Subject Classification:** 26D06; 33C05.

## Keywords

- sub-additivity
- hypergeometric functions
- inequalities

## 1. **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. [1]. In a recent article [2] 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 [2].

*F*(

*x*) =

*F*(

*a*,

*b*;

*c*;

*x*) when

*x*→ 1

^{-}, see (2.8), we define for 0

*< x*,

*y <*1,

*a*,

*b*,

*c >*0

Our task in this article 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 following theorem.

**Theorem 1.2**.

*For a*,

*b*,

*c >*0

*and*0

*< x*,

*y <*1

*let Q*

_{ F }

*be as in*(1.1)

*. Then*,

and the conclusion follows immediately. Similarly,

**Theorem 1.4**.

*For a*,

*b >*0,

*c > a*+

*b and*0

*< x*,

*y <*1

*, we have*

*with* $A=A\left(a,b,c\right)=\frac{\Gamma \left(c\right)\Gamma \left(c-a-b\right)}{\Gamma \left(c-a\right)\Gamma \left(c-b\right)}=F\left(a,b;c;1\right)$ *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 D*

_{ F }

*be as in*(1.1)

*. Then*,

*with R* = *R*(*a*, *b*) = -2*γ* - *ψ*(*a*) - *ψ*(*b*), *B* = *B*(*a*, *b*).

Both bounding constants are best possible.

In the sequel, we shall give a complete answer to an open question posed in [2].

## 2. **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.

*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

*difference equation*[1, 6.1.15]

*reflection property*[1, 6.1.17]

*γ*is the

*Euler-Mascheroni constant*given by

*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

*a*, 0) = 1 for

*a*≠ 0, and (

*a*,

*n*) is the

*shifted factorial function*or the

*Appell symbol*

for *n* ∈ ℕ\{0}, where ℕ = {0, 1, 2,...}.

*x*= 1 in the three cases

*a*+

*b < c*,

*a*+

*b*=

*c*, and

*a*+

*b > c*:

*a*,

*b*,

*c*), the functions

*F*(

*a*,

*b*;

*c*;

*x*) can be expressed in terms of well-known elementary function. A particular case that is often used in this article is [1, 15.1.3]

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, ∞) and that by (2.8) we see that it is onto [1, ∞) if *a* + *b* ≥ *c*. 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, ∞).

**Theorem 2.10**. [3],[[4], Theorem 1.52]

*For a*,

*b >*0,

*let B*=

*B*(

*a*,

*b*)

*be as in*(2.1)

*, and let R*=

*R*(

*a*,

*b*)

*be as in*(2.4)

*. Then the following are true*.

- (1)
*The function*${f}_{1}\left(x\right)\equiv \frac{F\left(a,b;a+b;x\right)-1}{\text{log}\left(1/\left(1-x\right)\right)}$*is strictly increasing from*(0, 1)*onto*(*ab*/(*a*+*b*), 1/*B*). - (2)
*The function f*_{2}(*x*) ≡*BF*(*a*,*b*;*a*+*b*;*x*) + log(1 -*x*)*is strictly decreasing from*(0, 1)*onto*(*R*,*B*). - (3)
*The function f*_{3}(*x*) ≡*BF*(*a*,*b*;*a*+*b*;*x*) + (1/*x*) log(1 -*x*)*is increasing from*(0, 1)*onto*(*B*- 1,*R*)*if a*,*b*∈ (0, 1). - (4)
*The function f*_{3}*is decreasing from*(0, 1)*onto*(*R*,*B*- 1)*if a*,*b*∈ (1, ∞). - (5)
*The function*${f}_{4}\left(x\right)\equiv \frac{xF\left(a,b;a+b;x\right)}{\text{log}\left(1/\left(1-x\right)\right)}$

*is decreasing from*(0, 1)

*onto*(1/

*B*, 1)

*if a*,

*b*∈ (0, 1).

- (6)
*If a*,*b >*1,*then f*_{4}*is increasing from*(0, 1)*onto*(1, 1/*B*). - (7)
*If a*=*b*= 1*, then f*_{4}(*x*) = 1*for all x*∈ (0, 1).

We also need the following refinement of some parts of Theorem 2.10.

**Lemma 2.11**. [[5], Cor. 2.14]

*For a*,

*b >*0

*, let B*=

*B*(

*a*,

*b*)

*be as in*(2.1)

*, and let R*=

*R*(

*a*,

*b*)

*be as in*(2.4)

*and denote*

*(1) If a* ∈ (0, ∞) *and b* ∈ (0, 1/*a*]*, then the function f is decreasing with range* (1/*B*, 1)*;*

*(2) If a* ∈ (1/2, ∞) *and b* ≥ *a*/(2*a* - 1)*, then f is increasing from* (0, 1) *to the range* (1, 1/*B*).

*(3) If a*∈ (0, ∞)

*and b*∈ (0, 1/

*a*]

*, then the function h defined by*

*is increasing from* (0, 1) *onto* (*B* - 1, *R*).

*(4) If a* ∈ (1/3, ∞) *and b* ≥ (1 + *a*)/(3*a* - 1)*, then h is increasing from* (0, 1) *onto* (*R*, *B* - 1).

For brevity, we write ℝ_{+} = (0, ∞).

**Lemma 2.12**.

*(Cf*. [4, 1.24, 7.42(1)]

*) (1) If E*(

*t*)/

*t is an increasing function on*ℝ

_{+}

*, then E is sub-additive, i.e., for each x*,

*y >*0

*we have that*

*(2) If E*(

*t*)/

*t decreases on*ℝ

_{+}

*, then E is a super-additive function, that is*

*for x*, *y* ∈ ℝ_{+}.

## 3. **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.

*g*in (2.9), we have

Based on this observation and a few computer experiments, we posed in [2] the following question:

**Question 3.1**. Fix

*c*,

*d >*0 and let

*g*(

*x*) =

*xF*(

*c*,

*d*;

*c*+

*d*;

*x*) for

*x*∈ (0, 1) and set

*x*,

*y*∈ (0, 1).

- (1)
For which values of

*c*and*d*, this function is bounded from below and above? - (2)
Is it true that

- a)
*Q*_{ g }(*x*,*y*) ≥ 1, if*cd*≤ 1? - b)
*Q*_{ g }(*x*,*y*) ≤ 1, if*c*,*d >*1? - c)Are there counterparts of Theorem 1.6 for the function${D}_{g}\left(x,y\right)=g\left(x\right)+g\left(y\right)-g\left(x+y-xy\right)?$

We shall give a complete answer to this question in the sequel.

Note first that the quotient *Q*_{
g
} is always bounded. Namely,

**Theorem 3.2**.

*For all c*,

*d >*0

*and all x*,

*y*∈ (0, 1)

*we have that*

A refinement of these bounds for some particular (*c*, *d*) pairs is given by the following two assertions.

**Theorem 3.3**.

*(1) For c*,

*d >*0,

*cd*≤ 1

*and x*,

*y*∈ (0, 1)

*we have*

*(2) For c*,

*d >*0, 1/

*c*+ 1/

*d*≤ 2

*and x*,

*y*∈ (0, 1)

*we have*

*c*,

*d >*0,

*cd*≤ 1, (

*c*,

*d*) ≠ (1, 1) we have

**Theorem 3.5**.

*For cd*≤ 1

*and x*,

*y*∈ (0, 1)

*we have*

We shall prove now the hypothesis from the second part of Question 3.1 under the condition 1/*c* + 1/*d* ≤ 2 in part (b) which, in particular, includes the case *c >* 1, *d >* 1.

**Theorem 3.6**.

*Fix c*,

*d >*0

*and let Q and g be as in Question 3.1*.

- (1)
*If cd*≤ 1*then Q*_{ g }(*x*,*y*) ≥ 1*for all x*,*y*∈ (0, 1). - (2)
*If*1/*c*+ 1/*d*≤ 2*, then Q*_{ g }(*x*,*y*) ≤ 1*for all x*,*y*∈ (0, 1).

Counterparts of Theorem 1.6 for the difference *D*_{
g
} are given in the next assertion.

**Theorem 3.7**.

*Fix c*,

*d >*0

*and let D be as in Question 3.1*.

- (1)
*If cd*≤ 1*, then*$0\le {D}_{g}\left(x,y\right)<\frac{2R\left(c,d\right)+1}{B\left(c,d\right)}-1$

*for all x*,

*y*∈ (0, 1).

- (2)
*If*1/*c*+ 1/*d*≤ 2*, then*$\frac{2R\left(c,d\right)+1}{B\left(c,d\right)}-1<{D}_{g}\left(x,y\right)\le 0$

*for all x*, *y* ∈ (0, 1).

Combining the results above, we obtain the following two-sided bounds for the quotient *Q*_{
g
}.

**Corollary 3.8**.

*Fix c*,

*d >*0

*and let Q be as in Question 3.1*.

- (1)
*If cd*≤ 1*, then*$1\le {Q}_{g}\left(x,y\right)<\text{min}\left\{B\left(c,d\right),2\right\}$

*for all x*,

*y*∈ (0, 1).

- (2)
*If*1/*c*+ 1/*d*≤ 2*, then*$B\left(c,d\right)<{Q}_{g}\left(x,y\right)\le 1$

*for all x*, *y* ∈ (0, 1).

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.

*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 3.8.

**Theorem 3.9**.

*Let*$S\left(t\right):=\frac{g\left(t\right)}{g\left(1-\sqrt{1-t}\right)}$,

*t*∈ (0, 1),

*with g*(

*t*):=

*tF*(

*a*,

*b*;

*a*+

*b*;

*t*),

*a*,

*b >*0.

- (1)
*If ab*≤ 1*, then*$1<S\left(t\right)\le 2.$ - (2)
*If*1/*a*+ 1/*b*≤ 2*, then*$2\le S\left(t\right)<\frac{2}{B\left(a,b\right)}.$

## 4. **Proofs**

### 4.1. **Proof of Theorem 1.2**

*F*(

*x*) =

*F*(

*a*,

*b*;

*c*;

*x*). Namely, for

*x*,

*y*∈ (0, 1), put

*z*=

*x*+

*y*-

*xy*,

*z*∈ (0, 1). Since

*F*(

*u*) is monotone increasing in

*u*, we obtain

The left-hand bound is trivial. □

### 4.2. **Proof of Theorem 1.4**

□

### 4.3. **Proof of Theorem 1.6**

where *y*, *y* ∈ (0, 1), is an independent parameter.

*x*)

*s*'(

*x*)

*s*(

*x*) is monotone decreasing on (0, 1) and, consequently,

Since *D*_{
F
}(0^{+}, 0^{+}) = 1 and *D*_{
F
}(1^{-}, 1^{-}) = *R*/*B*, cited bounds are best possible. □

### 4.4. **Proof of Theorem 3.2**

□

### 4.5. **Proof of Theorem 3.3**

for *u* ∈ (0, 1), *cd* ≤ 1.

The lower bound is proved in the same way.

Applying part (2) of Lemma 2.11, we bound *Q*_{
g
} similarly in the case 1/*c* + 1/*d* ≤ 2. □

**Remark 4.1**. From parts (1) and (3) of Lemma 2.11, we conclude that

for *c*, *d >* 0, *cd* ≤ 1, (*c*, *d*) ≠ (1, 1).

### 4.6. **Proof of Theorem 3.4**

*B*=

*B*(

*c*,

*d*),

*R*=

*R*(

*c*,

*d*) and

*L*= log(1/((1 -

*x*)(1 -

*y*)))

*>*0. By Lemma 2.11 (3) we have

*x*+

*y <*2(

*x*+

*y*-

*xy*) we obtain by (4.2)

□

### 4.7. **Proof of Theorem 3.6**

By the first part of Lemma 2.11, *f* is monotone decreasing for *cd* ≤ 1.

*< × < y <*1 we have

*x*=

*e*

^{-u}, 1 -

*y*=

*e*

^{-v};

*u*,

*v*∈ (0, ∞), we get that the inequality

holds whenever 0 *< u < v <* ∞.

*G*(

*t*)/

*t*is monotone decreasing, where

*G*is super-additive, that is

and the proof of the first part of Theorem 3.6 is complete.

The proof of the second part is similar. Note that the condition *c* ∈ (1/2, ∞), *d* ≥ *c*/(2*c* - 1) of Lemma 2.11 is equivalent to the condition 1/*c* + 1/*d* ≤ 2 of Theorem 3.6. □

### 4.8. **Proof of Theorem 3.7**

- (1)
The left-hand side of this inequality is a direct consequence of part (1) of Theorem 3.6.

*u*∈ (0, 1),

*cd*≤ 1, we get

*R*-

*B*+ 1

*>*0 and

*B*- 1

*>*0.

- (2)
To prove this part we shall use Lemma 2.11, part (4). Because

*d*≥*c*/(2*c*- 1) ≥ (*c*+ 1)/(3*c*- 1) and (1/2, ∞) ⊂ (1/3, ∞), we conclude that this assertion is valid under the condition 1/*c*+ 1/*d*≤ 2.

*u*∈ (0, 1), 1/

*c*+ 1/

*d*≤ 2, (

*c*,

*d*) ≠ (1, 1), we get

since parts (1) and (4) of Lemma 2.11 give *R* - *B* + 1 *<* 0 and *B* - 1 *<* 0.

Because the right-hand inequality is a consequence of Theorem 3.6, part (2), the proof is complete. □

### 4.9. **Proof of Theorem 3.9**

*z*=

*x*+

*y*-

*xy*=

*t*. Therefore,

The rest is an application of Corollary 3.8. □

## Declarations

### Acknowledgements

The authors are indebted to the referee for his/her constructive comments. The research of Matti Vuorinen was supported by the Academy of Finland, Project 2600066611. This project also supported Slavko Simić' visit to Finland.

## Authors’ Affiliations

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## Copyright

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