Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 159029, 11 pages
doi:10.1155/2008/159029
Research Article
Starlike and Convex Properties for
Hypergeometric Functions
Oh Sang Kwon1 and Nak Eun Cho2
1
2
Department of Mathematics, Kyungsung University, Busan 608-736, South Korea
Department of Applied Mathematics, Pukyong National University, Busan 608-737, South Korea
Correspondence should be addressed to Nak Eun Cho, necho@pknu.ac.kr
Received 13 February 2008; Accepted 18 June 2008
Recommended by Linda Sons
The purpose of the present paper is to give some characterizations for a Gaussian hypergeometric
function to be in various subclasses of starlike and convex functions. We also consider an integral
operator related to the hypergeometric function.
Copyright q 2008 O. S. Kwon and N. E. Cho. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let T be the class consisting of functions of the form
fz z −
∞
an zn ,
an ≥ 0,
1.1
n2
that are analytic and univalent in the open unit disk U {z : |z| < 1}. Let T∗ α and Cα
denote the subclasses of T consisting of starlike and convex functions of order α 0 ≤ α < 1,
respectively 1.
Recently, Bharati et al. 2 introduced the following subclasses of starlike and convex
functions.
Definition 1.1. A function f of the form 1.1 is in Sp Tα, β if it satisfies the condition
Re
zf z
fz
zf z
≥ α
− 1 β,
fz
and f ∈ UCTα, β if and only if zf ∈ Sp Tα, β.
α ≥ 0, 0 ≤ β < 1,
1.2
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International Journal of Mathematics and Mathematical Sciences
Definition 1.2. A function f of the form 1.1 is in PTα if it satisfies the condition
zf z
zf z
Re
α≥
− α, 0 < α < ∞,
fz
fz
1.3
and f ∈ CPTα if and only if zf ∈ PTα.
Bharati et al. 2 showed that Sp Tα, β T∗ α β/1 α, UCTα, β Cα β/1 α, PTα T∗ 1 − α 0 < α ≤ 1, and CPTα C1 − α 0 < α ≤ 1. In particular, we
note that UCT1, 0 is the class of uniformly convex functions given by Goodman 3 also see
4–6.
Let Fa, b; c; z be the Gaussian hypergeometric function defined by
Fa, b; c; z ∞
an bn
n0
cn 1n
zn ,
1.4
where c /
0, −1, −2, . . ., and λn is the Pochhammer symbol defined by
λn 1
if n 0,
λλ 1 · · · λ n − 1
if n ∈ N {1, 2, . . .}.
1.5
We note that Fa, b; c; 1 converges for Rec − a − b > 0 and is related to the Gamma function
by
Fa, b; c; 1 ΓcΓc − a − b
.
Γc − aΓc − b
1.6
Silverman 7 gave necessary and sufficient conditions for zFa, b; c; z to be in T∗ α and
Cα, and also examined a linear operator acting on hypergeometric functions. For the other
interesting developments for zFa, b; c; z in connection with various subclasses of univalent
functions, the readers can refer to the works of Carlson and Shaffer 8, Merkes and Scott 9,
and Ruscheweyh and Singh 10.
In the present paper, we determine necessary and sufficient conditions for zFa, b; c; z
to be in Sp Tα, β, UCTα, β, PTα, and CPTα. Furthermore, we consider an integral
operator related to the hypergeometric function.
2. Results
To establish our main results, we need the following lemmas due to Bharati et al. 2.
Lemma 2.1. i A function f of the form 1.1 is in Sp Tα, β if and only if it satisfies
∞
n1 α − α β an ≤ 1 − β.
2.1
n2
ii A function f of the form 1.1 is in UCTα, β if and only if it satisfies
∞
n n1 α − α β an ≤ 1 − β.
n2
2.2
O. S. Kwon and N. E. Cho
3
Lemma 2.2. i A function f of the form 1.1 is in PTα if and only if it satisfies
∞
n − 1 αan ≤ α.
2.3
n2
ii A function f of the form 1.1 is in CPTα if and only if it satisfies
∞
nn − 1 αan ≤ α.
2.4
n2
Theorem 2.3. i If a, b > −1, c > 0, and ab < 0, then zFa, b; c; z is in Sp Tα, β if and only if
c ≥ab1−
1 αab
.
1 − β
2.5
ii If a, b > 0 and c > a b 1, then F1 a, b; c; z z2 − Fa, b; c; z is in Sp Tα, β if and
only if
ΓcΓc − a − b
1 αab
1
≤ 2.
Γc − aΓc − b
1 − βc − a − b − 1
2.6
Proof. i Since
∞
ab ∞ a 1n−2 b 1n−2 n
a 1n−2 b 1n−2 n
ab z z − z ,
zFa, b; c; z z c n2 c 1n−2 1n−1
c n2 c 1n−2 1n−1
2.7
according to i of Lemma 2.1, we must show that
∞
a 1n−2 b 1n−2 c ≤ 1 − β.
n1 α − α β
c 1n−2 1n−1
ab
n2
2.8
Noting that λn λλ 1n−1 and then applying 1.6, we have
∞
a 1n b 1n
c 1n 1n1
n 21 α − α β
n0
1 α
∞
a 1n b 1n
n0
1 α
c 1n 1n
1 − β
∞
an bn
c ab n1 cn 1n
Γc 1Γc − a − b − 1
c ΓcΓc − a − b
1 − β
−1 .
Γc − aΓc − b
ab Γc − aΓc − b
2.9
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International Journal of Mathematics and Mathematical Sciences
Hence, 2.8 is equivalent to
Γc 1Γc − a − b − 1
c−a−b−1
c
c
1 α 1 − β
≤ 1 − β
0.
Γc − aΓc − b
ab
|ab| ab
2.10
Thus, 2.10 is valid if and only if 1 α 1 − βc − a − b − 1/ab ≤ 0 or, equivalently,
c ≥ a b 1 − 1 αab/1 − β.
ii Since
F1 a, b; c; z z −
∞
an−1 bn−1
n2
cn−1 1n−1
zn ,
2.11
by i of Lemma 2.1, we need only to show that
∞
n1 α − α β
n2
an−1 bn−1
≤ 1 − β.
cn−1 1n−1
2.12
Now,
∞
∞
∞
an−1 bn−1
nan bn
an bn
1 α
1 − β
n1 α − α β
c
1
c
1
cn 1n
n−1
n−1
n
n
n2
n1
n1
∞
∞
a 1n−1 b 1n−1
an bn
1 αab 1 − β
c
c 1n−1 1n−1
cn 1n
n1
n1
1 αab
ΓcΓc − a − b
1 − β − 1 − β.
Γc − aΓc − b c − a − b − 1
2.13
But this last expression is bounded above by 1 − β if and only if 2.6 holds.
Theorem 2.4. i If a, b > −1, ab < 0, and c > a b 2, then zFa, b; c; z is in UCTα, β if and
only if
1 αa2 b2 3 2α − βabc − a − b − 2 1 − βc − a − b − 22 ≥ 0.
2.14
ii If a, b > 0 and c > a b 2, then F1 a, b; c; z z2 − Fa, b; c; z is in UCTα, β if and
only if
3 2α − β
1 αa2 b2
ΓcΓc − a − b
ab
1 ≤ 2.
Γc − aΓc − b 1 − βc − a − b − 22
1−β
c−a−b−1
2.15
O. S. Kwon and N. E. Cho
5
Proof. i Since zF has the form 2.7, we see from ii of Lemma 2.1 that our conclusion is
equivalent to
∞
a 1n−2 b 1n−2
c
1 − β.
n n1 α − α β
≤
c
1
1
|ab|
n−2
n−1
n2
2.16
Writing n 2n 21 α − α β 1 αn 12 2 α − βn 1 1 − β, we see that
∞
a 1n b 1n
n 2 n 21 α − α β
c 1n 1n1
n0
1 α
∞
n 1
n0
∞
a 1n b 1n
a 1n b 1n
2 α − β
c 1n 1n
c 1n 1n
n0
∞
a 1n b 1n
1 − β
c 1n 1n1
n0
∞
∞
a 2n b 2n
a 1n b 1n
1 αa 1b 1 3 2α − β
c1
c
2
1
c 1n 1n
n
n
n0
n0
1 − β
2.17
∞
an bn
c ab n1 cn 1n
Γc 1Γc − a − b − 2
1 αa 1b 1 3 2α − βc − a − b − 2
Γc − aΓc − b
1 − βc
1−β
c − a − b − 22 −
.
ab
ab
This last expression is bounded above by 1 − βc/|ab| if and only if
1 αa 1b 1 3 2α − βc − a − b − 2 1−β
c − a − b − 22 ≤ 0,
ab
2.18
which is equivalent to 2.14.
ii In view of ii of Lemma 2.1, we need only to show that
∞
an−1 bn−1
n n1 α − α β
≤ 1 − β.
cn−1 1n−1
n2
2.19
Now,
∞
an1 bn1
n 2 n 21 α − α β
cn1 1n1
n0
1 α
∞
n0
n 2
2 an1 bn1
cn1 1n1
∞
a b
− α β n 2 n1 n1 .
cn1 1n1
n0
2.20
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International Journal of Mathematics and Mathematical Sciences
Writing n 2 n 1 1, we have
∞
n 2
n0
∞
n0
n 22
∞
∞
an1 bn1 an1 bn1 an1 bn1
,
cn1 1n1 n0 cn1 1n
cn1 1n1
n0
∞
∞
∞
a b
an1 bn1 an1 bn1 an1 bn1
n 1 n1 n1 2
cn1 1n1 n0
cn1 1n
c
1
cn1 1n1
n1
n
n0
n0
∞
an1 bn1
n1
cn1 1n−1
3
∞
an1 bn1
n0
cn1 1n
∞
an bn
n1
cn 1n
.
2.21
Substituting 2.21 into the right-hand side of 2.20, we obtain
∞
∞
∞
an2 bn2
an1 bn1
an1 bn1
3 2α − β
1 − β
.
1 α
cn2 1n
cn1 1n
cn1 1n1
n0
n0
n0
2.22
Since ank ak a kn , we write 2.22 as
1 αa2 b2 Γc 2Γc − a − b − 2
ab Γc 1Γc − a − b − 1
3 2α − β
c2
Γc − aΓc − b
c
Γc − aΓc − b
ΓcΓc − a − b
1 − β
−1 .
Γc − aΓc − b
2.23
By simplification, we see that the last expression is bounded above by 1 − β if and only if 2.15
holds.
Theorem 2.5. i If a, b > −1, c > 0, and ab < 0, then zFa, b; c; z is in PTα if and only if
c ≥ab1−
ab
.
α
2.24
ii If a, b > 0 and c > a b 1, then F1 a, b; c; z z2 − Fa, b; c; z is in PTα if and only
if
ΓcΓc − a − b
ab
1
≤ 2.
Γc − aΓc − b
αc − a − b − 1
2.25
Proof. i Since
zFa, b; c; z z ∞
∞
a 1n−2 b 1n−2 n
a 1n−2 b 1n−2 n
ab |ab| z z−
z ,
c n2 c 1n−2 1n−1
c n2 c 1n−2 1n−1
2.26
according to i of Lemma 2.2, we must show that
∞
a 1n−2 b 1n−2
c
n − 1 α
≤
α.
c
1
1
|ab|
n−2
n−1
n2
2.27
O. S. Kwon and N. E. Cho
7
Noting that λn λλ 1n−1 and then applying 1.6, we have
∞
n 1 α
n0
∞
∞
a 1n b 1n a 1n b 1n
an bn
c α
c 1n 1n1
c
1
1
ab
cn 1n
n
n
n0
n1
Γc 1Γc − a − b − 1
c ΓcΓc − a − b
α
−1 .
Γc − aΓc − b
ab Γc − aΓc − b
2.28
Hence, 2.27 is equivalent to
c−a−b−1
c
Γc 1Γc − a − b − 1
c
1α
≤α
0.
−
Γc − aΓc − b
ab
|ab| ab
2.29
Thus, 2.29 is valid if and only if 1 αc − a − b − 1/ab ≤ 0 or, equivalently, c ≥ a b 1 − ab/α.
ii Since
F1 a, b; c; z z −
∞
an−1 bn−1
cn−1 1n−1
n2
zn ,
2.30
by i of Lemma 2.2, we need only to show that
∞
n − 1 α
n2
an−1 bn−1
≤ α.
cn−1 1n−1
2.31
Now,
∞
∞
∞
a b
nan bn
an bn
n − 1 α n−1 n−1 α
c
1
c
1
cn 1n
n−1
n−1
n
n
n2
n1
n1
∞
∞
a 1n−1 b 1n−1
an bn
ab α
c n1 c 1n−1 1n−1
cn 1n
n1
ab
ΓcΓc − a − b
α − α.
Γc − aΓc − b c − a − b − 1
2.32
But this last expression is bounded above by α if and only if 2.25 holds.
Theorem 2.6. i If a, b > −1, ab < 0, and c > a b 2, then zFa, b; c; z is in CPTα if and only
if
a2 b2 2 αabc − a − b − 2 αc − a − b − 22 ≥ 0.
2.33
ii If a, b > 0 and c > a b 2, then F1 a, b; c; z z2 − Fa, b; c; z is in CPTα if and
only if
a2 b2
ΓcΓc − a − b
ab
2α
1 ≤ 2.
Γc − aΓc − b αc − a − b − 22
α
c−a−b−1
2.34
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International Journal of Mathematics and Mathematical Sciences
Proof. i Since zF has the form 2.26, we see from ii of Lemma 2.2 that our conclusion is
equivalent to
∞
a 1n−2 b 1n−2
c
α.
nn − 1 α
≤
c
1
1
|ab|
n−2
n−1
n2
2.35
Writing n 2n 1 α n 12 1 αn 1 α, we see that
∞
n 2n 1 α
n0
a 1n b 1n
c 1n 1n1
∞
∞
∞
a 1n b 1n
a 1n b 1n
a 1n b 1n
n 1
1 α
α
c 1n 1n
c 1n 1n
c 1n 1n1
n0
n0
n0
∞
∞
∞
a 2n b 2n
a 1n b 1n
an bn
a 1b 1 c 2 α
α
c1
c 2n 1n
c 1n 1n
ab n1 cn 1n
n0
n0
α
αc
Γc 1Γc − a − b − 2
a 1b 1 2 αc − a − b − 2 c − a − b2 −
.
Γc − aΓc − b
ab
ab
2.36
This last expression is bounded above by αc/|ab| if and only if a 1b 1 2 αc − a −
b − 2 α/abc − a − b − 12 ≤ 0, which is equivalent to 2.33.
ii In view of ii of Lemma 2.2, we need only to show that
∞
nn − 1 α
n2
an−1 bn−1
≤ α.
cn−1 1n−1
2.37
Now,
∞
∞
∞
an1 bn1 a b
an1 bn1
n 2 n 2 − 1 − α
n 22
− 1 − α n 2 n1 n1 .
c
1
c
1
cn1 1n1
n1
n1
n1
n1
n0
n0
n0
2.38
Substituting 2.21 into the right-hand side of 2.38, we obtain
∞
an2 bn2
n0
cn2 1n
∞
∞
an1 bn1
an bn
2 α
α
.
c
1
cn 1n
n1
n
n0
n1
2.39
Since ank ak a kn , we may write 2.39 as
a2 b2 Γc 2Γc − a − b − 2
ab Γc 1Γc − a − b − 1
ΓcΓc − a − b
2 α
α
−1 .
c2
Γc − aΓc − b
c
Γc − aΓc − b
Γc − aΓc − b
2.40
By simplification, we see that the last expression is bounded above by α if and only if 2.34
holds.
O. S. Kwon and N. E. Cho
9
3. An integral operator
In the next theorems, we obtain similar-type results in connection with a particular integral
operator Ga, b; c; z acting on Fa, b; c; z as follows:
Ga, b; c; z z
Fa, b; c; tdt.
3.1
0
Theorem 3.1. Let a, b > −1, ab < 0, and c > max{0, a b}. Then,
i Ga, b; c; z defined by 3.1 is in Sp Tα, β if and only if
α βc − 12
Γc 1Γc − a − b 1 α α βc − a − b
≤ 0;
−
Γc − aΓc − b
ab
a − 12 b − 12
a − 12 b − 12
3.2
ii Ga, b; c; z defined by 3.1 is in PTα if and only if
α − 1c − a2
Γc 1Γc − a − b 1
α − 1c − a − b
≤ 0.
−
Γc − aΓc − b
ab
a − 12 b − 12
a − 12 b − 12
3.3
Proof. i Since
Ga, b; c; z z −
∞
a 1n−2 b 1n−2 n
|ab| z ,
c n2 c 1n−2 1n
3.4
by i of Lemma 2.1, we need only to show that
∞
a 1n−2 b 1n−2
c
≤ 1 − β
.
n1 α − α β
c 1n−2 1n
|ab|
n2
3.5
Now,
∞
a 1n b 1n
n 21 α − α β
c 1n 1n2
n0
1 α
∞
a 1n b 1n
n0
c 1n 1n1
− α β
∞
an bn
c ab n1 cn 1n1
Γc 1Γc − a − b 1 α α βc − a − b
−
Γc − aΓc − b
ab
a − 12 b − 12
α βc − 12
c
− 1 − β
a − 12 b − 12
ab
≤ 1 − β
which is equivalent to 3.2.
c
,
|ab|
3.6
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International Journal of Mathematics and Mathematical Sciences
ii According to i of Lemma 2.2, it is sufficient to show that
∞
a 1n−2 b 1n−2
c
n − 1 α
≤α
.
c
1
1
|ab|
n−2
n
n2
3.7
Now,
∞
n 1 α
n0
a 1n b 1n
c 1n 1n2
∞
a 1n b 1n
α − 1
c
1
1
c 1n 1n2
n
n1
n0
n0
c ΓcΓc − a − b
−1
ab Γc − aΓc − b
c − 1
Γc − 1Γc − a − b 1
c
−1 −1
α − 1
ab a − 1b − 1
Γc − aΓc − b
α − 1c − 12
α − 1c − a − b
c
Γc 1Γc − a − b 1
−
−α
Γc − aΓc − b
ab
a − 12 b − 12
a − 12 b − 12
ab
∞
a 1n b 1n
3.8
c
,
|ab|
which is equivalent to 3.3.
Now, we observe that Ga, b; c; z ∈ UCTα, βCPTα if and only if zFa, b; c; z ∈
Sp Tα, βPTα. Thus, any result of functions belonging to the class Sp Tα, βPTα about
zF leads to that of functions belonging to the class UCTα, βCPTα. Hence, we obtain the
following analogues to Theorems 2.3 and 2.5.
≤α
Theorem 3.2. Let a, b > −1, ab < 0, and c > a b 2. Then,
i Ga, b; c; z defined by 3.1 is in UCTα, β if and only if
c ≥ab1−
1 αab
;
1 − β
3.9
ii Ga, b; c; z defined by 3.1 is in CPTα if and only if
c ≥ab1−
ab
.
α
3.10
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11
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