Starlike and Convex Properties for Hypergeometric Functions

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.


Introduction
Let T be the class consisting of functions of the form a n z n , a n ≥ 0, 1.1 that are analytic and univalent in the open unit disk U {z : |z| < 1}.Let T * α and C α denote the subclasses of T consisting of starlike and convex functions of order α 0 ≤ α < 1 , respectively 1 .
Recently, Bharati et In particular, we note that UCT 1, 0 is the class of uniformly convex functions given by Goodman 3 also see 4-6 .
Let F a, b; c; z be the Gaussian hypergeometric function defined by where c / 0, −1, −2, . .., and λ n is the Pochhammer symbol defined by We note that F a, b; c; 1 converges for Re c − a − b > 0 and is related to the Gamma function by Silverman 7 gave necessary and sufficient conditions for zF a, b; c; z to be in T * α and C α , and 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 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 zF a, b; c; z to be in S p T α, β , UCT α, β , PT α , and CPT α .Furthermore, we consider an integral operator related to the hypergeometric function.

Results
To establish our main results, we need the following lemmas due to Bharati et according to i of Lemma 2.1, we must show that Noting that λ n λ λ 1 n−1 and then applying 1.6 , we have

2.9
International Journal of Mathematics and Mathematical Sciences Hence, 2.8 is equivalent to Thus, 2.10 is valid if and only if 1 α ii Since 11 by i of Lemma 2.1, we need only to show that

2.13
But this last expression is bounded above by 1 − β if and only if 2.6 holds.

2.15
Proof.i Since zF has the form 2.7 , we see from ii of Lemma 2.1 that our conclusion is equivalent to Writing

2.17
This last expression is bounded above by 1 − β c/|ab| if and only if which is equivalent to 2.14 .
ii In view of ii of Lemma 2.1, we need only to show that

International Journal of Mathematics and Mathematical Sciences
Writing n 2 n 1 1, we have a n b n c n 1 n .

2.21
Substituting 2.21 into the right-hand side of 2.20 , we obtain

2.22
Since a n k a k a k n , we write 2.22 as

2.23
By simplification, we see that the last expression is bounded above by 1 − β if and only if 2.15 holds.

2.26
according to i of Lemma 2.2, we must show that

2.28
Hence, 2.27 is equivalent to

2.32
But this last expression is bounded above by α if and only if 2.25 holds.

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

2.36
This last expression is bounded above by αc/|ab| if and only if

2.38
Substituting 2.21 into the right-hand side of 2.38 , we obtain Since a n k a k a k n , we may write 2.39 as

2.40
By simplification, we see that the last expression is bounded above by α if and only if 2.34 holds.

An integral operator
In the next theorems, we obtain similar-type results in connection with a particular integral operator by i of Lemma 2.1, we need only to show that which is equivalent to 3.2 .

International Journal of Mathematics and Mathematical Sciences
ii According to i of Lemma 2.2, it is sufficient to show that

Theorem 2 . 4 .
i If a, b > −1, ab < 0, and c > a b 2, then zF a, b; c; z is in UCT α, β if and only if

Theorem 2 . 5 .≥ a b 1 ii
i If a, b > −1, c > 0, and ab < 0, then zF a, b; c; z is in PT α if and only if c If a, b > 0 and c > a b 1, then F 1 a, b; c; z z 2 − F a, b; c; z is in PT α if and only if

Theorem 2 . 6 .
i If a, b > −1, ab < 0, and c > a b 2, then zF a, b; c; z is in CPT α if and only if

0 F 1 Theorem 3 . 1 .
G a, b; c; z acting on F a, b; c; z as follows: G a, b; c; z z a, b; c; t dt.3.Let a, b > −1, ab < 0, and c > max{0, a b}.Then, i G a, b; c; z defined by 3.1 is in S p T α, β if and only if

5 . 3 . 2 .
observe that G a, b; c; z ∈ UCT α, β CPT α if and only if zF a, b; c; z ∈ S p T α, β PT α .Thus, any result of functions belonging to the class S p T α, β PT α about zF leads to that of functions belonging to the class UCT α, β CPT α .Hence, we obtain the following analogues to Theorems 2.3 and 2.Theorem Let a, b > −1, ab < 0, and c > a b 2.Then, i G a, b; c; z defined by 3.1 is in UCT α, β if and only if c ≥ a b 1 − 1 α ab 1 − β ; 3.9 ii G a, b; c; z defined by 3.1 is in CPT α if and only if c ≥ a b 1 − ab α .3.10 al. 2 introduced the following subclasses of starlike and convex functions.Definition 1.1.A function f of the form 1.1 is in S p T α, β if it satisfies the condition International Journal of Mathematics and Mathematical Sciences Definition 1.2.A function f of the form 1.1 is in PT α if it satisfies the condition and f ∈ UCT α, β if and only if zf ∈ S p T α, β .and f ∈ CPT α if and only if zf ∈ PT α .Bharati et al. 2 showed that S p T α, β

al. 2 . Lemma 2.1. i
A function f of the form 1.1 is in S p T α,β if and only if it satisfies