A PROPERTY OF THE HYPERGEOMETRIC
MEAN VALUE1
B. C. CARLSON AND M. D. TOBEY
1. Introduction.
The
mean
of order
t of the positive
values
x
= (xi, x2, ■ ■ ■ , xn) with positive
weights
w=iwi,
Wi, ■ ■ ■ , wH),
~^Zwi= 1, is Mtix, w) = i ^Zwix'i)11', an increasing function of the real
parameter
t. As shown recently
[l], [3], Mt is a limiting case of a
homogeneous mean constructed
from a hypergeometric
function of n
variables.
The hypergeometric
mean Mit, c; x, w) is an increasing
function of t for any positive value of the second parameter
c and
reduces to Mt as e—>0. The present paper is concerned with its de-
pendence on the positive c-parameter.
It is shown (Theorem 4) that
Mit, c) is an increasing or decreasing
function of c according as
/<1
or t>l.
The
hypergeometric
function
Ri~t,
cw, x) =
[Mit, c; x, w)]', already known [l] to be log convex in t, is shown
(Theorems 5 and 6) to be concave in c if 0<t<l,
convex in c if 1 <t
<2, and log convex in c if t<0 or t = 2, 3, ■ ■ ■ . Log convexity in c
for every t>l is conjectured
but not proved.
Monotony in c will be proved first for the hypergeometric
^-function by using an integral representation
[l, Equation
(2.4)] constructed as follows. The quantity (SM»X>)! is integrated with weight
function P over all possible choices of the positive weights u satisfying ^Zu{=l.
Because m„=1— «i— • • • —un-i, we have an (» —1)fold integral
(1.1)
where
il^i^n)}.
with respect
Ri-t,
du' = duidui
to Ui, uit ■ ■ ■ , un-i:
b, x) = j
■ ■ ■ dun-i
( £ Uix)j Pib, u)du',
and
E=
{(«i,
w2, • ■ ■ , un-i)\ui>0
The weight function
(1.2)
Pib, u) = -
I I Ui
Vibi) ■ ■ ■ Yibn)
tl
satisfies JePib, u)du' = l. We shall be concerned with the case in
which the positive
parameters
b = Q>i, • • • , bn) = icwi, ■ ■ ■ , cwn)
= cw all vary in proportion to the positive parameter c. If the positive constants w are normalized by zZwi= 1> then c= ^Zbt, but this
Received by the editors June 15, 1966.
1 Work performed
sion.
in the Ames Laboratory
of the U. S. Atomic Energy
255
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Commis-
256
B. C. CARLSONAND M. D. TOBEY
[April
normalization
of w is inessential and sometimes inconvenient.
Although R( —t, cw, x) may be denned for complex t, c, w, and x,
we shall be interested in the real domain with one exception: it is
convenient
to consider complex c in order to justify differentiating
R with respect to c by taking the derivative under the integral sign.
Assuming the variables x and the weights w to be positive and t to be
real, we observe that the integrand of (1.1) is analytic in c and continuous in c and u at every point of A. If Re c^5>0,
then | JJu™'~1\
= IT^i**-1 ar,d Js(^2iUiXi)t\[ufi~ldu'
converges uniformly (if it is
improper) by the Weierstrass Af-test. Hence this integral is analytic
in c and may be differentiated under the integral sign [4]. Multiplying by a ratio of gamma functions, we see that R( —t, cw, x) is analytic in c if Re c> 0 and may be differentiated
under the integral sign.
2. Monotony in c. We shall prove first that the A-function in two
variables is monotonic in c and subsequently
remove the restriction
to two variables by induction. For both parts of the proof we shall
need the following lemma.
Lemma 1. Let G(u) satisfy G(u)>0 if 0<a<u<p<l,
G(u)<0 if
0<u<a
or /3<w<l, and flL(u)G(u)du
= Qfor every linear function L.
If C is a strictly convex function of u, then /lC(u)G(u)du
<0. 1/ C is
strictly concave, the inequality is reversed.
Proof.
Choose L(u) = C(a)(P-u)/(fl-a)
C is strictly
0<u<a
convex,
then
or &<u<l.
C(u) <L(u)
+ C(fi)(u-a)/(fl-a).
if a<M</3
Hence foC(u)G(u)du<foL(u)G(u)du
Theorem
1. Assume « = 2, x,->0 and wt>0
and e>0, then
dR(-t,
cw, x)/dc < 0,
with reversed inequality
Proof.
and C(u)>L(u)
(i=l,
If
if
= 0.
2). 1/ »i^x2
(/ < 0 or I > 1),
tf 0<t<l.
We differentiate
(1.1) with respect to c to obtain
dR( —t, cw, x)/dc = f
J 0
[uxi + (1 — u)xi]lG(u)du,
where
G(u) = dP(cw, u)/dc = P(cw, u)[wi log u + w2 log(l — u) +?A],
K being independent of u. Now P is positive in (0,1) and G is negative
when u is sufficiently close to 0 or 1. However, G must be positive
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i968]
A PROPERTY OF THE HYPERGEOMETRIC MEAN VALUE
257
somewhere in (0,1) because fgPicw, u)du = 1 implies that f0lGiu)du = 0.
The quantity in brackets is concave in u and cannot have more than
two zeros. Hence G satisfies the first two conditions
of Lemma 1.
Moreover, j~\uPicw, u)du = wi/iwi+w2) is independent of c, so that
f0luGiu)du = 0. Hence f01Liu)Giu)du = Q for every linear function L.
We obtain the desired inequality
by applying Lemma 1 with C(u)
= [uxi + il— u)xi]', observing that C is strictly convex in u if t<0
orJ> 1 and strictly concave in u if 0<t<l.
We shall extend this theorem to an arbitrary number of variables
by induction,
using a previously known result [2, Equation
(7.8)]
which we shall now prove more directly. The proof requires the two
following formulas
[l, Equations
(2.,)mM-^S^^-^f-f,
(7, N)
(3.1) and (3.2)]:
mi\ • • • mn\
where y='zZ°i> (°> m)=Yia+m)/Yia),
and the summation extends
over all nonnegative
integers mi, m2, • ■ ■ , mn whose sum is N; and
Ri-t,
A i-t, N)
b, x) = £ ——
i^-i)
AT=0
Ri-N,
b, 1 - x),
A!
(0 < Xi < 1, 1 ^ i g n).
Theorem
2. Assume Xi>0 and bi>0 il^i^n),
Pih, b2; u) =
Tibi)Tibi)
u
and let
(1 - «)
.
Then
Ri-t, b, x) = f P(bu b2;u)Ri-l; bi + b2,b», • • • , bn;
(2.3;i
J 0
uxx + (1 — u)Xi, x3, • • • , xn)du.
Proof.
Since both sides of (2.3) are homogeneous
in x, we may sup-
pose that 0<£,-<l
il^i^n).
By (2.1), (2.2), and the identity
1—uxi —(1 —u)xi = uil— Xi)+ (1— m)(1—Xi), the right side of (2.3)
becomes
f \uPib b-u)YY
(2.4)
Jo
"
ivd
■[«(1 - xi) + (!-«)(!-
{~1'N) (h+^'Wih^s)
iy,N)
■■■jbn,mn)
M\m3\---mnl
x2)]Mil - a,)"* •••(!-
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x.)»»,
258
B. C. CARLSONAND M. D. TOBEY
where y = h+
tive integers
(l^i^n)
and
for all «G[0,
only the part
f
[April
■ • ■ +b„ and the last summation is over all nonnegaM, m3, ■ • ■ , mn whose sum is N. Since 0<x,-<l
7>0, the series converges absolutely
and uniformly
l], and we may integrate term by term. Considering
which depends
on u, we have, by (1.1) and (2.1),
[«(1 - xi) + (1 - «)(1 - x2)]MP(bi,b2;u)du
J 0
= R( —M; bi, b2; 1 — xu 1 — x2)
Ml
_^ (bu mi)(b2, m2)
-C1
(bi + b2, M)
w»i!«5!
=-2-
- »i)mi(l - x2)m',
where the summation
is over all nonnegative
integers mi, m2 whose
sum is M. Substituting
this expression in (2.4) and applying
(2.1)
and (2.2) completes the proof.
While extending
Theorem
1 to an arbitrary
number
of variables, we use the notation
xmBl^= max{xi,
x2, ■ ■ ■ , x„} and xmin
= min{*i,
x2, ■ ■ ■ , xn}.
Theorem
3. Assume
c>0,
Xm»x>*min>0,
and
W{>0
(l^i^n).
Then
dR(-l,
cw, x)/dc < 0,
with reversed inequality
(i < 0 or t>
1),
i/ 0<t<l.
Proof. The proof is by induction on the number of variables, Theorem 1 being the case w = 2. We put bi = cwi (l^i^n)
in (2.3) and
differentiate
with respect to c. Differentiation
under the integral sign
can be shown to be permissible
if Re c>0 by arguments
similar to
those used earlier in discussing the analyticity
of R( —t, cw, x). Hence
dR( —I, cw, x)
dc
= j
R( —t; cwi + cw2, cwz, • ■ ■ , cwn; uxx + (1 — u)x2, xz, ■ ■ ■ . xn)
J o
• [dP(cwi, cw2; u)/dc]du
+
I
P(cwu cw2; u)
^ o
■[dR( —t; cwi + cw^ cws, • ■ ■ , cwn; uxi
+ (1 — u)x2, x3, • • ■ , x„)/dc]du.
By (1.1), R( —t, b, x) is strictly
convex in each of its arguments
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Xi,
i968]
A PROPERTY OF THE HYPERGEOMETRIC MEAN VALUE
x2, • ■ ■ , x„ if /<0
or i>l
and strictly
concave
if 0</<
259
1. Therefore
C(u) = R( —t; cwi + cw2, cw^ ■ ■ ■ , cwn; uxi + (1 — u)x2, x3, ■ ■ • , x„)
has the same strict convexity or concavity in u provided that xi^x2.
As shown in the proof of Theorem 1, G(u) = (d/dc)P(cwi,
cwi\ u)
satisfies the conditions of Lemma 1. Hence the first of the two integrals above has the required sign if Xi?^;
if Xi = X2, then C(u) is independent of u and the first integral vanishes. The second integral contains the derivative of an AM unction with n —1 variables and therefore has the required sign by the inductive hypothesis.
We now restate Theorem 3 in terms of the hypergeometric
mean
value [l, Equations (1.2) and (2.6)], defined for any positive valuesx,
positive weights w(^,Wi=l),
real /, and positive c by
M(t, c; x, w) = [R(-t,
(2.5)
M(0, c;x,w)
(t ^ 0),
— lim M(t, c;x,w)
«-»o
= exp
Theorem
cw, x)]1",
c>0,
UiXij P(cw, u)du'
xmaX>xmin>0,
.
wt>0
(l^i^n),
and
^Wi=l.
Then M(t, c; x, w) is a strictly increasing
t<l and a strictly decreasing /unction of c if t~>l.
/unction
o/ c if
Proof.
4. Assume
j logf ^
If t^O the result follows immediately from (2.5) and The-
orem 3. Since M(0, c;x,w) is homogeneous in x, we may suppose that
0<x,<l
(l^i^n)
and hence we may use the expansion
[l, Equa-
tion (3.3)]
oo
log M(Q, c;x,w)
= -
X) A"1A(-A7, cw, l-x).
AT=1
By (2.1), R( —1, cw, 1 —x) = ^Wi(l— xt) is independent
of c. The-
orem 3 implies that R( —N, cw, 1 —x) is a strictly decreasing function
of c for N = 2, 3, ■ ■ ■ , and hence M(0, c; x, w) is a strictly increasing
function of c.
3. Convexity in c. The proof of Theorem 3 gave no information
about the convexity of R( —t, cw, x) with respect to c. We shall now
consider an alternative
proof which unfortunately
applies only to
restricted values of t but has the advantage of providing information
about convexity.
The alternative
proof rests on a binomial theorem for A-polynomials which follows at once from (1.1):
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260
B. C. CARLSONAND M. D. TOBEY
[April
Ri-N, b,x+X) = A/A^\
J2[ )\N-mRi-m,b,x),
(3.1)
where x+\=
(#i+X, x2+\, • • • , x„+X). We shall need also two
properties
of i?( —N, b, x) which are evident from (2.1). First,
Ri —N; bi, ■ • ■ , b„; Xi, ■ • ■ , xn) is unchanged
by permutations
of
the subscripts 1,2, • • • , n; and secondly,
ibi+
■ ■ ■ + bn, N)Ri-N;
= ibi+
■ ■ ■ +h,
bu ■ ■ ■, bn; xu ■ ■ ■ , xk, 0, ■ • • , 0)
N)Ri-N;
bu ■ ■ ■, bk; xlt • • • , xk).
Theorem
5. Assume c>0, xmax>xmin>0,
and w,->0 (lgigw).
Then, for A = 2, 3, • • • , i?( —N, cw, x) is a strictly decreasing and
strictly log convex function of c.
Proof. If fie) is log convex when c is positive, and if h is a positive
constant, then fQic) is also log convex in c. Hence the normalization
of w does not affect the theorem and we may assume without loss of
generality that
£w,-=l.
The proof is by induction on n, the number of variables. By permuting subscripts
if necessary,
we may assume when w = 2 that
£i>X2>0.
In (3.1) we replace x by x— X and choose X = ^2:
x
Ri — N;cwi,cw2;x1,x2)
»/N\
= Zj[
jv_m
,
}x2
Ri—m;cwi,cWi;xi
— x2,Q)
) *2
(xi — x2)micwi, m)/ic, m),
m=o \mj
= zZ (
where the last equality
follows from (2.1) with y = c. If we write
icwi, m)
cwiicwi + 1) ■ • ■ icwi + m — 1)
(c, m)
cic + 1) • • • (c + m — 1)
[1
— Wi~\
wi-\-—••■
c+ 1J
wi+im-1)—■-
L
1 — Wi 1
c + m — 1J
,
each factor is positive, decreasing, and log convex in c. Since these
properties are preserved under multiplication and addition, the theorem is true for n = 2.
For any number of variables we may assume (by permuting subscripts
if necessary)
if j^k,
for some k such that l^k^n
that
with X = xmin, and using
xmin=xn=xn-i=::
■ ■ • =xk+i
and
Xj>xm\n
—1. Applying (3.1) as before,
(3.2), we have
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i968]
A PROPERTY OF THE HYPERGEOMETRIC MEAN VALUE
R( —N, cw, x) = Zj [
261
) ^N~mR(—m; cwu ■ • • , cw„;
n-o \m/
xi — X, ■ • • , xk — X, 0, • • • , 0)
= Z) (
) \N~mR(-m;
cwh ■ ■ ■ , cwk;
m=o \m/
xi — X, • • • , xk — X)
(cwi + ■ ■ ■ + cwk, m)
(c,m)
The ratio (cwi+ • ■ -+cwk, m)/(c, m) is decreasing and log convex in
c as before since Wi+ • ■ •+w„ = l>wi+
• ■ -+wk. If we make the
inductive
hypothesis
that the theorem is true for k variables
(2^k^n
—1), then R( —m; cwi, • ■ ■ , cwk; xi— X, • • • , xk—X)
also is positive, decreasing, and log convex in c. (If k = l, then
R(—m, cwi, Xi—X) = (xi—X)"* is positive and independent
of c.) The
truth of the theorem for n variables now follows because products and
sums of positive, decreasing, and log convex functions are still positive, decreasing, and log convex.
Theorem
6. Assume
c>0,
xmai>xmin>0,
and
Wi>0
(l^i^n).
Then R(—t, cw, x) is a strictly decreasing and strictly log convex function of c if / < 0, a strictly increasing and strictly concave function of c
if 0<t<l,
and a strictly decreasing and strictly convex function of c if
Kt<2.
Proof.
We may assume
R( —t, cw, x) is homogeneous
as before that ^3W«= I and also, since
of degree t in x, that 0<x,<l
(1 ^i^n).
From (2.2) we have
^, (-*, A)
R( —t, cw, x) = y. -R(
^o
A!
—N, cw, 1 — x)
n
= l-tJ2
»-i
Wi(l- x{)
- (2 - t, N - 2)
+ t(t - 1) Z-'
,
R(~N, cw, 1 - *).
N=i
If t<2, then (2-t, N-2)>0
is decreasing
Nl
for N=2,3,
and log convex by Theorem
• • - . Hence the last series
5, and its product
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with
262
B. C. CARLSONAND M. D. TOBEY
tit— 1) is log convex if t<0 or l<t<2
and concave if 0</<l.
If
1 <t<2,
however, the sum of the two terms which are independent
of c may be negative. The sum of a negative constant and a log convex
function is certainly convex but, even though positive, it may or may
not be log convex.
If x and w are fixed and if t — 1 is positive and sufficiently small, the
part independent
of c is positive and log convexity holds, as it does
also (by Theorem 5) if t — 1 is a positive integer. This suggests the conjecture that Ri —t, cw, x) is log convex in c for every t>l.
References
1. B. C. Carlson, A hypergeometric mean value, Proc. Amer. Math.
Soc. 16 (1965),
759-766.
2. -,
Lauricella's hypergeometric function Fd, J- Math. Anal. Appl. 7 (1963),
452-470.
3. -,
Some inequalities for hypergeometric functions,
Proc. Amer. Math.
Soc.
17 (1966), 32-39.
4. E. C. Titchmarsh,
The theory of functions,
2nd ed., Oxford Univ. Press, Oxford,
1939;pp. 99-100.
Iowa State
University
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