1oumal of Applied Mathematics and Stochastic Analysis
7, Number 1, Spring 1994, 79-89
INCLUSION RELATIONS BETWEEN CLASSES
OF HYPERGEOMETRIC FUNCTIONS
O.P. AHUJA
of Papua New Guinea
Department of Mathematics
Box 220, University Post Office
University
Papua, NEW GUINEA
M. JAHANGIRI
California State University
Department of Mathematics
Bakersfield, CA 93311 USA
ABSTILCT
The use of hypergeometric functions in univalent function theory
received special attention after the surprising application of such
functions by de Branges in the proof of the 70-year old Bieberbach
Conjecture. In this paper we consider certain classes of analytic functions
and examine the distortion and containment properties of generalized
hypergeometric hnctions under some operators in these classes.
Key words:
Generalized hypergeometr]c functions, analytic
functions.
AMS (MOS) sub]cct classifications:
Primary 33A30; Secon-
dary 30C45.
1. INTRODUCTION
The expansions and generating functions involving associated Lagurre,
Jacobi Polynomials, Bessel functions and their generalizations such as
hypergeometric functions of one and several variables occur frequently in the
seemingly diverse fields of Physics, Engineering, Statistics, Probability,
Operations Research and other branches of applied mathematics (see e.g. Exton
[7] and Schiff [19]). The use of hypergeometric functions in univalent function
theory received special attention after the surprising application of such functions
by de Branges [6] in the proof of the Bieberbach Conjecture [2]; also see [1].
i’Received"
September, 1993. Revised: January, 1994.
2This work was completed while the second author was a Visiting Scholar at the University of
California, Davis.
Printed in the U.S.A. (C) 1994 by North Atlantic Science Publishing Company
79
O.P. AHUJA and M. JAI-IANGII
80
Merkes-Scott [11], Carlson-$haffer [4] and Ruscheweyh-Singh [181 studied the
starlikeness of certain hypergeometric functions. Miller-Mocanu [12] found a
univalence criterion for such functions. The second author and Silvia [8] and
more recently Noor [14] studied the behavior of certain hypergeometric functions
under various operators. In the present paper we consider certain classes of
analytic functions and examine the distortion and containment properties of
generalized hypergeometric functions under some operators in these classes.
Let p and q be natural numbers such that p<_q+l. For aj
(j = 1,2,...,p) and
(k = 1,2,...,q) complex numbers such tha Z 0,
-1,- 2,... (k = 1, 2,..., q), let ,Fq(z), that is
pFq(tl’a2"’"aP;/l’fl2"’"flq;Z)’-n=OE
()()-..() zA
(fll)n(fl2)n" "(#q)n nl.
denote the generalized hypergeomeric function. Here
symbol defined by
+ n)
(A),=..-D(,kF(X)’
i,
[
[ A(A+Z)...(A+n-Z),
(A).
(1.1)
is the Pochhammer
if n-0
if n
1,2,
It is known [9, p. 43] that the series given by (1.1) converges absolutely for
]z[ <oc if p<q+l, and for zU-{z:[z] <1} if p=q+l. Thus for
p<_q+l,
F(,
,
., ; 5,, 5, ., &; z)
,
where .A denotes the class of functions hat are analytic in U.
For oFq(z) defined by (1.1), le
()._ ()._ ...(%)_
for all z
(1.2)
U, where p _< q + 1.
Let S be the family of functions of the form
f(z) = z -t-.
E anZ
that are analytic and univalent in U. For
A by
(z e u)
f S,
(.a)
define the convolution operator
Inclusion Relations Between Classes
of Hypergeometric Functions
A(Cel, o2,..., Op; ill, f12,’" flq; f):
",
-
81
paqf.
The operator "," denotes the Hadmard product or convolution of two power
series
b,.,z ’
a,.,z ’ and g(z)=
f(z) =
n=0
n=0
that is
(f* g)(z) = a,.,b,z r’.
Thus for f of the form (1.3) in S, we may write
oo
where p
(Ol)n- I((R2)n -1"" "((Y’p)n-1
anzn(z e U),
_< q + 1.
We recall that a function is convex in U if it is univalent conformal
mapping of U onto a convex domain. It is well-known that I) is convex in U if
and only if
+
>o
(z e u)
’
O. Also, a function f is said to be close-to-convex in U if there exists
convex function (I) in U such that Re(f’(z)/ff2’(z)) > 0 (z E U). For 0 a < 1, we
define the following subclasses of A:
and
{ ,Fq e A: Re ,Fq(z) >
R(a, r): = {f e A" Ref’(z) > a for
S(a): = {f A: Re(f(z)/z)>
A(p, q; a):
T: = {f e S: f(z) = z--
akz
k
z
e U,
k=2
T*(a)" = {f e T: Re(z f’(z)/f(z)) >
C(a): = {f e T: Re(1 + zf"(z)/f’(z)) >
ak
> O},
O.P. AHUJA and M. JArIANGI
82
2. MAIN RESULTS
We use the following lemma due to Chen [5] to prove our first theorem.
Lemma 1:
If f S(a),
then
if O <_r < l/2,
Re f’(z) >_
a
2at 2 + (2a- 1)r4
if l/2 < r < l.
(i ,’)
Let
Theorem 1:
then
(i)
ForO_< [zl -r<l/2, z -+Gq+ 6 A(p+l,q+l;X), where
(ii)
For1 zl
-r
< l, z-X+Gq+x A(P+l,q+l;Y), where
Y=
(1 O)r4
3’(1 r) ’
7
>- 1.
We observe that
Proof:
z
(i i/7) ,G(z) +
,G’(z) : + G + (z).
Thus by an application of Lemma 1, we have
G’q(z))
Re(z + 1Gq +i(z)) = (1 1/7)ie(z -’ Gv(z)) +
-
}Re(
>_ (I- I/7)a d
i
+ 2(2a- 1)r+ (2a- l)r
7(i + )
=X, if0_<
and
.
Izl =r<l/2,
+ Gq + (z)) >_ (1 1/7)a +
a
2ar
+ (2a 1)r4
7( 1 r2)
Inclusion Relations Between Classes
=Y, ill/2_<
To prove our next theorem
MacGregor [10].
of Hypergeometric Functions
Izl
83
=r<l.
we need the
following lemma due to
Lemma 2:
f,(z) < +
z
and
+ 2oa(Z +
z
z
I) < If(z)! <
2tog(X
z
z
I).
Let
Theorem 2:
-r(prq(z)) = ’z "[’-i
/f .r(,Fq) e R(0,1), then
(i)
pFq(z)[<
1
+-(i
(ill
F(z)[ >_
1
---(1 -llz[
Jt-lFq(t)
i’ll’
(7>_ I).
dt
0
+
log(1
l,
Z I))"
z
-" 1- ’7=1z1= foe(1 it-
Note that
Proof:
(I- 1/"/).r(vFq(z))-F z(q(,Fq(z))= z ,Fq(z).
Since
Re(q’.r(vFq(z)) > O,
we obtain from
(2.1) and Lemma 2
Fq()I_<
z
=-1+- ( i
1
and
Fq(z)]>_ (I
1/-r)
-
1-"/log(l_lz
lki +
z
>-(1-1/7)+2(1-1/7) log(l +
z
),
)-(l + )
z
O.P. AHUJA and M. JAHANGI
84
Theorem 3:
If oFg A(p, q; a), 0 <_ a < 1,
then for 7 >- 1,
We observe that
Proof:
+ Fq + (c, a,..., at,
3’; 3x, fl,...,
flq, 3’ + 1; z) =
z7_
i
f
t
0
Now by applying a result of Own [15] that if f S(a), then
f(z) <
Z
1
-
f(t)dt.
+(1--2 )1zl
1-lzl
we obtain
3’
t
0
7
1+(I --20)Izl
1
/((1-I- (1- 2)
Next
_7)dt
z
I)
we use a result of Nikolaeva and Repnina
[13] concerning the convex
combination of certain analytic functions to prove our Theorem 4.
Lemma 3:
Let f e R(O, 1). Denote
h0=
Let
#3
(1 + v/i
Ft,(z) (1 #)f(z) + l, zf’(z),
Then
0_<#_<1.
h
c/(1 c).
F. e R(a, r) where
ifh>_ho
r
ifO<h<ho.
Inclusion Relations Between Classes
of Hypergeometc Functions
85
If Gq R(a, 1), 0 < a < 1, then the function
+Gq+x(z) = z+Fq+x(a,a,...,a,l +7; x,,...,#,7;z)
Theorem 4:
is in
R(a, r) where
(it)
r
.
= r(,7) =
g 0 g. < /( + ( + ( + a))) fo
Proof:
Noting that
and comparing
manipulations.
wigh
Lemma 3, the
If Gq R(c, 1),
univalence of p + iGq + for 7 >- 1.
Theorem 5"
If
Corollary:
is convex,
follows by simple algebraic
then r = r(a,7 ) is also the radius
of order a(O _< a < 1), then so
r,+Fq+(al,a2,...,ap,7 + 1;/,52,...,flq,7 + 2;z),
close-to-convex,
z
resulg
or starlike
for Re(7) > 0.
Proof:
Consider the general transform
7(f(z))
7z,+ 1
f
tT_
f(t)dt, (Re(7) > 0).
0
Then
.,(z .F(z))
+2"7
’(z pFq(t))dt
zn+l
n!
is
of
O.P. AHUJA and M. JAIGI1%I
86
where
H() =
"r+ lz,
[16] to be convex in U. It follows from the work of Ruscheweyh and
Sheil-$mall [17] that the function (I).r(z Fg) is convex, close-o-convex, or sarlike
is known
of order a whenever
f is such. Now he result follows because we may write
.r(z vFq(z))= z
.
=,o
(a)"(a)""’(av)"(’7 + 1),, z..y.
(/),,()...(),( + ), !
Our last theorem deals wih hypergeomeric functions with negative
coefficients. Its proof uses the following lemma which is due to Silverman [20].
Lemma 4:
Let
f(z)=z(i)
Eaz, a>O.
(n a)a. <_ 1 a,
f e T*(a).
(ii) f e C(a). n=2
E n(n a)a. <_ l a,
(iii) f e T*(a)=[a. _<’-",=. (n _> 2),
(iv) f e C(a)= a, <_ ,(-%) (n >_ 2).
Theorem 6:
For"
f(z)
the function
z-
a,z k e T*(a),
Inclusion Relations Between Classes
of Hypergeometric Functions
87
E (- ) > , + ,
P
./=1
Proof:
For
f(z) = z-and using
akz
k
eT
(1.5), we have
where
d,
(Ol)k- l(O2)k- 1)" .(Op + 1)k-1
(/l)k- l(2)k- 1’" "(p)k-l(1)k 1 ak>O.
In view of Lemma 4, the function given by (2.3) is in T*(a)if and only if
1--o:
_(
=
(a)!a):.,(a+ !)t;
-_o
()(&)...(G) ]
,+F,(a,a2,...,a,+;fl,fl2,...,fl,;1)- 1.
The desired result follows because the +
Izl = l if
(( )k=l
see
1
F series is absolutely convergent for
,, ,)
+
> o;
[9, p. 44].
Remark 1"
Using (ii) and (iv)in Lemma 4, we can similarly prove
that Theorem 6 holds for C(a), that is,
O.P. AHUJA and M. JAHANGI
88
if and only if the conditions of Theorem 6 hold.
In view of the generalization of the Gaussian summation
Remark 2:
formula for p = 2, 3,..., determined in [3], the condition (2.2) may be expressed
&S
< sr()...r(, + )r( + )r( + )
Ao)
(;=s)(+ S)
r(z,)r(z;)...r(z)r()
(s)
where
AP)’s are given by some lengthy expressions in [3] and
p
=
E (/- )- + > 0.
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[1]
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[3]
Bhring, W., Generalized hypergeometric functions at unit argument, Proc. Amer.
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Inclusion Relations Between Classes
of Hypergeometric Functions
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