arxiv.org · arxiv:0712.3856v1 [math.ca] 22 dec 2007 file: om845.tex, 9 oct. 2001 topics in special...

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TOPICS IN SPECIAL FUNCTIONS

G. D. ANDERSON

1, M. K. VAMANAMURTHY, AND M. VUORINEN

Abstra t. The authors survey re ent results in spe ial fun tions, par-

ti ularly the gamma fun tion and the Gaussian hypergeometri fun tion.

1. Introdu tion

Conformal invariants are powerful tools in the study of quasi onformal

mappings, and many of these have expressions in terms of spe ial fun tions.

For instan e, the distortion results in geometri fun tion theory, su h as the

quasi onformal S hwarz Lemma, involve spe ial fun tions. A frequent task

is to simplify ompli ated inequalities, so as to larify the dependen e on im-

portant parameters without sa ri ing sharpness. For these reasons we were

led to study, as an independent subje t, various questions for spe ial fun -

tions su h as monotoneity properties and majorants/minorants in terms of

rational fun tions. These new inequalities gave rened versions of some las-

si al distortion theorems for quasi onformal maps. The lasses of fun tions

that o ur in lude omplete ellipti integrals, hypergeometri fun tions, and

Euler's gamma fun tion. The main part of our resear h is summarized in

[AVV5.

In the later development most of our resear h has involved appli ations

to geometri properties of quasi onformal maps. However, some of the ques-

tions on erning spe ial fun tions, raised in [AVV1, [AVV3, and [AVV5,

relate to spe ial fun tions whi h are useful in geometri fun tion theory in

general, not just to quasi onformal maps. In this survey our goal is to review

the latest developments of the latter type, due to many authors [A1[A9,

[AlQ1, AlQ2, AW, BPR1, BPR2, BPS, BP, EL, K1, K2, Ku.

The methods used in these studies are based on lassi al analysis. One

of the te hni al tools is the Monotone l'Hpital's Rule, stated in the next

paragraph, whi h played an important role in our work [AVV4[AVV5.

The authors dis overed this result in [AVV4, unaware that it had been used

earlier (without the name) as a te hni al tool in dierential geometry. See

[Ch, p. 124, Lemma 3.1 or [AQVV, p. 14 for relevant remarks.

1.1. Lemma. For −∞ < a < b < ∞, let g and h be real-valued fun tions

that are ontinuous on [a, b] and dierentiable on (a, b), with h′ 6= 0 on (a, b).

1991 Mathemati s Subje t Classi ation. Primary 33-02, 33B15, 33C05. Se ondary

33C65, 33E05.

Key words and phrases. Spe ial fun tions, hypergeometri fun tions, gamma fun tion,

beta fun tion, Euler-Mas heroni onstant, ellipti integrals, generalized ellipti integrals,

mean values, arithmeti -geometri mean.

1

This paper is an outgrowth of an invited talk given by the rst author at the 18th

Rolf Nevanlinna Colloquium in Helsinki, Finland, in August 2000.

1

http://arxiv.org/abs/0712.3856v1

2 G. D. ANDERSON, M. K. VAMANAMURTHY, AND M. VUORINEN

If g′/h′ is stri tly in reasing (resp. de reasing) on (a, b), then the fun tions

g(x)− g(a)

h(x)− h(a)and

g(x) − g(b)

h(x) − h(b)

are also stri tly in reasing (resp. de reasing) on (a, b).

Graphing of the fun tions and omputer experiments in general played an

important role in our work. For instan e, the software that omes with the

book [AVV5 provides omputer programs for su h experiments.

We begin this survey by dis ussing some re ent results on the gamma

fun tion, in luding monotoneity and onvexity properties and lose approx-

imations for the Euler-Mas heroni onstant. Hypergeometri fun tions have

a very entral role in this survey. We give here a detailed proof of the so-

alled Elliott's identity for these fun tions following an outline suggested by

Andrews, Askey, and Roy in [AAR, p. 138. This identity ontains, as a spe-

ial ase, the lassi al Legendre Relation and has been studied re ently in

[KV and [BPSV. After this we dis uss mean values, a topi related to om-

plete ellipti integrals and their estimation, and we present several sharp

approximations for omplete ellipti integrals. We display inequalities for

hypergeometri fun tions that generalize the Landen relation, and on lude

the paper with a remark on re ent work of geometri mapping properties of

hypergeometri fun tions as a fun tion of a omplex argument.

This survey does not over re ent work on the appli ations of spe ial

fun tions to the hange of distan e under quasi onformal maps. For this

subje t the interested reader may onsult [AVV5.

2. The Γ and Ψ fun tions

Throughout this paper Γ will denote Euler's gamma fun tion, dened by

Γ(z) =

∫

∞

0e−ttz−1 dt, Re z > 0,

and then ontinued analyti ally to the nite omplex plane minus the set

of nonpositive integers. The re urren e formula Γ(z + 1) = z Γ(z) yields

Γ(n+1) = n! for any positive integer n. We also use the fa t that Γ(1/2) =√π. The beta fun tion is related to the gamma fun tion by B(a, b) =

Γ(a)Γ(b)/Γ(a + b). The logarithmi derivative of the gamma fun tion will

be denoted, as usual, by

Ψ(z) ≡ d

dzlog Γ(z) =

Γ′(z)

Γ(z).

The Euler-Mas heroni onstant γ is dened as (see [A2, [TY, [Y)

γ ≡ limn→∞

Dn = 0.5772156649 . . . ; Dn ≡n∑

k=1

1

k− log n.

Then Ψ(1) = Γ′(1) = −γ and Ψ(1/2) = −γ − 2 log 2. For a survey of the

gamma fun tion see [G, and for some inequalities for the gamma and psi

fun tions see [A1.

TOPICS IN SPECIAL FUNCTIONS 3

2.1. Approximation of the Euler-Mas heroni onstant. The onver-

gen e of the sequen e Dn to γ is very slow (the speed of onvergen e is

studied by Alzer [A2). D. W. DeTemple [De studied a modied sequen e

whi h onverges faster and proved

1

24(n + 1)2< Rn − γ <

1

24n2, where Rn ≡

n∑

k=1

1

k− log

(

n+1

2

)

.

Now let

h(n) = Rn − γ, H(n) = n2h(n), n > 1.

Sin e Ψ(n) = −γ − 1/n +∑n

k=1 1/k, we see that

H(n) = (Rn − γ)n2 = (Ψ(n) + 1/n− log(n+1

2))n2.

Some omputer experiments led M. Vuorinen to onje ture that H(n) in-

reases on the interval [1,∞) from H(1) = −γ + 1 − log(3/2) = 0.0173 . . .to 1/24 = 0.0416 . . .. E. A. Karatsuba proved in [K1 that for all integers

n > 1,H(n) < H(n + 1), by lever use of Stirling's formula and Fourier

series. Moreover, using the relation γ = 1− Γ′(2) she obtained, for k ≥ 1,

−ck ≤ γ − 1 + (log k)12k+1∑

r=1

d(k, r)−12k+1∑

r=1

d(k, r)

r + 1≤ ck,

where

ck =2

(12k)!+ 2k2e−k, d(k, r) = (−1)r−1 kr+1

(r − 1)!(r + 1),

giving exponential onvergen e. Some omputer experiments also seem to

indi ate that ((n + 1)/n)2H(n) is a de reasing onvex fun tion.

2.2. Gamma fun tion and volumes of balls. Formulas for geometri

obje ts, su h as volumes of solids and ar lengths of urves, often involve

spe ial fun tions. For example, if Ωn denotes the volume of the unit ball

Bn = x : |x| < 1 in Rn, and if ωn−1 denotes the (n − 1)-dimensional

surfa e area of the unit sphere Sn−1 = x : |x| = 1, n > 2, then

Ωn =πn/2

Γ((n/2) + 1); ωn−1 = nΩn.

It is well known that for n > 7 both Ωn and ωn de rease to 0 ( f. [AVV5,

2.28). However, neither Ωn nor ωn is monotone for n on [2,∞). On the other

hand, Ω1/(n logn)n de reases to e−1/2

as n → ∞ [AVV1, Lemma 2.40(2).

Re ently H. Alzer [A4 has obtained the best possible onstants a, b, A, B,α, β su h that

aΩn

n+1

n+1 6 Ωn 6 bΩn

n+1

n+1,√

n+A

2π6

Ωn−1

Ωn6

√

n+B

2π,

(

1 +1

n

)α

6Ω2n

Ωn−1Ωn+16

(

1 +1

n

)β


for all integers n > 1. He showed that a = 2/√π = 1.12837 . . ., b =

√e =

1.64872 . . . , A = 1/2, B = π/2 − 1 = 0.57079 . . . , α = 2 − (log π)/ log 2 =0.34850 . . . , β = 1/2. For some related results, see [KlR.

2.3. Monotoneity properties. In [AnQ it is proved that the fun tion

(2.4) f(x) ≡ log Γ(x+ 1)

x log x

is stri tly in reasing from (1,∞) onto (1−γ, 1). In parti ular, for x ∈ (1,∞),

(2.5) x(1−γ)x−1 < Γ(x) < xx−1.

The proof required the following two te hni al lemmas, among others:

2.6. Lemma. The fun tion

g(x) ≡∞∑

n=1

n− x

(n+ x)3

is positive for x ∈ [1, 4).

2.7. Lemma. The fun tion

(2.8) h(x) ≡ x2Ψ′(1 + x)− xΨ(1 + x) + log Γ(1 + x)

is positive for all x ∈ [1,∞).

It was onje tured in [AnQ that the fun tion f in (2.4) is on ave on

(1,∞).

2.9. Horst Alzer [A2 has given an elegant proof of the monotoneity of the

fun tion f in (2.4) by using the Monotone l'Hpital's Rule and the onvolu-

tion theorem for Lapla e transforms. In a later paper [A3 he has improved

the estimates in (2.5) to

(2.10) xα(x−1)−γ < Γ(x) < xβ(x−1)−γ , x ∈ (0, 1),

where α ≡ 1 − γ = 0.42278 . . . , β ≡ 12

(

π2/6− γ)

= 0.53385 . . . are best

possible. If x ∈ (1,∞), he also showed that (2.10) holds with best onstants

α ≡ 12

(

π2/6− γ)

= 0.53385 . . . , β ≡ 1.

2.11. Elbert and Laforgia [EL have shown that the fun tion g in Lemma 2.6

is positive for all x > −1. They used this result to prove that the fun tion

h in Lemma 2.7 is stri tly de reasing from (−1, 0] onto [0,∞) and stri tly

in reasing from [0,∞) onto [0,∞). They also showed that f ′′ < 0 for x > 1,thus proving the Anderson-Qiu onje ture [AnQ, where f is as in (2.4).

2.12. Berg and Pedersen [BP have shown that the fun tion f in (2.4) is not

only stri tly in reasing from (0,∞) onto (0, 1), but is even a (non onstant)

so- alled Bernstein fun tion. That is, f > 0 and f ′is ompletely monotoni ,

i.e., f ′ > 0, f ′′ < 0, f ′′′ > 0, . . . . In parti ular, the fun tion f is stri tly

in reasing and stri tly on ave on (0,∞).In fa t, they have proved the stronger result that 1/f is a Stieltjes trans-

form, that is, an be written in the form

1

f(x)= c+

∫

∞

0

dσ(t)

x+ t, x > 0,


where the onstant c is non-negative and σ is a non-negative measure on

[0,∞) satisfying∫

∞

0

dσ(t)

1 + t< ∞.

In parti ular, for 1/f they have shown by using Stirling's formula that c = 1.Also they have obtained dσ(t) = H(t)dt, where H is the ontinuous density

H(t) =

tlog |Γ(1− t)|+ (k − 1) log t

(log |Γ(1 − t)|)2 + (k − 1)2π2, t ∈ (k − 1, k), k = 1, 2, . . . ,

0 , t = 1, 2, . . . .

Here log denotes the usual natural logarithm. The density H(t) tends to 1/γas t tends to 0, and σ has no mass at 0.

2.13. In The Lost Notebook and Other Unpublished Papers of Ramanujan

[Ra1, the Indian mathemati al genius, appears the following re ord:

“Γ(1 + x) =√π(x

e

)x

8x3 + 4x2 + x+θx30

1/6,

where θx is a positive proper fra tion

θ0 =30

π3= .9675

θ1/12 = .8071 θ7/12 = .3058

θ2/12 = .6160 θ8/12 = .3014

θ3/12 = .4867 θ9/12 = .3041

θ4/12 = .4029 θ10/12 = .3118

θ5/12 = .3509 θ11/12 = .3227

θ6/12 = .3207 θ1 = .3359

θ∞ = 1.

Of ourse, the values in the above table, ex ept θ∞, are irrational and

hen e the de imals should be nonterminating as well as nonre urring. The

re ord stated above has been the subje t of intense investigations and is

reviewed in [BCK, page 48 (Question 754). This note of Ramanujan led the

authors of [AVV5 to make the following onje ture.

2.14. Conje ture. Let

G(x) = (e/x)xΓ(1 + x)/√π

and

H(x) = G(x)6 − 8x3 − 4x2 − x =θx30

.

Then H is in reasing from (1,∞) into (1/100, 1/30) [AVV5, p. 476.

2.15. In a ni e pie e of work, E. A. Karatsuba [K2 has proved the above

onje ture. She did this by representing the fun tion H(x) as an integral for

whi h she was able to nd an asymptoti development. Her work also led to


an interesting asymptoti formula for the gamma fun tion:

Γ(x+ 1) =√π(x

e

)x(

8x3 + 4x2 + x+1

30− 11

240x+

79

3360x2+

3539

201600x3

− 9511

403200x4− 10051

716800x5+

47474887

1277337600x6+

a7x7

+ · · ·+ anxn

+∆n+1(x))1/6

,

where ∆n+1(x) = O( 1xn+1 ), as x → ∞, and where ea h ak is given expli itly

in terms of the Bernoulli numbers.

3. Hypergeometri fun tions

Given omplex numbers a, b, and c with c 6= 0,−1,−2, . . . , the Gaussian

hypergeometri fun tion is the analyti ontinuation to the slit plane C \[1,∞) of

(3.1) F (a, b; c; z)=2F1(a, b; c; z)≡∞∑

n=0

(a, n)(b, n)

(c, n)

zn

n!, |z| < 1.

Here (a, 0) = 1 for a 6= 0, and (a, n) is the shifted fa torial fun tion

(a, n) ≡ a(a+ 1)(a + 2) · · · (a+ n− 1)

for n = 1, 2, 3, . . ..The hypergeometri fun tion w = F (a, b; c; z) in (3.1) has the simple

dierentiation formula

(3.2)

d

dzF (a, b; c; z) =

ab

cF (a+ 1, b + 1; c + 1; z).

The behavior of the hypergeometri fun tion near z = 1 in the three ases

a+ b < c, a+ b = c, and a+ b > c, a, b, c > 0, is given by

(3.3)

F (a, b; c; 1) = Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) , a+ b < c,

B(a, b)F (a, b; a + b; z) + log(1− z)

= R(a, b) +O((1− z) log(1− z)),

F (a, b; c; z) = (1− z)c−a−bF (c− a, c− b; c; z), c < a+ b,

where R(a, b) = −2γ −Ψ(a)−Ψ(b), R(a) ≡ R(a, 1− a), R(12 ) = log 16, andwhere log denotes the prin ipal bran h of the omplex logarithm. The above

asymptoti formula for the zero-balan ed ase a+ b = c is due to Ramanujan

(see [As, [Be1). This formula is implied by [AS, 15.3.10.

The asymptoti formula (3.3) gives a pre ise des ription of the behavior

of the fun tion F (a, b; a + b; z) near the logarithmi singularity z = 1. Thissingularity an be removed by an exponential hange of variables and the

transformed fun tion will be nearly linear.

3.4. Theorem. [AQVV For a, b > 0, let k(x) = F (a, b; a+b; 1−e−x), x > 0.Then k is an in reasing and onvex fun tion with k′((0,∞)) = (ab/(a + b),Γ(a+ b)/(Γ(a)Γ(b))).

3.5. Theorem. [AQVV Given a, b > 0, and a + b > c, d ≡ a + b − c, the

fun tion ℓ(x) = F (a, b; c; 1 − (1 + x)−1/d), x > 0, is in reasing and onvex,

with ℓ′((0,∞)) = (ab/(cd), Γ(c)Γ(d)/(Γ(a)Γ(b))).


3.6. Gauss ontiguous relations and derivative formula. The six fun -

tions F (a±1, b; c; z), F (a, b±1; c; z), F (a, b; c±1; z) are alled ontiguous to

F (a, b; c; z). Gauss dis overed 15 relations between F (a, b; c; z) and pairs of

its ontiguous fun tions [AS, 15.2.1015.2.27, [R2, Se tion 33. If we apply

these relations to the dierentiation formula (3.2), we obtain the following

useful formulas.

3.7. Theorem. For a, b, c > 0, z ∈ (0, 1), let u = u(z) = F (a − 1, b; c; z),v = v(z) = F (a, b; c; z), u1 = u(1− z), v1 = v(1− z). Then

zdu

dz= (a− 1)(v − u),(3.8)

z(1 − z)dv

dz= (c− a)u+ (a− c+ bz)v,(3.9)

and

(3.10)

ab

cz(1− z)F (a+ 1, b+ 1; c + 1; z) = (c− a)u+ (a− c+ bz)v.

Furthermore,

(3.11) z(1−z)d

dz

(

uv1+u1v−vv1)

=(1−a−b)[

(1−z)uv1−zu1v−(1−2z)vv1)]

.

Formulas (3.8)-(3.10) in Theorem 3.7 are well known. See, for example,

[AAR, 2.5.8. On the other hand, formula (3.11), whi h follows from (3.8)-

(3.9) is rst proved in [AQVV, 3.13 (4).

Note that the formula

(3.12) z(1− z)dF

dz= (c− b)F (a, b− 1; c; z) + (b− c+ az)F (a, b; c; z)

follows from (3.9) if we use the symmetry property F (a, b; c; z) = F (b, a; c; z).

3.13. Corollary. With the notation of Theorem 3.7, if a ∈ (0, 1), b = 1−a <c, then

uv1 + u1v − vv1 = u(1) =(Γ(c))2

Γ(c+ a− 1)Γ(c− a+ 1).

4. Hypergeometri differential equation

The fun tion F (a, b; c; z) satises the hypergeometri dierential equation

(4.1) z(1 − z)w′′ + [c− (a+ b+ 1)z]w′ − abw = 0.

Kummer dis overed solutions of (4.1) in various domains, obtaining 24 in

all; for a omplete list of his solutions see [R2, pp. 174, 175.

4.2. Lemma. (1) If 2c = a+ b+1 then both F (a, b; c; z) and F (a, b; c; 1− z)satisfy (4.1) in the lens-shaped region z : 0 < |z| < 1, 0 < |1 − z| < 1.(2) If 2c = a+ b+ 1 then both F (a, b; c; z2) and F (a, b; c; 1 − z2) satisfy the

dierential equation

(4.3) z(1− z2)w′′ + [2c − 1− (2a+ 2b+ 1)z2]w′ − 4abzw = 0

in the ommon part of the disk z : |z| < 1 and the lemnis ate z : |1−z2| <1.


Proof. By Kummer ( f. [R2, pp. 174-177), the fun tions F (a, b; c; z)and F (a, b; a+ b+1− c; 1− z) are solutions of (4.1) in z : 0 < |z| < 1 and

z : 0 < |1 − z| < 1, respe tively. But a + b + 1 − c = c under the stated

hypotheses. The result (2) follows from result (1) by the hain rule.

4.4. Lemma. The fun tion F (a, b; c;√1− z2) satises the dierential equa-

tion

Z3(1− Z)zw′′ − Z(1− Z) + [c− (a+ b+ 1)Z]Zz2w′ − abz3w = 0,

in the subregion of the right half-plane bounded by the lemnis ate r2 =2cos(2ϑ), −π/4 ≤ ϑ ≤ π/4, z = reiϑ. Here Z =

√1− z2, where the

square root indi ates the prin ipal bran h.

Proof. From (4.1), the dierential equation for w = F (a, b; c; t) is givenby

t(1− t)d2w

dt2+ [c− (a+ b+ 1)t]

dw

dt− abw = 0.

Now put t =√1− z2. Then

dz

dt= − t

z,dt

dz= −z

t,d2t

dz2= − 1

t3

and

dw

dt= − t

z

dw

dz,d2w

dt2=

t2

z2d2w

dz2− 1

z3dw

dz.

So

t(1− t)[ t2

z2w′′ − 1

z3w′

]

+[

c− (a+ b+ 1)t](

− t

z

)

w′ − abw = 0.

Multiplying through by z3 and repla ing t by Z ≡√1− z2 gives the result.

If w1 and w2 are two solutions of a se ond order dierential equation, then

their Wronskian is dened to be W (w1, w2) ≡ w1w′

2 − w2w′

1.

4.5. Lemma. [AAR, Lemma 3.2.6 If w1 and w2 are two linearly independent

solutions of (4.1), then

W (z) = W (w1, w2)(z) =A

zc(1− z)a+b−c+1,

where A is a onstant.

(Note the misprint in [AAR, (3.10), where the oe ient x(1 − x) is

missing from the rst term.)

4.6. Lemma. If 2c = a+ b+ 1 then, in the notation of Theorem 3.7,

(4.7) (c− a)(uv1 + u1v) + (a− 1)vv1 = A · z1−c(1− z)1−c.

Proof. If 2c = a+ b+ 1 then by Lemma 4.2(1), both v(z) and v(1 − z)are solutions of (4.1). Sin e W (z) = W (v1, v)(z) = v′(z)v1(z) − v(z)v′1(z),we have

z(1− z)W (z) = z(1− z)(v′v1 − vv′1)

= (c− a)(uv1 + u1v) + (2a+ b− 2c)vv1

= (c− a)(uv1 + u1v) + (a− 1)vv1.


Next, sin e 2c = a + b + 1, Lemma 4.5 shows that zc(1 − z)cW (z) = A,and the result follows.

Note that in the parti ular ase c = 1, a = b = 12 the right side of (4.7) is

onstant and the result is similar to Corollary 3.13. This parti ular ase is

Legendre's Relation (5.3), and this proof of it is due to Duren [Du.

4.8. Lemma. If a, b > 0, c ≥ 1, and 2c = a + b + 1, then the onstant Ain Lemma 4.6 is given by A = (Γ(c))2/(Γ(a)Γ(b)). In parti ular, if c = 1then Lemma 4.6 redu es to Legendre's Relation (5.8) for generalized ellipti

integrals.

Proof. The idea of the proof is to repla e the possibly unbounded hyper-

geometri fun tions in formula (4.7) by bounded or simpler ones. Therefore

we onsider three ases.

Case (1): c ≥ 2. Now a+ b ≥ c+ 1 ≥ 3. By (3.3) or [AS, 15.3.3, we have

u(z) = (1−z)2−cF (c+1−a, c−b; c; z), u1(z) = z2−cF (c+1−a, c−b; c; 1−z),

v(z) = (1− z)1−cF (c− a, c− b; c; z), v1(z) = z1−cF (c− a, c− b; c; 1 − z).

Hen e

A = (c− a)[(1− z)F (c + 1− a, c− b; c; z)F (c − a, c− b; c; 1 − z)

+zF (c+ 1− a, c− b; c; 1 − z)F (c − a, c− b; c; z)]

+(a− 1)F (c − a, c− b; c; z)F (c − a, c− b; c; 1 − z).

Now, sin e a+ b− c = c− 1, letting z → 0, from (3.3) we get

A = (c− a)Γ(c)Γ(c− 1)

Γ(a)Γ(b))+ (a− 1)

Γ(c)Γ(c − 1)

Γ(a)Γ(b)

= (c− 1)Γ(c)Γ(c − 1)

Γ(a)Γ(b)=

(Γ(c))2

Γ(a)Γ(b),

as laimed.

Case (2): 1 < c < 2. Now, 1 < c < a+ b < c+ 1 < 3. Then

A = (c−a)[(1−z)c−1u(z)F (c−a, c−b; c; 1−z)+zc−1u1(z)F (c−a, c−b; c; z)]

+(a− 1)F (c − a, c− b; c; z)F (c − a, c− b; c; 1 − z).

Now letting z → 0, from (3.3), as in Case (1), we get

A = (c− a)Γ(c)Γ(c− 1)

Γ(a)Γ(b)+ (a− 1)

Γ(c)Γ(c − 1)

Γ(a)Γ(b)

= (c− 1)Γ(c)Γ(c − 1)

Γ(a)Γ(b)=

(Γ(c))2

Γ(a)Γ(b),

as laimed.

Case (3): c = 1. Now a+ b = 1. Then

A = (1− a)[u(z)v1(z) + u1(z)v(z) − v(z)v1(z)]

= (1− a)u1(z)v(z) + (1− a)v1(z)[u(z) − v(z)].


From [R1, Ex. 21(4), p.71 we have

u(z)− v(z) = F (a− 1, b; c; z) − F (a, b; c; z)

=c− b

czF (a, b; c + 1; z) − zF (a, b; c; z),

so that

u(z)− v(z)

z=

c− b

cF (a, b; c + 1; z) − F (a, b; c; z) → −b/c,

as z → 0. Also, by (3.3), zv1(z) → 0 as z → 0. Hen e, letting z → 0, we get

A = (1− a)u1(1) = (1− a)Γ(c)Γ(c + 1− a− b)

Γ(c+ 1− a)Γ(c− b)

= (1− a)(Γ(c))2

(1 − a)Γ(a)Γ(b)=

(Γ(c))2

Γ(a)Γ(b),

as laimed.

Note that, in Case (3), Γ(c) = Γ(1) = 1,Γ(b) = Γ(1−a), and thus by [AS,

6.1.17 A = 1/(Γ(a)Γ(1 − a)) = (sinπa)/π.

For rational triples (a, b, c) there are numerous ases where the hyper-

geometri fun tion F (a, b; c; z) redu es to a simpler fun tion (see [PBM).

Other important parti ular ases are generalized ellipti integrals, whi h we

will now dis uss. For a, r ∈ (0, 1), the generalized ellipti integral of the rst

kind is given by

Ka = Ka(r) =π

2F (a, 1− a; 1; r2)

= (sinπa)

∫ π/2

0(tan t)1−2a(1− r2 sin2 t)−a dt,

K′

a = K′

a(r) = Ka(r′).

We also dene

µa(r) =π

2 sin(πa)

K′

a(r)

Ka(r), r′ =

√

1− r2.

The invariant of the linear dierential equation

(4.9) w′′ + pw′ + qw = 0,

where p and q are fun tions of z, is dened to be

I ≡ q − 1

2p′ − 1

4p2

( f. [R2,p.9). If w1 and w2 are two linearly independent solutions of (4.9),

then their quotient w ≡ w2/w1 satises the dierential equation

Sw(z) = 2I,

where Sw is the S hwarzian derivative

Sw ≡(

w′′

w′

)

′

− 1

2

(

w′′

w′

)2

and the primes indi ate dierentiations ( f. [R2, pp. 18,19).


From these onsiderations and the fa t that Ka(r) and K′

a(r) are linearlyindependent solutions of (4.3) (see [AQVV, (1.11)), it follows that w = µa(r)satises the dierential equation

Sw(r) =−8a(1− a)

(r′)2+

1 + 6r2 − 3r4

2r2(r′)4.

The generalized ellipti integral of the se ond kind is given by

Ea = Ea(r) ≡π

2F (a− 1, 1 − a; 1; r2)

= (sinπa)

∫ π/2

0(tan t)1−2a(1− r2 sin2 t)1−a dt

E′

a = E′

a(r) = Ea(r′),

Ea(0) =π

2, Ea(1) =

sin(πa)

2(1 − a).

For a = 12 , Ka and Ea redu e to K and E, respe tively, the usual ellipti

integrals of the rst and se ond kind, respe tively. Likewise µ1/2(r) = µ(r),the modulus of the well-known Grötzs h ring in the plane [LV.

4.10. Corollary. The generalized ellipti integrals Ka and Ea satisfy the dif-

ferential equations

r(r′)2d2Ka

dr2+ (1− 3r2)

dKa

dr− 4a(1− a)rKa = 0,(4.11)

r(r′)2d2Ea

dr2+ (r′)2

dEa

dr+ 4(1− a)2rEa = 0,(4.12)

respe tively.

Proof. These follow from (4.3).

For a = 12 these redu e to well-known dierential equations [AVV5, pp.

474-475, [BF.

5. Identities of Legendre and Elliott

In geometri fun tion theory the omplete ellipti integrals K(r) and E(r)play an important role. These integrals may be dened, respe tively, as

K(r) = π2F (12 ,

12 ; 1; r

2), E(r) = π2F (12 ,−1

2 ; 1; r2),

for −1 < r < 1. These are Ka(r) and Ea(r), respe tively, with a = 12 . We

also onsider the fun tions

K′ = K

′(r) = K(r′), 0 < r < 1,

K(0) = π/2, K(1−) = +∞,

and

E′ = E

′(r) = E(r′), 0 6 r 6 1,

where r′ =√1− r2. For example, these fun tions o ur in the following

quasi onformal ounterpart of the S hwarz Lemma [LV:


5.1. Theorem. For K ∈ [1,∞), let w be a K-quasi onformal mapping of

the unit disk D = z : |z| < 1 into the unit disk D′ = w : |w| < 1 with

w(0) = 0. Then

|w(z)| 6 ϕK(|z|),where

(5.2) ϕK(r) ≡ µ−1( 1

Kµ(r)

)

and µ(r) ≡ πK′(r)

2K(r).

This result is sharp in the sense that for ea h z ∈ D and K ∈ [1,∞) there is

an extremal K-quasi onformal mapping that takes the unit disk D onto the

unit disk D′with w(0) = 0 and |w(z)| = ϕK(|z|) (see [LV, p. 63).

It is well known [BF that the omplete ellipti integrals K and E satisfy

the Legendre relation

(5.3) EK′ + E

′K−KK

′ =π

2.

For several proofs of (5.3) see [Du.

In 1904, E. B. Elliott [E ( f. [AVV3) obtained the following generalization

of this result.

5.4. Theorem. If a, b, c > 0 and 0 < x < 1 then

(5.5) F1F2 + F3F4 − F2F3 =Γ(a+ b+ 1)Γ(b+ c+ 1)

Γ(a+ b+ c+ 32 )Γ(b+

12)

.

where

F1 = F

(

1

2+ a,−1

2− c; 1 + a+ b;x

)

,

F2 = F

(

1

2− a,

1

2+ c; 1 + b+ c; 1− x

)

,

F3 = F

(

1

2+ a,

1

2− c; 1 + a+ b;x

)

,

F4 = F

(

− 1

2− a,

1

2+ c; 1 + b+ c; 1 − x

)

.

Clearly (5.3) is a spe ial ase of (5.5), when a = b = c = 0 and x =r2. For a dis ussion of generalizations of Legendre's Relation see Karatsuba

and Vuorinen [KV and Balasubramanian, Ponnusamy, Sunanda Naik, and

Vuorinen [BPSV.

Elliott proved (5.5) by a lever hange of variables in multiple integrals.

Another proof was suggested without details in [AAR, p. 138, and here we

provide the missing details.

Proof of Theorem 5.4. In parti ular, let y1 ≡ F3, y2 ≡ x−a−b(1 −x)b+cF2. Then by [R2, pp. 174, 175 or [AAR, (3.2.12), (3.2.13), y1 and y2are linearly independent solutions of (4.1).

By (3.12),

(5.6) x(1−x)y′1 =(

a+ b+ c+1

2

)

F1 + [−(a+ b+ c+1

2) + (a+

1

2)x]F3,


and by (3.9),

x(1− x)y′2 =x(1− x)[

− (a+ b)x−a−b−1(1− x)b+c

− (b+ c)x−a−b(1− x)b+c−1]

F2

− x−a−b(1− x)b+c[(

a+ b+ c+1

2

)

F4

+[

− (a+ b+ c+1

2

)

+(

c+1

2

)

(1− x))F2

]

.

(5.7)

Multiplying (5.7) by y1 and (5.6) by y2 and subtra ting, we obtain

x(1− x)(y2y′

1 − y1y′

2) =(

a+ b+ c+1

2

)

x−a−b(1− x)b+cF1F2

+[

−(

a+ b+ c+1

2

)

+(

a+1

2

)

x]


+ x(1− x)[(a+ b)x−a−b−1(1− x)b+c

+ (b+ c)x−a−b(1− x)b+c−1]F2F3

+ x−a−b(1− x)b+c(

a+ b+ c+1

2

)

F3F4

+ x−a−b(1− x)b+c[

− (a+ b+ c+1

2

)

+(

c+1

2

)

(1− x)]

F2F3

=(

a+ b+ c+1

2

)


+ x−a−b(1− x)b+c[

− (a+ b+ c+1

2

)

+(

a+1

2

)

x+ (a+ b)(1− x)

+ (b+ c)x−(

a+ b+ c+1

2

)

+(

c+1

2

)

(1− x)]

F2F3

+ x−a−b(1− x)b+c(

a+ b+ c+1

2

)

F3F4

=(

a+ b+ c+1

2

)


+ x−a−b(1− x)b+c[

−(

a+ b+ c+ 12

)]

F2F3

+(

a+ b+ c+1

2

)

x−a−b(1− x)b+cF3F4.

So

x(1− x)W (y2, y1) =A

xa+b(1− x)−b−c

= x−a−b(1− x)b+c(

a+ b+ c+1

2

)

[F1F2 + F3F4 − F2F3]

by Lemma 4.5. Thus

F1F2 + F3F4 − F2F3 = A,

where A is a onstant.


Now, by (3.3),

F1F2 tends to F(1

2+ a,−1

2− c; a+ b+ 1; 1

)

=Γ(a+ b+ 1)Γ(b+ c+ 1)

Γ(b+ 12)Γ(a+ b+ c+ 3

2 )

as x → 1, sin e 12 + a+ (−1

2 − c) = a− c < a+ b+ 1.Next

F3F4 − F3F2 = F3(F4 − F2),

where F4 − F2 ∼ onst · (1− x)2 +O((1 − x)3), and

F3 =Γ(a+ b+ 1)Γ(b + c+ 1)

Γ(b+ 12 )Γ(a+ b+ c+ 1

2)

if a+ 12 + 1

2 − c < a+ b+ 1, or −c < b, i.e., b > 0 or c > 0. If −c = b = 0,then

F3 =R(a+ 1

2 ,12 − c)

B(a+ 12 ,

12 − c)

+O((1− x) log(1− x))

by (3.3). In either ase the produ t F3(F4 − F2) tends to 0 as x → 1. Thethird ase a+ 1

2+12−c > a+b+1 is impossible sin e we are assuming that b, c

are nonnegative. Thus A = Γ(a+b+1)Γ(b+c+1)/(Γ(b+ 12)Γ(a+b+c+ 3

2)),as desired.

The generalized ellipti integrals satisfy the identity

(5.8) EaK′

a + E′

aKa −KaK′

a =π sin(πa)

4(1− a).

This follows from Elliott's formula (5.5) and ontains the lassi al relation

of Legendre (5.3) as a spe ial ase.

Finally, we re ord the following formula of Kummer [Kum, p. 63, Form.

30:

F (a, b; a+ b− c+ 1; 1− x)F (a+ 1, b+ 1; c + 1;x)

+c

a+ b− c+ 1F (a, b; c;x)F (a + 1, b+ 1; a+ b− c+ 2; 1 − x)

= Dx−c(1− x)c−a−b−1, D =Γ(a+ b− c+ 1)Γ(c+ 1)

Γ(a+ 1)Γ(b+ 1).

This formula, like Elliott's identity, may be rewritten in many dierent ways

if we use the ontiguous relations of Gauss. Note also the spe ial ase c =a+ b− c+ 1.

6. Mean values

The arithmeti -geometri mean of positive numbers a, b > 0 is the limit

AGM(a, b) = lim an = lim bn,

where a0 = a, b0 = b, and for n = 0, 1, 2, 3, ...,

an+1 = A(an, bn) ≡ (an + bn)/2, bn+1 = G(an, bn) ≡√

anbn,

are the arithmeti and geometri means of an and bn, resp. For a mean value

M , we also onsider the t-modi ation dened as

Mt(a, b) = M(at, bt)1/t.


For example, the power mean of a, b > 0 is

At(a, b) =

(

at + bt

2

)1/t

,

and the logarithmi mean is

L(a, b) =a− b

log(b/a).

The power mean is the t-modi ation of the arithmeti mean A1(a, b).

The onne tion between mean values and ellipti integrals is provided by

Gauss's amazing result

AGM(1, r′) =π

2K(r).

This formula motivates the question of nding minorant/majorant fun tions

for K(r) in terms of mean values. For a xed x > 0 the fun tion t 7→Lt(1, x), t > 0, in reases with t by [VV, Theorem 1.2 (1). The two-sided

inequality

L3/2(1, x) > AG(1, x) > L(1, x)

holds; the se ond inequality was pointed out in [CV, and the rst one, due

to J. and P. Borwein [BB2, proves a sharp estimate settling a question raised

in onne tion with [VV. Combined with the identity above, this inequality

yields a very pre ise inequality for K(r).Several inequalities between mean values have been proved re ently. See,

for instan e, [AlQ2, [QS, [S1, [S2, [S3, [T, [C, and [Br.

Finally, we remark that the arithmeti -geometri mean, together with

Legendre's Relation, played a entral role in a rapidly onverging algorithm

for the number π in [Sa. See also [BB1, H, Le, Lu.

7. Approximation of ellipti integrals

E ient algorithms for the numeri al evaluation of K(r) and E(r) are

based on the arithmeti -geometri mean iteration of Gauss. This fa t led to

some lose majorant/minorant fun tions for K(r) in terms of mean values in

[VV.

Next, let a and b be the semiaxes of an ellipse with a > b and e entri ity

e =√a2 − b2/a, and let L(a, b) denote the ar length of the ellipse. Without

loss of generality we take a = 1. In 1742, Ma laurin ( f. [AB) determined

that

L(1, b) = 4E(e) = 2π · 2F1(12 ,−1

2 ; 1; e2).

In 1883, Muir ( f. [AB) proposed that L(1, b) ould be approximated by

the expression 2π[(1 + b3/2)/2]2/3. Sin e this expression has a lose resem-

blan e to the power mean values studied in [VV, it is natural to study the

sharpness of this approximation. Close numeri al examination of the error

in this approximation led Vuorinen [V2 to onje ture that Muir's approxi-

mation is a lower bound for the ar length. Letting r =√1− b2, Vuorinen

asked whether

(7.1)

2

πE(r) = 2F1

(

12 ,−1

2 ; 1; r2)

>

(1 + (r′)3/2

2

)2/3


for all r ∈ [0, 1].In [BPR1 Barnard and his oauthors proved that inequality (7.1) is true.

In fa t, they expanded both fun tions into Ma laurin series and proved that

the dieren es of the orresponding oe ients of the two series all have the

same sign.

Later, the same authors [BPR2 dis overed an upper bound for E that

omplements the lower bound in (7.1):

(7.2)

2

πE(r) = 2F1

(

12 ,−1

2 ; 1; r2)

6

(1 + (r′)2

2

)1/2, 0 6 r 6 1.

See also [BPS.

In [BPR2 the authors have onsidered 13 histori al approximations (by

Kepler, Euler, Peano, Muir, Ramanujan, and others) for the ar length of an

ellipse and determined a linear ordering among them. Their main tool was

the following Lemma 7.3 on generalized hypergeometri fun tions. These

fun tions are dened by the formula

pFq(a1, a2, · · · , ap; b1, b2, · · · , bq; z) ≡ 1 +

∞∑

n=1

Πpi=1(ai, n)

Πqj=1(bj , n)

· zn

n!,

where p and q are positive integers and in whi h no denominator parameter

bj is permitted to be zero or a negative integer. When p = 2 and q = 1, thisredu es to the usual Gaussian hypergeometri fun tion F (a, b; c; z).

7.3. Lemma. Suppose a, b > 0. Then for any ǫ satisfying ab1+a+b < ǫ < 1,

3F2(−n, a, b; 1 + a+ b, 1 + ǫ− n; 1) > 0

for all integers n > 1.

7.4. Some inequalities for K(r). At the end of the pre eding se tion we

pointed out that upper and lower bounds an be found for K(r) in terms

of mean values. Another sour e for the approximation of K(r) is based on

the asymptoti behavior at the singularity r = 1, where K(r) has logarith-mi growth. Some of the approximations motivated by this aspe t will be

dis ussed next.

Anderson, Vamanamurthy, and Vuorinen [AVV2 approximated K(r) bythe inverse hyperboli tangent fun tion arth, obtaining the inequalities

(7.5)

π

2

(

arth r

r

)1/2

< K(r) <π

2

arth r

r,

for 0 < r < 1. Further results were proved by Laforgia and Sismondi [LS.

Kühnau [Ku and Qiu [Q proved that, for 0 < r < 1,

9

8 + r2<

K(r)

log(4/r′).

Qiu and Vamanamurthy [QVa proved that

K(r)

log(4/r′)< 1 +

1

4(r′)2 for 0 < r < 1.


Several inequalities for K(r) are given in [AVV5, Theorem 3.21. Later Alzer

[A3 showed that

1 +( π

4 log 2− 1)

(r′)2 <K(r)

log(4/r′),

for 0 < r < 1. He also showed that the onstants

14 and π/(4 log 2) − 1 in

the above inequalities are best possible.

For further renements, see [QVu1, (2.24) and [Be.

Alzer and Qiu [AlQ1 have written a related manus ript in whi h, besides

proving many inequalities for omplete ellipti integrals, they have rened

(7.5) by proving that

π

2

(arth r

r

)3/4< K(r) <

π

2

arth r

r.

They also showed that 3/4 and 1 are the best exponents for (arth r)/r on

the left and right, respe tively.

One of the interesting tools of these authors is the following lemma of

Biernaki and Krzy» [BK (for a detailed proof see [PV1):

7.6. Lemma. Let rn and sn, n = 1, 2, . . . be real numbers, and let the power

series R(x) =∑

∞

n=1 rnxnand S(x) =

∑

∞

n=1 snxnbe onvergent for |x| <

1. If sn > 0 for n = 1, 2, . . ., and if rn/sn is stri tly in reasing (resp.

de reasing) for n = 1, 2, . . ., then the fun tion R/S is stri tly in reasing

(resp. de reasing) on (0, 1).

7.7. Generalized ellipti integrals. For the ase of generalized ellipti

integrals some inequalities are given in [AQVV. B. C. Carlson has intro-

du ed some standard forms for ellipti integrals involving ertain symmetri

integrals. Approximations for these fun tions an be found in [CG.

8. Landen inequalities

It is well known ( f. [BF) that the omplete ellipti integral of the rst

kind satises the Landen identities

K

(

2√r

1 + r

)

= (1 + r)K(r), K

(

1− r

1 + r

)

=1 + r

2K

′(r).

Re all that K(r) = π2F (12 ,

12 ; 1; r

2). It is thus natural to onsider, as sug-

gested in [AVV3, the problem of nding an analogue of these formulas for

the zero-balan ed hypergeometri fun tion F (a, b; c; r) for a, b, c > 0 and

a+ b = c, at least when the parameters (a, b, c) are lose to (12 ,12 , 1). From

(3.3) it is lear that F (a, b; c; r2) has a logarithmi singularity at r = 1, ifa, b > 0, c = a + b ( f. [AAR). Some renements of the growth estimates

were given in [ABRVV and [PV1.

Qiu and Vuorinen [QVu1 proved the following Landen-type inequalities:

For a, b ∈ (0, 1), c = a+ b,

F

(

a, b; c;

(

2√r

1 + r

)2)

6 (1 + r)F (a, b; c; r2)

6 F

(

a, b; c;

(

2√r

1 + r

)2)

+1

B(R− log 16)


and

1 + r

2F (a, b; c; 1 − r2) 6 F

(

a, b; c;

(

1− r

1 + r

)2)

61 + r

2

[

F (a, b; c; 1 − r2) +1

B(R− log 16)

]

,

with equality in ea h instan e if and only if a = b = 12 . Here B = B(a, b),

the beta fun tion, and R = R(a, b) = −2γ−Ψ(a)−Ψ(b), where Ψ is as given

in Se tion 2.

9. Hypergeometri series as an analyti fun tion

For rational triples (a, b, c) the hypergeometri fun tion often an be ex-

pressed in terms of elementary fun tions. Long lists with su h triples on-

taining hundreds of fun tions an be found in [PBM. For example, the

fun tions

f(z) ≡ zF (1, 1; 2; z) = − log(1− z)

and

g(z) ≡ zF (1, 1/2; 3/2; z2) =1

2log

(

1 + z

1− z

)

have the property that they both map the unit disk into a strip domain.

Observing that they both orrespond to the ase c = a + b one may ask

(see [PV1, PV2) whether there exists δ > 0 su h that zF (a, b; a + b; z) andzF (a, b; a + b; z2) with a, b ∈ (0, δ) map into a strip domain.

Membership of hypergeometri fun tions in some spe ial lasses of univa-

lent fun tions is studied in [PV1, PV2, BPV2.

Referen es

[AS M. Abramowitz and I. A. Stegun, eds: Handbook of Mathemati al Fun -

tions with Formulas, Graphs and Mathemati al Tables, Dover, New York, 1965.

[AB G. Almkvist and B. Berndt: Gauss, Landen, Ramanujan, the arithmeti -

geometri mean, ellipses, pi, and the Ladies Diary, Amer. Math. Monthly 95

(1988), 585608.

[A1 H. Alzer: On some inequalities for the gamma and psi fun tions,Math. Comp.

66 (1997), 373389.

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ANDERSON:

Department of Mathemati s

Mi higan State University

East Lansing, MI 48824, USA

email: andersonmath.msu.edu

FAX: +1-517-432-1562

VAMANAMURTHY:


University of Au kland

Au kland, NEW ZEALAND

email: vamanamumath.au kland.nz

FAX: +649-373-7457

VUORINEN:


University of Helsinki

P.O. Box 4 (Yliopistonkatu 5)

FIN-00014, FINLAND

e-mail: vuorinen s .fi

FAX: +358-9-19123213

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