Research Article | Open Access

# An Extension to the Owa-Srivastava Fractional Operator with Applications to Parabolic Starlike and Uniformly Convex Functions

**Academic Editor:**Shaher Momani

#### Abstract

Let be the class of analytic functions in the open unit disk 𝕌. We define by where is the fractional derivative of of order . If , then a function in is said to be in the class if is a parabolic starlike function. In this paper, several properties and characteristics of the class are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class-preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved.

#### 1. Introduction and Definitions

Let be the class of functions analytic in the open
unit disk and let be the subclass of consisting of functions of the formand be the class of functions in of the formLet be the subclass of consisting of functions of the formA function in is said to be *uniformly convex* in if is a univalent convex function along with the
property that, for every circular arc contained in ,
with center also in ,
the image curve is a convex arc. The class of uniformly convex
functions is denoted by (for details, see [1]). It is well known from
[2, 3] thatCondition (1.4) implies
thatlies in the interior of the
parabolic regionfor every value of .
A function in is said to be in the class of *parabolic
starlike functions*, denoted by (cf. [3]), if

Let the function be given bywhere is the *Pochhammer symbol* defined
byFurther, let (cf. [4, 5])In terms of *Hadamard product
or convolution,
*
note that is the identity operator andIt is well known that if ,
then maps into itself. We also need the following
definitions of a fractional derivative.

*Definition 1.1 (cf. [5, 6], see also [7, 8]). * Let the
function be analytic in a simply connected domain of
the -plane containing the origin. The fractional
derivative of of order is defined by
where the multiplicity of is removed by requiring to be real when .

Using Definition 1.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [5] introduced the operator defined byNote that .

Corresponding to the operator defined in (1.13), Srivastava and Mishra [9] studied the class of functions satisfying the inequality

In Definition 1.2, we generalize the Owa-Srivastava operator defined in (1.13) as follows.

*Definition 1.2. *Let be in .
One defines an operator bywhere is the fractional derivative of of order .

From Definition 1.2, we note that

In the present paper, we study a class of analytic functions, related to , , and , using the operator defined in Definition 1.2.

*Definition 1.3. *Let ,
where be the class of functions satisfying the inequality

It follows that

*Remark 1.4. * if and only if is uniformly convex function.

Using the definition of , we start with proving sandwich theorem related to the fractional derivative. Then, we investigate several properties and characteristics of the general class using similar techniques to [9]. These include subordination, inclusions and characterization, growth theorems, and class-preserving operators (like the Hadamard product and various integral transforms).

#### 2. Sandwich Theorem

In order to
prove our sandwich result, we need first to recall
the principle of subordination between analytic functions, let the functions and
be in .
We say that is subordinate to or is superordinate to in ,
written as ,
if is univalent in ,Let and let .
If and are univalent and satisfies the first-order differential
superordination,then is a solution of the differential
superordination (2.2). An analytic function is called a *subordination* if for all satisfying (2.2). A univalent subordinant that satisfies for all subordinations of (2.2) is said to be the best subordinant.
An analytic function is said to be dominant if for all satisfyingA univalent dominant that satisfies for all dominants of (2.3) is said to be the best dominant.

We also need the following definition and lemma.

*Definition 2.1 (see [10, page 817, Definition 2]). *
Denoted by ,
the set of all functions that are analytic and injective on ,
whereand are such that for .

Lemma 2.2 (see [11]). * Let be two nonzero univalent functions in ,
and let .
Further assume that and for , is starlike univalent in . If , , is the th derivative of and
is univalent in ,
then**implies**and are, respectively, the best subordinant and
the best dominant.*

As an application of Lemma 2.2, we prove the following theorem.

Theorem 2.3. *Let be two nonzero univalent functions in ,
and let , and .
Further, assume that and are starlike univalent in .
If ,**then**implies**and are, respectively, the best subordinant and
the best dominant.*

*Proof. *Let be defined as in Definition 1.2, where and .
Then from (1.17), we have for ,This yieldsBy applying Lemma 2.2 for and ,
we get the result.

Putting in Theorem 2.3, we get the following corollary.

Corollary 2.4. *Let be two nonzero univalent functions in ,
and let , .
Further, assume that and are starlike univalent in .
If ,**then**implies**and are, respectively, the best subordinant and
the best dominant.*

In particular, for , Corollary 2.4 reduces to the following remark.

*Remark 2.5. *Let be two nonzero univalent functions in ,
and assume that and are starlike univalent in .
For ,
if , is univalent in and ,
thenimpliesand are, respectively, the best subordinant and
the best dominant.

*Remark 2.6. *Taking in Theorem 2.3 yields that Corollary
2.4 and
Remark 2.5 are also hold true for ,
where .

#### 3. Some Properties of the Class

We need the following results in our investigation of the class .

Lemma 3.1 (see [12]). * Let and be univalent convex functions in .
Then the Hadamard product is also univalent convex in .*

Lemma 3.2 (see [13]). * Let and be univalent convex functions in .
Also let and .
Then .*

Lemma 3.3 (see [12]). * Let each of the functions and be univalent starlike of order .
Then, for every function ,**where denotes the closed convex hull.*

Theorem 3.4. *If and ,
then*

*Proof. *Let .
ThenAlso it is known that (cf.
[14])Since is a convex region, using Lemma 3.3, we
getThus, .
This completes the proof of Theorem 3.4.

Corollary 3.5. *Let and .
Then**In particular, the functions in are parabolic starlike and they are uniformly
convex when .*

Corollary 3.6. *Let and .
Then *

It can be verified that the Riemann map of onto the region , satisfying and , is given by

We define the function by

Theorem 3.7. *Let and let be defined by (3.8). Then is a convex univalent function. Furthermore,
if ,
then*

*Proof. *We first note thatwhere each member of the
Hadamard product in (3.10) is known to be a convex univalent function (cf.
[2, 14]). Therefore, by
Lemma 3.1, is a univalent convex function. Next, if ,
thenThus, there exists a function satisfying the Schwarz Lemma such
thatSince is a univalent convex function, a result of
[15] (see also
[16, page 50])
yieldsIt now follows from a known
result of [14, page
508, Theorem 2] thatThe proof of Theorem 3.7 is
evidently completed.

*Remark 3.8. *(i) Letting or equal to zero in Theorem 3.7, we immediately
obtain a subordination result due to Srivastava and Mishra (see [9]).

(ii) Taking in Theorem 3.7, we get a result of [2,
page 169, Theorem 3].

Theorem 3.9. *Let .
If ,
then**where is defined by (3.8). Equality holds true in
(3.15) and (3.16) for some if and only if is a rotation of .*

*Proof. *Let .
Then, by Theorem 3.7 and the *Lindelöf* principle of subordination, we
getSince is a univalent convex function and has real
coefficients, is a convex region symmetric with respect to
real axis. Hence,Thus, (3.17) gives the assertion
(3.15) of Theorem 3.9. Also, we readily have the assertion
(3.16) of
Theorem 3.9.
The sharpness in (3.15) and
(3.16) is also a consequence of the principle of
subordination. This completes the proof of Theorem 3.9.

Corollary 3.10. *Let ,
where .
Then,**The result is sharp.*

*Remark 3.11. *(i) Letting or equal to zero in Theorem 3.9, we obtain a result
due to Srivastava and Mishra (see [9]).

(ii) Taking in Theorem 3.9, we get a result of [2, page 170,
Corollary 3].

Next, we investigate characterization for to be in the class . We need first the following lemma.

Lemma 3.12. *If , where ,
then*

*Proof. *Suppose .
We can writeThen, there exists an integer such thatFor ,
we haveSince ,
there exists a real number , ,
such that .
Hence, is not univalent.

Theorem 3.13. *Let .
Then, a function if and only if*

*Proof. *First, considerwhere .
Hence, if (3.24) holds, then the above expression is less than ,
and consequentlyConversely, if and is real, we getLet along the real axis, then we
getUsing Lemma 3.12, we
haveTherefore, the denominator in
(3.28) is positive, and hence
(3.24) holds. This completes the proof of Theorem
3.13.

*Remark 3.14. *Theorem 3.13 is sharp for functions
of the form

Corollary 3.15. *If ,
then**In particular, if and only if and if and only if *

Corollary 3.16. *If , where ,
then*

Corollary 3.17. *If where ,
then*

*Proof. *Let , where .
Clearly,Therefore,

*Remark 3.18. *Under the hypothesis of Corollary 3.17, lies in a disc centered at the origin with
radius given byIn particular, we have

(i) if ,
then lies in a disc centered at the origin with
radius (ii) if ,
then lies in a disc centered at the origin with
radius ;(iii)if ,
then lies in a disc centered at the origin with
radius ;(iv)if ,
then lies in a disc centered at the origin with
radius .

Consequently, let be defined byThen, belonging to the corresponding class implies lies in a disc centered at the origin with radius given by

#### 4. Class-Preserving Operators and Transforms

Theorem 4.1. *Let be univalent starlike function of order and let .
Then,**In particular, if is univalent starlike function of order and ,
then*

*Proof. *Let be univalent starlike function of order and .
By definition,The commutative and associative
properties of the *Hadamard product* yieldTherefore, using Lemma 3.3, we
getThis completes the proof of
Theorem 4.1.

Taking and in Theorem 4.1, then we have the following corollary.

Corollary 4.2. *If and ,
then .
In particular, if and ,
then .
Moreover, if and ,
then .*

Taking or , and in Theorem 4.1, then we have the following corollary.

Corollary 4.3. *If and ,
then .
In particular, if and ,
then .
Moreover, if and ,
then .*

Corollary 4.4 (see [9]). * If and ,
then**In particular, if and ,
then*

Corollary 4.5 (see [3]). * If and ,
then .
In particular, if and ,
then .*

Theorem 4.6. *Let and ,
where .
Then, .*

*Proof. *The proof of Theorem 4.6 is similar
to that of Theorem 4.1. Let and .
We first note that