STARLIKE AND CONVEXITY PROPERTIES FOR
HYPERGEOMETRIC FUNCTIONS
p-VALENT
R. M. EL-ASHWAH, M. K. AOUF and A. O. MOUSTAFA
∞ (a) (b)
P
n
n n
z , we place conditions on
n=0 (c)n (1)n
p
a, b and c to guarante that z F (a, b; c; z) will be in various subclasses of p-valent starlike and p-valent
convex functions. Operators related to the hypergeometric function are also examined.
Abstract. Given the hypergeometric function F (a, b; c; z) =
1. Introduction
Let S(p) be the class of functions of the form:
(1)
f (z) = z p +
∞
X
ap+n z p+n
(p ∈ N = {1, 2, . . .})
n=1
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which are analytic and p-valent in the unit disc U = {z : |z| < 1}. A function f (z) ∈ S(p) is called
p-valent starlike of order α if f (z) satisfies
0 zf (z)
(2)
Re
>α
f (z)
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Received November 11, 2008.
2000 Mathematics Subject Classification. Primary 30C45.
Key words and phrases. p-valent; starlike; convex; hypergeometric function.
for 0 ≤ α < p, p ∈ N and z ∈ U . By S ∗ (p, α) we denote the class of all p-valent starlike functions
of order α. By Sp∗ (α) denote the subclass of S ∗ (p, α) consisting of functions f (z) ∈ S(p) for which
0
zf (z)
(3)
f (z) − p < p − α
for 0 ≤ α < p, p ∈ N and z ∈ U . Also a function f (z) ∈ S(p) is called p-valent convex of order α
if f (z) satisfies
zf 00 (z)
(4)
Re 1 + 0
>α
f (z)
for 0 ≤ α < p, p ∈ N and z ∈ U . By K(p, α) we denote the class of all p-valent convex functions
of order α. It follows from (2) and (4) that
(5)
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zf 0 (z)
∈ S(p, α).
p
Also by Kp (α) denote the subclass of K(p, α) consisting of functions f (z) ∈ S(p) for which
00
1 + zf (z) − p < p − α
(6)
0
f (z)
for 0 ≤ α < p, p ∈ N and z ∈ U .
By T (p) we denote the subclass of S(p) consisting of functions of the form:
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(7)
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f (z) ∈ K(p, α) ⇔
f (z) = z p −
∞
X
n=1
ap+n z p+n
(ap+n ≥ 0; p ∈ N ).
By T ∗ (p, α), Tp∗ (α), C(p, α) and Cp (α) we denote the classes obtained by taking interesctions,
respectively, of the classes S ∗ (p, α), Sp∗ (α), K(p, α) and Kp (α) with the class T (p)
T ∗ (p, α) = S ∗ (p, α) ∩ T (p),
Tp∗ (α) = Sp∗ (α) ∩ T (p),
C(p, α) = K(p, α) ∩ T (p),
and
Cp (α) = Kp (α) ∩ T (p).
The class S ∗ (p, α) was studied by Patil and Thakare [5]. The classes T ∗ (p, α) and C(p, α) were
studied by Owa [4].
For a, b, c ∈ C and c 6= 0, −1, −2, . . . , the (Gaussian) hypergeometric function is defined by
(8)
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F (a, b; c; z) =
∞
X
(a)n (b)n n
z
(c)n (1)n
n=0
(z ∈ U ),
where (λ)n is the Pochhammer symbol defined, in terms of the Gamma function Γ, by
Γ(λ + n)
1
(n = 0)
(9)
(λ)n =
=
λ(λ + 1) · . . . · (λ + n − 1) (n ∈ N ).
Γ(λ)
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The series in (8) represents an analytic function in U and has an analytic continuation throughout
the finite complex plane except at most for the cut [1, ∞). We note that F (a, b; c; 1) converges for
Re(a − b − c) > 0 and is related to the Gamma function by
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F (a, b; c; 1) =
Γ(c)Γ(c − a − b)
.
Γ(c − a)Γ(c − b)
Corresponding to the function F (a, b; c; z) we define
(11)
hp (a, b; c; z) = z p F (a, b; c; z).
We observe that for a function f (z) of the form (1), we have
(12)
hp (a, b; c; z) = z p +
∞
X
(a)n−p (b)n−p n
z .
(c)n−p (1)n−p
n=p+1
In [7] Silverman gave necessary and sufficient conditions for zF (a, b; c; z) to be in T ∗ (1, α) = T ∗ (α)
and C(1, α) = C(α) and has also examined a linear operator acting on hypergeometric functions.
For the other interesting developments for zF (a, b; c; z) in connection with various subclasses of
univalent functions, the reader can refer to the works of Carlson and Shaffer [1], Merkes and Scott
[3] and Ruscheweyh and Singh [6].
In the present paper, we determine necessary and sufficient conditions for hp (a, b; c; z) to be in
T ∗ (p, α) and C(p, α). Furthermore, we consider an integral operator related to the hypergeometric
function.
2. Main Results
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To establish our main results we shall need the following lemmas.
Lemma 1 ([4]). Let the function f (z) defined by (1).
(i) A sufficient condition for f (z) ∈ S(p) to be in the class Sp∗ (α) is that
∞
X
n=p+1
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(n − α) |an | ≤ (p − α).
(ii) A sufficient condition for f (z) ∈ S(p) to be in the class Kp (α) is that
∞
X
n
(n − α) |an | ≤ p − α.
p
n=p+1
Lemma 2 ([4]). Let the function f (z) be defined by (7). Then
(i) f (z) ∈ T (p) is in the class T ∗ (p, α) if and only if
∞
X
(n − α)an ≤ p − α.
n=p+1
(ii) f (z) ∈ T (p) is in the class C(p, α) if and only if
∞
X
n
(p − α)an ≤ p − α.
p
n=p+1
Lemma 3 ([2]). Let f (z) ∈ T (p) be defined by (7). Then f (z) is p-valent in U if
∞
X
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(p + n)ap+n ≤ p.
n=1
In addition, f (z) ∈ Tp∗ (α) ⇔ f (z) ∈ T ∗ (p, α), f (z) ∈ Kp (α) ⇔ f (z) ∈ K(p, α) and f (z) ∈
⇔ f (z) ∈ S ∗ (p, α).
Sp∗ (α)
Theorem 1. If a, b > 0 and c > a + b + 1, then a sufficient condition for hp (a, b; c; z) to be in
Sp∗ (α), 0 ≤ α < p, is that
Γ(c)Γ(c − a − b)
ab
(13)
1+
≤ 2.
Γ(c − a)Γ(c − b)
(p − α)(c − a − p − 1)
Condition (13) is necessary and sufficient
z p (2 − F (a, b; c; z)) to be in T∗ (p, α)(Tp∗ (α)).
for
Fp
defined
by
Fp (a, b; c; z)
=
∞
P
(a)n−p (b)n−p n
z , according to Lemma 1(i), we only need
n=p+1 (c)n−p (1)n−p
Proof. Since hp (a, b; c; z) = z p +
to show that
∞
X
n=p+1
(n − α)
(a)n−p (b)n−p
≤ p − α.
(c)n−p (1)n−p
Now
(14)
∞
X
n=p+1
(n − α)
∞
∞
X
X
(a)n−p (b)n−p
(a)n (b)n
(a)n (b)n
=
+ (p − α)
.
(c)n−p (1)n−p
(c)
(1)
(c)n (1)n
n
n−1
n=1
n=1
Noting that (λ)n = λ(λ + 1)n−1 and then applying (10), we may express (14) as
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∞
∞
X
(a)n (b)n
ab X (a + 1)n−1 (b + 1)n−1
+ (p − α)
c n=1 (c + 1)n−1 (1)n−1
(c)n (1)n
n=1
ab Γ(c + 1)Γ(c − a − b − 1)
Γ(c)Γ(c − a − b)
=
+ (p − α)
−1
c
Γ(c + a)Γ(c − b)
Γ(c − a)Γ(c − b)
Γ(c)Γ(c − a − b)
ab
=
+ p − α − (p − α).
Γ(c − a)Γ(c − b) c − a − b − 1
But this last expression is bounded above by p − α if and only if (13) holds.
∞
P
(a)n−p (b)n−p n
Since Fp (a, b; c; z) = z p −
z , the necessity of (13) for Fp to be in Tp∗ (α) and
n=p+1 (c)n−p (1)n−p
T ∗ (p, α) follows from Lemma 2(i).
Remark 1. Condition (13) with α = 0 is both necessary and sufficient for Fp to be in the class
Tp∗ .
In the next theorem, we find constraints on a, b and c that lead to necessary and sufficient
conditions for hp (a, b; c; z) to be in the class T ∗ (p, α).
Theorem 2. If a, b > −1, c > 0 and ab < 0, then a necessary and sufficient condition for
ab
. The condition c ≥ a + b + 1 − ab
hp (a, b; c; z) to be in T ∗ (p, α)(Tp∗ (α)) is that c ≥ a + b + 1 − p−α
p
is necessary and sufficient for hp (a, b; c; z) to be in Tp∗ .
Proof. Since
hp (a, b; c; z) = z p +
∞
ab X (a + 1)n−p−1 (b + 1)n−p−1 n
z
c n=p+1
(c + 1)n−p−1 (1)n−p
X
ab ∞ (a + 1)n−p−1 (b + 1)n−p−1 n
p
= z − z ,
c n=p+1
(c + 1)n−p−1 (1)n−p
= zp +
(15)
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∞
X
(a)n−p (b)n−p n
z
(c)n−p (1)n−p
n=p+1
according to Lemma 2(i), we must show that
(16)
∞
X
n=p+1
(n − α)
c
(a + 1)n−p−1 (b + 1)n−p−1
≤ (p − α).
(c + 1)n−p−1 (1)n−p
ab
Note that the left side of (16) diverges if c ≤ a + b + 1. Now
∞
X
(n + p + 1 − α)
n=0
(a + 1)n (b + 1)n
(c + 1)n (1)n+1
∞
∞
X
(a + 1)n (b + 1)n
c X (a)n (b)n
+ (p − α)
(c + 1)n (1)n
ab n=1 (c)n (1)n
n=0
c Γ(c)Γ(c − a − b)
Γ(c + 1)Γ(c − a − b − 1)
+ (p − α)
−1
=
Γ(c − a)Γ(c − b)
ab Γ(c − a)Γ(c − b)
=
Hence, (16) is equivalent to
(17)
Γ(c + 1)Γ(c − a − b − 1)
(c − a − b − 1)
1 + (p − α)
Γ(c − a)Γ(c − b)
ab
c
c
+
= 0.
≤ (p − α)
|ab| ab
Thus, (17) is valid if and only if
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1 + (p − α)
(c − a − b − 1)
≤ 0,
ab
or, equivalently,
ab
.
p−α
Another application of Lemma 2(i) when α = 0 completes the proof of Theorem 2.
c≥a+b+1−
Our next theorems will parallel Theorems 1 and 2 for the p-valent convex case.
Theorem 3. If a, b > 0 and c > a + b + 2, then a sufficient condition for hp (a, b; c; z) to be in
Kp (α), 0 ≤ α < p, is that
Γ(c)Γ(c − a − b)
(2p + 1 − α)
ab
1 +
Γ(c − a)Γ(c − b)
p(p − α)
c−a−b−1
(18)
(a)2 (b)2
+
≤ 2.
p(p − α)(c − a − b − 2)2
Condition (18) is necessary and sufficient for Fp (a, b; c; z) = z p (2 − F (a, b; c; z)) to be in C(p, α)
(Cp (α)).
Proof. In view of Lemma 1(ii), we only need to show that
∞
X
(a)n−p (b)n−p
(n − α)
≤ p(p − α).
(c)n−p (1)n−p
n=p+1
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Now
∞
X
(n + p + 1)(n + p + 1 − α)
n=0
=
∞
X
(n + 1)2
n=0
(a)n+1 (b)n+1
(c)n+1 (1)n+1
∞
X
(a)n+1 (b)n+1
(a)n+1 (b)n+1
+ (2p − α)
(n + 1)
(c)n+1 (1)n+1
(c)n+1 (1)n+1
n=0
∞
X
(a)n+1 (b)n+1
+ p(p − α)
(c)n+1 (1)n+1
n=0
=
∞
X
n=0
(n + 1)
∞
X
(a)n+1 (b)n+1
(a)n+1 (b)n+1
+ (2p − α)
(c)n+1 (1)n
(c)n+1 (1)n
n=0
+ p(p − α)
=
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∞
∞
X
X
(a)n+1 (b)n+1
(a)n+1 (b)n+1
+ (2p + 1 − α)
(c)
(1)
(c)n+1 (1)n
n+1
n−1
n=1
n=0
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+ p(p − α)
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(19)
∞
X
(a)n+1 (b)n+1
(c)n+1 (1)n+1
n=0
=
∞
X
(a)n+1 (b)n+1
(c)n+1 (1)n+1
n=0
∞
∞
X
X
(a)n+2 (b)n+2
(a)n+1 (b)n+1
+ (2p + 1 − α)
(c)n+2 (1)n
(c)n+1 (1)n
n=0
n=0
+ p(p − α)
∞
X
(a)n (b)n
.
(c)n (1)n
n=1
Since (a)n+k = (a)k (a + k)n , we may write (19) as
(a)2 (b)2 Γ(c + 2)Γ(c − a − b − 2)
ab
+ (2p + 1 − α)
(c)2
Γ(c + a)Γ(c − b)
c
Γ(c + 1)Γ(c − a − b − 1)
Γ(c)Γ(c − a − b)
·
+ p(p − α)
−1 .
Γ(c − a)Γ(c − b)
Γ(c − a)Γ(c − b)
Upon simplification, we see that this last expression is bounded above by p(p−α) if and only if (18)
holds. That (18) is also necessary for Fp to be in C(p, α)(Cp (α)) follows from Lemma 2(ii).
Theorem 4. If a, b > −1, ab < 0 and c > a + b + 2, then a necessary and sufficient condition
for hp (a, b; c; z) to be in C(p, α)(Cp (α)) is that
(20)
(a)2 (b)2 + (2p + 1 − α)ab(c − a − b − 2) + p(p − α)(c − a − b − 2)2 ≥ 0.
Proof. Since hp (a, b; c; z) has the form (15), we see from Lemma 2(ii) that our conclusion is
equivalent to
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(21)
∞
X
n=p+1
n(n − α)
c
(a + 1)n−p−1 (b + 1)n−p−1
≤ p(p − α).
(c + 1)n−p−1 (1)n−p
ab
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Note that c > a + b + 2 if the left-hand side of (21) converges. Writing
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(n + p + 1)(n + p + 1 − α) = (n + 1)2 + (2p − α)(n + 1) + p(p − α),
we see that
∞
X
n(n − α)
n=p+1
=
=
∞
X
n=0
∞
X
(a + 1)n−p−1 (b + 1)n−p−1
(c + 1)n−p−1 (1)n−p
(n + p + 1)(n + p + 1 − α)
(n + 1)
n=0
+ p(p − α)
=
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∞
X
(a + 1)n (b + 1)n
(a + 1)n (b + 1)n
+ (2p − α)
(c + 1)n (1)n
(c + 1)n (1)n
n=0
∞
X
(a + 1)n (b + 1)n
(c + 1)n (1)n+1
n=0
∞
∞
X
(a + 1)(b + 1) X (a + 2)n (b + 2)n
(a + 1)n (b + 1)n
+ (2p + 1 − α)
(c + 1)
(c
+
2)
(1)
(c + 1)n (1)n
n
n
n=0
n=0
+ p(p − α)
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(a + 1)n (b + 1)n
(c + 1)n (1)n+1
∞
c X (a)n (b)n
ab n=1 (c)n (1)n
Γ(c + 1)Γ(c − a − b − 2)
[(a + 1)(b + 1) + (2p + 1 − α)(c − a − b − 2)
Γ(c − a)Γ(c − b)
p(p − α)
p(p − α)c
+
(c − a − b − 2)2 −
.
ab
ab
c
p(p − α) if and only if
This last expression is bounded above by ab
=
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(a + 1)(b + 1) + (2p + 1 − α)(c − a − b − 2) +
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p(p − α)
(c − a − b − 2)2 ≤ 0,
ab
which is equivalent to (20).
Putting p = 1 in Theorem 4, we obtain the following corollary.
Corollary 1. If a, b > −1, ab < 0 and c > a + b + 2, then a necessary and sufficient condition
for h1 (a, b; c; z) to be in C(1, α)(C(α)) is that
(a)2 (b)2 + (3 − α)ab(c − a − b − 2) + (1 − α)(c − a − b − 2)2 ≥ 0.
Remark 2. We note that Corollary 1 corrects the result obtained by Silverman [7, Theorem
4].
3. Integral Operator
In this section, we obtain similar results in connection with a particular integral operator
Gp (a, b; c; z) acting on F (a, b; c; z) as follows
Zz
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Gp (a, b; c; z) = p
tp−1 F (a, b; c; z)dt
0
(22)
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= zp +
∞ X
n=1
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We note that
zG0p
p
Theorem 5.
= hp .
p
n+p
(a)n (b)n n+p
z
.
(c)n (1)n
(i) If a, b > 0 and c > a + b, then a sufficient condition for Gp (a, b; c; z) defined by (22) to be
in S ∗ (p) is that
Γ(c)Γ(c − a − b)
≤ 2.
Γ(c)Γ(c − b)
(23)
(ii) If a, b > −1, c > 0, and ab < 0, then Gp (a, b; c; z) defined by (22) is in T (p) or S(p) if only
if c > max{a, b}.
Proof. Since
p
Gp (a, b; c; z) = z +
∞ X
n=1
we note that
∞
X
n=1
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(n + p)
p
n+p
(a)n (b)n n+p
z
,
(c)n (1)n
∞
X
(a)n (b)n
(a)n (b)n
=p
(c)n (1)n
(c)n (1)n
n=1
"∞
#
X (a)n (b)n
=p
−1
(c)n (1)n
n=0
is bounded above by p if and only if (23) holds.
To prove (ii), we apply Lemma 3 to
∞
|ab| X p (a + 1)n−p−1 (b + 1)n−p−1 n
p
Gp (a, b; c; z) = z −
z .
c n=p+1 n
(c + 1)n−p−1 (1)n−p
It suffices to show that
Close
p
n+p
∞
X
(a + 1)n−p−1 (b + 1)n−p−1
c
≤
(c + 1)n−p−1 (1)n−p
|ab|
n=p+1
or, equivalently,
∞
∞
X
(a + 1)n (b + 1)n
c X (a)n (b)n
c
.
=
≤
(c
+
1)
(1)
ab
(c)
(1)
|ab|
n
n+1
n
n
n=0
n=1
But this is equivalent to
Γ(c)Γ(c − a − b)
− 1 ≥ −1,
Γ(c − a)Γ(c − b)
which is true if and only if c > max{a, b}. This completes the proof of Theorem 5.
Now Gp (a, b; c; z) ∈ Kp (α)(K(p, α)) if and only if
z 0
G (a, b; c; z) = hp (a, b; c; z) ∈ Sp∗ (α)(S ∗ (p, α)).
p p
This follows upon observing that
zG0p
z
1
= hp , G00p = h0p − G0p , and so
p
p
p
1+
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zh0p
zG00p
=
.
Gp
hp
Thus any p-valent starlike about hp leads to a p-valent convex about Gp . Thus from Theorems 1
and 2, we have
Theorem 6.
(i) If a, b > 0 and c > a+b+1, then a sufficient condition for Gp (a, b; c; z) defined in Theorem 5
to be in Kp (α)(0 ≤ α < p) is that
ab
Γ(c)Γ(c − a − b)
1+
≤ 2.
Γ(c − a)Γ(c − b)
(p − α)(c − a − b − 1)
(ii) If a, b > −1, ab < 0, and c > a + b + 2, then a necessary and sufficient condition for
Gp (a, b; c; z) to be in C(p, α)(Cp (α)) is that
c≥a+b+1−
ab
.
(p − α)
Remark 3. Putting p = 1 in all the above results, we obtain the results obtained by Silverman
[7].
Acknowledgment. The authors thank the referees for their valuable suggestions to improve
the paper.
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1. Carlson B. C. and Shaffer D. B., Starlike and prestarlike hypergeometric functions, J. Math. Anal. Appl. 15
(1984), 737–745.
2. Chen M.-P., Multivalent functions with negative coefficients in the unit disc, Tamkang J. Math. 17(3) (1986),
127–137.
3. Merkes E. and Scott B. T., Starlike hypergeometric functions, Proc. Amer. Math. Soc. 12 (1961), 885–888.
4. Owa S., On certain classes of p-valent functions with negative coefficients, Simon Stevin, 59 (1985), 385–402.
5. Patil D. A. and Thakare N. K., On convex hulls and extreme points of p-valent starlike and convex classes with
applications, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 27(75) (1983), 145–160.
6. Ruscheweyh St. and Singh V., On the order of starlikeness of hypergeometric functions, J. Math. Anal. Appl.
113 (1986), 1–11.
7. Silverman H., Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl. 172 (1993),
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R. M. El-Ashwah, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt,
e-mail: r elashwah@yahoo.com
M. K. Aouf, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt, e-mail:
mkaouf127@yahoo.com
A. O. Moustafa, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt,
e-mail: adelaeg254@yahoo.com
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