Applications of Multivalent Functions Associated with Generalized Fractional Integral Operator ()
1. Introduction
Let 
denote the class of functions 
of the form
(1.1)
which are analytic in the open unit disk 
Also let f and g be analytic in 
with
. Then we say that f is subordinate to g in
, written 
or
, if there exists the Schwarz function w, analytic in 
such that
, 
and
. We also observe that
if and only if
whenever 
is univalent in
.
Let a, b and c be complex numbers with
. Then the Gaussian/classical hypergeometric function 
is defined by
(1.2)
where 
is the Pochhammer symbol defined, in terms of the Gamma function, by
(1.3)
The hypergeometric function 
is analytic in 
and if a or b is a negative integer, then it reduces to a polynomial.
For each A and B such that
, let us define the function
(1.4)
It is well known that
, for
, is the conformal map of the unit disk onto the disk symmetrical respect to the real axis having the center 
and the radius
. The boundary circle cuts the real axis at the points 
and
.
Many essentially equivalent definitions of fractional calculus have been given in the literature (cf., e.g. [2,3]). We state here the following definition due to Saigo [4] (see also [1,5]).
Definition 1. For
, 
, the fractional integral operator 
is defined by
(1.5)
where 
is the Gaussian hypergeometric function defined by (1.2) and 
is taken to be an analytic function in a simply-connected region of the z-plane containing the origin with the order
for
, and the multiplicity of 
is removed by requiring that 
to be real when
.
The definition (1.5) is an interesting extension of both the Riemann-Liouville and Erdélyi-Kober fractional operators in terms of Gauss’s hypergeometric functions.
With the aid of the above definition, Owa, Saigo and Srivastava [1] defined a modification of the fractional integral operator 
by
(1.6)
for 
and
. Then it is observed that 
also maps 
onto itself as follows:
(1.7)
We note that
, where the operator 
was introduced and studied by Jung, Kim and Srivastava [6] (see also [7]).
It is easily verified from (1.7) that
(1.8)
The identity (1.8) plays an important and significant role in obtaining our results.
Recently, by using the general theory of differential subordination, several authors (see, e.g. [7-9]) considered some interesting properties of multivalent functions associated with various integral operators. In this manuscript, we shall derive some subordination properties of the fractional integral operator 
by using the technique of differential subordination.
2. Main Results
In order to establish our results, we shall need the following lemma due to Miller and Mocanu [10].
Lemma 1. Let 
be analytic and convex univalent in 
with
, and let 
be analytic in
. If
(2.1)
then for 
and
,
(2.2)
We begin by proving the following theorem.
Theorem 1. Let
, 
, 
, 
, 
and
, and let
. Suppose that
(2.3)
where
(2.4)
and 
is given by (1.3).
1) If
, then
(2.5)
2) If 
and
, then
(2.6)
The result is sharp.
Proof. 1) If we set
then, from (1.7) we see that
(2.7)
For 
and
, it follows from (2.3) that
(2.8)
which implies that
2) Let
(2.9)
Then the function 
is analytic in
. Using (1.8) and (2.9), we have
(2.10)
From (2.5), (2.9) and (2.10) we obtain
Thus, by applying Lemma 1, we observe that
or
(2.11)
where 
is analytic in 
with 
and
. In view of 
and 
, we conclude from (2.11) that
(2.12)
Since 
for 
and
, from (2.12) we see that the inequality (2.6) holds.
To prove sharpness, we take 
defined by
For this function we find that
and
Hence the proof of Theorem 1 is evidently completed.
Theorem 2. Let
, 
, 
, 
, 
and
. Suppose that
, 
and 
. If the sequence 
is nondecreasing with
(2.13)
where 
is given by 
and satisfies the condition
, then
(2.14)
and
(2.15)
Each of the bounds in (2.14) and (2.15) is best possible for
.
Proof. We prove the bound in (2.14). The bound in (2.15) is immediately obtained from (2.14) and will be omitted. Let
Then, from (1.7) we observe that
where, for convenience,
It is easily seen from (2.4) and (2.13) that 
and
(2.16)
Hence, by applying (2.3) and (2.16), we have
which readily yields the inequality (2.14).
If we take
, then
This show that the bound in (2.14) is best possible for each m, which proves Theorem 2.
Finally, we consider the generalized Bernardi-LiveraLivingston integral operator 
defined by (cf. [11-13])
(2.17)
Theorem 3. Let
, 
, 
, 
, 
, 
and
and let
. Suppose that
(2.18)
where
and 
is given by (1.3).
1) If
, then
(2.19)
2) If 
and
, then
(2.20)
The result is sharp.
Proof. 1) If we put
then, from (1.7) and (2.17) we have
Therefore, by using same techniques as in the proof of Theorem 1 1), we obtain the desired result.
2) From (2.17) we have
(2.21)
Let
(2.22)
Then, by virtue of (2.21), (2.22) and (2.19), we observe that
Hence, by applying the same argument as in the proof of Theorem 1 2), we obtain (2.20), which evidently proves Theorem 3.
3. Acknowledgements
This work was supported by Daegu National University of Education Research grant in 2011.