AbstractTurán-type inequalities for combinations of Kummer functions involving Φ(a±ν,c±ν,x) and Φ(a,c±ν,x) have been recently investigated in [Á. Baricz, Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math. 26 (3) (2008) 279–293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of special functions, Constr. Approx. 26 (2007) 1–9]. In the current paper, we resolve the corresponding Turán-type and closely related mean inequalities for the additional case involving Φ(a±ν,c,x). The application to modeling credit risk is also summarized

Barnard, Roger W.

Gordy, Michael B.

Richards, Kendall C.

Elsevier - Publisher Connector 

A note on Turán type and mean inequalities for the Kummer function 

J. Math. Anal. Appl. 349 (2009) 259–263Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa
A note on Turán type and mean inequalities for the Kummer function
Roger W. Barnard a, Michael B. Gordy b,1, Kendall C. Richards c,∗
a Department of Mathematics, Texas Tech University, Lubbock, TX 79409, United States
b Division of Research and Statistics, Federal Reserve Board, Washington, DC 20551, United States
c Department of Mathematics, Southwestern University, Georgetown, TX 78627, United States
a r t i c l e i n f o a b s t r a c t
Article history:
Received 20 June 2008
Available online 22 August 2008
Submitted by B.C. Berndt
Keywords:
Kummer conﬂuent hypergeometric function
Turán-type inequality
Mean
Generalized hypergeometric function
Turán-type inequalities for combinations of Kummer functions involving Φ(a ± ν, c ± ν, x)
and Φ(a, c ± ν, x) have been recently investigated in [Á. Baricz, Functional inequalities
involving Bessel and modiﬁed Bessel functions of the ﬁrst kind, Expo. Math. 26 (3)
(2008) 279–293; M.E.H. Ismail, A. Laforgia, Monotonicity properties of determinants of
special functions, Constr. Approx. 26 (2007) 1–9]. In the current paper, we resolve the
corresponding Turán-type and closely related mean inequalities for the additional case
involving Φ(a ± ν, c, x). The application to modeling credit risk is also summarized.
© 2008 Elsevier Inc. All rights reserved.
1. Introduction
The Kummer conﬂuent hypergeometric function is given by
Φ(α,β, x) ≡
∞∑
n=0
(α)nxn
(β)nn! ,
where (α)n is the Pochhammer symbol deﬁned by (α)n ≡ (α + n)/(α) = α(α + 1) · · · (α + n − 1) for n ∈ N, (α)0 = 1,
(α + 1)−1 = 1α , and the Gamma function is (x) ≡
∫∞
0 t
x−1e−t dt, for x> 0.
Inequalities involving contiguous Kummer conﬂuent hypergeometric functions of the form Φ(a ± ν, c ± ν, x) and
Φ(a, c ± ν, x) were presented in Theorem 2 of [4] and Theorem 2.7 of [11]. These inequalities are of the Turán type [15] in
the case that ν = 1. In the present note, we resolve the remaining Turán-type case involving Φ(a ± 1, c, x) and extend it
to include Φ(a ± ν, c, x), ν ∈ N. We then establish a closely related mean inequality that provides simultaneous upper and
lower bounds for Φ(a, c, x). Turán-type inequalities, which are of independent interest, also have important applications in
Information Theory (as demonstrated by McEliece, Reznick, and Shearer in their paper [12]) and in modeling credit risk, as
summarized below.
In particular, Carey and Gordy [9] model a lending relationship in which the bank has an option to foreclose upon the
borrower at any time. Following the seminal models of Merton [13] and Black and Cox [6], it is assumed that the value
of the ﬁrm’s assets follows a geometric Brownian motion. It is shown that the bank’s optimal foreclosure threshold solves
a ﬁrst order condition involving a ratio of contiguous Kummer functions, which implies that a Turán-type inequality for
the Kummer function arises naturally in studying the comparative statics of the model. A proof of this key Turán-type
* Corresponding author.
E-mail addresses: roger.w.barnard@ttu.edu (R.W. Barnard), michael.gordy@frb.gov (M.B. Gordy), richards@southwestern.edu (K.C. Richards).
1 The opinions expressed here are those of the authors, and do not reﬂect the views of the Board of Governors or its staff.0022-247X/$ – see front matter © 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2008.08.024
260 R.W. Barnard et al. / J. Math. Anal. Appl. 349 (2009) 259–263Fig. 1. Graphs of A(x) = A(Φ(a+1,a+b, x),Φ(a−1,a+b, x)); Φ(x) = Φ(a,a+b, x); and G(x) = G(Φ(a+1,a+b, x),Φ(a−1,a+b, x)); with a = 1, b = 1.5,
and ν = 1.
inequality is established in this paper. (For general background on applications of the Kummer function in economic theory
and econometrics, see [1].)
2. Main results
Theorem 1. Suppose a,b > 0. Then for any ν ∈ N with a,b ν − 1
Φ(a,a + b, x)2 > Φ(a + ν,a + b, x)Φ(a − ν,a + b, x), (1)
for all nonzero x ∈ R. Moreover, these expressions coincide with value 1 when x = 0 and asymptotically for any x when b → ∞.
Corollary 2. Suppose a > 0 and c + 1> 0 with c = 0. Then for any ν ∈ N with a ν − 1,
Φ(a, c, x)2 Φ(a − ν, c, x)Φ(a + ν, c, x), (2)
for all x> 0.
The next result begins with the well-known arithmetic mean-geometric mean inequality,
A(α,β) ≡ α + β
2
>
√
αβ ≡ G(α,β) for α,β distinct and positive, (3)
which has many interesting reﬁnements and applications (e.g., see [7,8]). Corollary 3 is a reﬁnement of inequality (3) with
α = Φ(a + ν,a + b, x) and β = Φ(a − ν,a + b, x) (see illustrated special case in Fig. 1).
Corollary 3. Suppose ν ∈ N and a,b ν . Then for all nonzero x ∈ R
A
(
Φ(a + ν,a + b, x),Φ(a − ν,a + b, x))
> Φ(a,a + b, x)
> G
(
Φ(a + ν,a + b, x),Φ(a − ν,a + b, x)). (4)
It is also interesting to compare these results with the elegant Theorem 2.3 and open problems in [5] regarding Turán-
type and arithmetic mean-geometric mean inequalities involving the Gaussian hypergeometric function 2F1.
3. Proofs
Proof of Theorem 1. First assume x> 0. For c > −1, c = 0, deﬁne
fν(x) ≡ Φ(a, c, x)2 − Φ(a + ν, c, x)Φ(a − ν, c, x).
We will make use of the following contiguous relation (see [10, p. 1013]):
Φ(α + 1, β, x) − Φ(α,β, x) = x
β
Φ(α + 1, β + 1, x). (5)
Subtracting and adding a term to fν+1(x) − fν(x) and applying this contiguous relation, we have that
R.W. Barnard et al. / J. Math. Anal. Appl. 349 (2009) 259–263 261fν+1(x) − fν(x) = Φ(a + ν, c, x)Φ(a − ν, c, x) − Φ(a + ν + 1, c, x)Φ(a − ν − 1, c, x)
= Φ(a − ν, c, x)(Φ(a + ν, c, x) − Φ(a + ν + 1, c, x))
+ Φ(a + ν + 1, c, x)(Φ(a − ν, c, x) − Φ(a − ν − 1, c, x))
= Φ(a − ν, c, x)
(−x
c
)
Φ(a + ν + 1, c + 1, x) + Φ(a + ν + 1, c, x)
(
x
c
)
Φ(a − ν, c + 1, x)
= x
c
gν(x),
where
gν(x) ≡ Φ(a + ν + 1, c, x)Φ(a − ν, c + 1, x) − Φ(a − ν, c, x)Φ(a + ν + 1, c + 1, x).
The Cauchy product reveals
gν(x) =
∞∑
n=0
n∑
k=0
(a + ν + 1)k(a − ν)n−k
k!(n − k)!
(
1
(c)k(c + 1)n−k −
1
(c)n−k(c + 1)k
)
xn
=
∞∑
n=1
n∑
k=0
(a + ν + 1)k(a − ν)n−k
k!(n − k)!
(
(c + k) − (c + n − k)
c(c + 1)n−k(c + 1)k
)
xn
= 1
c
∞∑
n=1
n∑
k=0
Tn,k(2k − n)xn,
where Tn,k ≡ (a+ν+1)k(a−ν)n−kk!(n−k)!(c+1)n−k(c+1)k . If n is even, then
n∑
k=0
Tn,k(2k − n) =
n/2−1∑
k=0
Tn,k(2k − n) +
n∑
k=n/2+1
Tn,k(2k − n)
=
n/2−1∑
k=0
Tn,k(2k − n) +
n/2−1∑
k=0
Tn,n−k
(
2(n − k) − n)
=
[(n−1)/2]∑
k=0
(Tn,n−k − Tn,k)(n − 2k),
where [·] denotes the greatest integer function. Similarly, if n is odd, then
n∑
k=0
Tn,k(2k − n) =
[(n−1)/2]∑
k=0
(Tn,n−k − Tn,k)(n − 2k).
Therefore,
fν+1(x) − fν(x) = x
c
gν(x) = x
c2
∞∑
n=1
[(n−1)/2]∑
k=0
(Tn,n−k − Tn,k)(n − 2k)xn. (6)
Simplifying, we ﬁnd that
Tn,n−k − Tn,k = (a + ν + 1)n−k(a − ν)k − (a + ν + 1)k(a − ν)n−kk!(n − k)!(c + 1)n−k(c + 1)k
= (a + ν + 1)k(a − ν)k
k!(n − k)!(c + 1)n−k(c + 1)k
(
(a + ν + 1)n−k
(a + ν + 1)k −
(a − ν)n−k
(a − ν)k
)
= (a + ν + 1)k(a − ν)k
k!(n − k)!(c + 1)n−k(c + 1)k
(
h(a + ν + 1) − h(a − ν)), (7)
where h(β) ≡ (β)n−k
(β)k
. For β > 0 and n − k > k (i.e., for [(n − 1)/2] k), the logarithmic derivative of h satisﬁes
h′(β)
h(β)
= Ψ (β + n − k) − Ψ (β + k) > 0,
where Ψ ≡ ′/ is the digamma function. Hence, h is increasing under the conditions stated. This fact together with (6)
and (7) yield
262 R.W. Barnard et al. / J. Math. Anal. Appl. 349 (2009) 259–263fν+1(x) − fν(x) = x
c2
∞∑
n=1
[(n−1)/2]∑
k=0
(Tn,n−k − Tn,k)(n − 2k)xn > 0 (8)
when a ν  0, since x> 0 and c + 1> 0, c = 0.
Thus, (8) implies that
fν+1(x) =
(
fν+1(x) − fν(x)
)+ ( fν(x) − fν−1(x))+ · · · + ( f1(x) − f0(x))> 0,
for a ν > ν − 1 · · · 0 and f0(x) = 0. Replacing ν by ν − 1, we conclude that, for x> 0,
fν(x) > 0, for each ν ∈ N satisfying a ν − 1. (9)
Moreover, under these conditions, fν is absolutely monotonic on (0,∞) (i.e., f (n)ν (x) > 0 for n = 0,1,2, . . .). With c = a + b,
this proves Theorem 1 for the case that x> 0.
Now suppose x< 0, a,b > 0, and ν ∈ N with a,b ν − 1. Taking advantage of the available symmetry with c = a+ b, we
can interchange a and b in (1) to arrive at
Φ(b,a + b,−x)2 − Φ(b + ν,a + b,−x)Φ(b − ν,a + b,−x) > 0.
Kummer’s transformation [2, p. 191], Φ(α,β,−x) = e−xΦ(β − α,β, x), yields
e−2xΦ(a,a + b, x)2 − e−2xΦ(a − ν,a + b, x)Φ(a + ν,a + b, x) > 0.
Thus, (1) also holds for x< 0. 
Proof of Corollary 2. It follows from the proof of Theorem 1 that (9) holds for x > 0 under the conditions that c + 1 > 0,
c = 0. 
Proof of Corollary 3. First suppose x  0 and let a,b  ν , ν ∈ N. The ﬁrst inequality in (4) is a direct consequence of the
fact that A((a + ν)n, (a − ν)n) = (a)n for n = 0,1 and
A
(
(a + ν)n, (a − ν)n
)
> (a)n for all n 2,
which follows by induction. Thus,
A
(
Φ(a + ν,a + b, x),Φ(a − ν,a + b, x))= ∞∑
n=0
A((a + ν)n, (a − ν)n)xn
(a + b)nn!
>
∞∑
n=0
(a)nxn
(a + b)nn! = Φ(a,a + b, x).
For x 0, the second inequality in (4) follows by taking the square-root across (1), which is allowed since the right-hand
side of (1) is nonnegative when a,b  ν .
Now suppose x< 0 with a,b  ν . Interchanging a and b in (4), we have
A
(
Φ(b + ν,a + b,−x),Φ(b − ν,a + b,−x))
> Φ(b,a + b,−x) > G(Φ(b + ν,a + b,−x),Φ(b − ν,a + b,−x)).
Kummer’s transformation and the homogeneity of A and G yield
e−x A
(
Φ(a − ν,a + b, x),Φ(a + ν,a + b, x))
> e−xΦ(a,a + b, x) > e−xG(Φ(a − ν,a + b, x),Φ(a + ν,a + b, x)).
Thus, (4) also holds for x< 0. 
4. Concluding remarks
The proof of Theorem 1 can also be used to verify cases when the Turán-type inequality reverses. For example, if a < 0
and c + 1< 0 with [a] = [c + 1] and c not a negative integer, then
Φ(a, c, x)2 < Φ(a + 1, c, x)Φ(a − 1, c, x) for all x> 0.
To see this, take ν = 0 in (6) and then simplify to ﬁnd that
Φ(a, c, x)2 − Φ(a + 1, c, x)Φ(a − 1, c, x) = x
ac2
∞∑( [(n−1)/2]∑ (a)k(a)n−k(n − 2k)2
(c + 1)k(c + 1)n−kk!(n − k)!
)
xn. (10)n=1 k=0
R.W. Barnard et al. / J. Math. Anal. Appl. 349 (2009) 259–263 263The result follows by noting that (a)m and (c + 1)m will have the same signs under the stated conditions (unless (a)m = 0
for some m 2). Hence
1
ac2
[(n−1)/2]∑
k=0
(a)k(a)n−k(n − 2k)2
(c + 1)k(c + 1)n−kk!(n − k)!  0
for all n ∈ N. Moreover, the ﬁrst nonzero term in the series in (10) simpliﬁes to x2
c2(c+1) < 0.
Finally, we note that the techniques of proof presented here can be used to obtain a result similar to (4) with Φ = 1F1
replaced by the generalized hypergeometric function p Fq , where
p Fq(a1, . . . ,ap;b1, . . . ,bq; x) ≡
∞∑
n=0
(a1)n · · · (ap)nxn
(b1)n · · · (bq)nn! .
It can be shown that if p  q + 1, α > 1, bi > 0 for i = 1, . . . ,q, and ai > bi for i = 1, . . . , p − 1, then for all x > 0 in the
interval of convergence,
A
(
p Fq(α + 1,ai;b j; x), p Fq(α − 1,ai;b j; x)
)
> p Fq(α,ai;b j; x) > G
(
p Fq(α + 1,ai;b j; x), p Fq(α − 1,ai;b j; x)
)
(11)
where p Fq(α,ai;b j; x) ≡ p Fq(α,a1, . . . ,ap−1;b1, . . . ,bq; x).
Of particular interest is the case that p = 2 and q = 1. In this case, inequality (11) completes the results of M.E.H. Ismail
and A. Laforgia [11] and of Á. Baricz [3,5] regarding the Gaussian hypergeometric function 2F1. See for example Theo-
rems 2.13 and 2.14 in [11] and Theorem 2.17 in [3].
The ﬁrst inequality in (11) follows as in Theorem 1. The second inequality in (11) follows by using a generalized version
of (5) (see [14, Identity 30, p. 440]) to reveal that
F (x) ≡ p Fq(α,ai;b j; x)2 − p Fq(α + 1,ai;b j; x)p Fq(α − 1,ai;b j; x)
= x
∏p−1
i=1 a
2
i
α
∏q
i=1 b
2
i
∞∑
n=1
( [(n−1)/2]∑
k=0
(α)k(α)n−k
∏p−1
i=1 ((ai + 1)k(ai + 1)n−k)
k!(n − k)!∏qi=1((bi + 1)k(bi + 1)n−k) Rn,k
)
xn,
where
Rn,k =
(∏q
i=1(bi + n − k)∏p−1
i=1 (ai + n − k)
−
∏q
i=1(bi + k)∏p−1
i=1 (ai + k)
)
(n − 2k).
For n − k > k, the positivity of Rn,k (and hence F ) follows when r 	→
∏q
i=1(bi+r)∏p−1
i=1 (ai+r)
is increasing, which is the case under the
stated conditions on the ai ’s and bi ’s.
Acknowledgment
The authors wish to acknowledge the referee’s comments and suggestions which enhanced this paper.
References
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A note on Turán type and mean inequalities for the Kummer function

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