Some Properties of Fractional Calculus and Linear Operators Associated with Certain Subclass of Multivalent Functions
We investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivatives, and some well-known integral operators. In special cases, the results presented in this paper provide new approaches to several previously known results.
Let denote the class of functions of the form which are analytic in the open unit disc Also, let denote the subclass of consisting of all functions of the form For functions , given by the Hadamard product (or convolution) of and is defined by The Ruscheweyh derivative of of order is defined by where is given by (1.1) and . The Ruscheweyh derivative has been studied by several authors; for example, see [1, 2].
For , , and , let consist of functions so that for some . In , the authors obtained four containment results for the class . We denote . The class was studied by Swaminathan [4–6], Barnard et al. , Kim and Rønning , and others.
In the present paper, we investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivative, and some well-known integral operators defined on the class . In special cases, the results presented here provide new approaches to some previously known results.
Remark 1.1. Throughout this section, we assume that .
2. Definitions and Lemmas
For the function given by (1.2), we define It is easy to verify that
Lemma 2.1. Let the function be given by (1.2). Then if and only if for some .
Proof. Using the fact that if and only if , it suffices to show that where is defined by (2.1). Letting and assuming (2.3), we obtain where . By (2.3), the desired inequality (2.4) follows at once. Conversely, if , then or, equivalently (2.4). This yields which implies that Squaring the above inequality, choosing the value of on the half line and letting through this line, we obtain Hence we get which reduces to So the desired inequality (2.3) follows upon using (2.5).
Setting and in Lemma 2.1, we get the following result.
Throughout this paper, we define
As an immediate consequence of Lemma 2.1, we have the following corollary.
Corollary 2.3. Let the function be defined by (1.2). If , then for some .
Let and (, ) be complex numbers such that for . The generalized hypergeometric function is given by where denotes the Pochhammer symbol defined by The operator has recently been studied by several authors; for example, [3, 5]. For , the above series give rise to the Gaussian hypergeometric series .
In , Hohlov introduced the convolution operator by Motivated by the operator , the authors in  defined the convolution operators and as follows: where , and . For , the operator was introduced in .
In Section 3, we will make use of the following well-known fractional calculus operators , and . For an analytic function defined in a simply connected region of the complex -plane containing the origin, these operators are defined as follows (See [1, 10]): where multiplicity of is removed by requiring to be real when ; where the multiplicity of is removed, as in the definition of ; By virtue of (2.21), (2.22), (2.23) and in terms of Gamma function, it is wellknown (see for details ) that
where , , and for .
In Section 4, we will investigate the integral operator defined by where , and . For and , the operator was first defined by Bernardi . Later on several authors studied the operator ; for example, see [1, 5].
3. Distortion Inequalities of Convolution Operators
We next obtain distortion inequalities for the fractional operaters and .
Theorem 3.2. Suppose and . If , then for some , one has where , , , and the operator was defined by (2.19).
Proof. By using (2.19), we deduce that Then where Since is a decreasing function of , when , then Also, according to Lemma 2.1 and , we have for some . Then for some . From (3.7) and (3.9), we obtain In view of (3.11), we conclude that for some , and for some , which yield (3.5).
By letting and in Theorem 3.2, we deduce the following consequence.
Corollary 3.3. If , then for , , and some
Theorem 3.4. Let , and . If , then for some . The operator was defined by (2.19).
Proof. In view of (2.19) and (2.22), we have where Since is a decreasing function of , it follows that By using (3.11), (3.18), and (3.20), we get for some . The last two inequalities yield (3.16) and (3.17), respectively.
Letting , and in Theorem 3.4, we get the following result.
Corollary 3.5. Let be defined by (2.19). If , then for some , , , and .
We next prove the distortion theorems involving fractional calculus and generalized convolution operator defined by (2.20).
Theorem 3.6. Suppose , , and . Also, let , and . If , then for some . Here, the operator is defined by (2.20).
Proof. By making use of (2.20), we have It is easy to verify that This implies that where Since is a decreasing function of , when , and , we get From Lemma 2.1 and , we obtain for some . It follows from (3.26) and (3.28) that for some , which yield (3.23).
We state an obvious variant of Theorem 3.6 as follows.
Corollary 3.7. Let the function defined by (1.2) be in the class . Also let , , and . Then for some , , and .
Theorem 3.8. Let , and . Also, let , and . If , then for some .
Next we prove the following.
Theorem 3.9. Let , and . Also, let , and . If , then for some .
Proof. We have where Therefore So, from (3.25), we have where is defined by (3.27). Since is a decreasing function of , when and , then From (3.37), (3.38), and Lemma 2.1, we find that for some . The above inequalities lead us to the desired inequalities (3.33).
The proof of the following theorem is similar to Theorem 3.9, and so it is omitted here.
Theorem 3.10. Let , and . Also, let , and . If , then for some .
Corollary 3.11. Let , , , and . If , then for some and . Furthermore for some and .
Remark 3.12. Under the hypothesis of Corollary 3.11, and are included in disks with its center at origin and radii and , respectively, given by
4. Distortion Inequalities of Integral Operator
In this section, we obtain the distortion theorems involving the integral operator of functions in the class and fractional calculus operator.
Theorem 4.1. Let , and . If then for some . The operator is defined by (2.21).
Proof. Using the definition (2.25), for function of the form (1.2), we have So Therefore, we obtain where Since is a decreasing function of , when , then By using (3.11), (4.4), and (4.6), we get for some , which prove the inequalities (4.1).
The proof of the following theorem is similar to Theorem 4.1, and so it is omitted here.
Theorem 4.2. Let , , and . If , then for some .
The authors thank the referee for some useful suggestions for improvement of the article.
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