Research Article | Open Access
Certain Properties of Multivalent Functions Associated with the Dziok-Srivastava Operator
By making use of the techniques of the differential subordination, we derive certain properties of -valent functions associated with the Dziok-Srivastava operator.
Let denote the class of functions of the form which are analytic in the open unit disk . We write .
Suppose that and are analytic in . We say that the function is subordinate to in , or superordinate to in , and we write or , if there exists an analytic function in with and , such that . If is univalent in , then the following equivalence relationship holds true (see [1–3]):
For functions given by we define the Hadamard product (or convolution) of and by
For complex parameters and , the generalized hypergeometric function is defined (see ) by the following infinite series: where is the Pochhammer symbol defined, in terms of the Gamma function , by Corresponding a function defined by Dziok and Srivastava  considered a linear operator defined by the following Hadamard product: If is given by (1), then we have where To make the notation simple, we write It easily follows from (9) or (10) that It should be remarked that the linear operator is a generalization of many other linear operators considered earlier. In particular, for we have the following observations:(i), where the linear operator was investigated by Hohlov ;(ii), where the linear operator was studied by Goel and Sohi . In the case when , is the Ruscheweyh derivative of (see );(iii), where is the generalized Bernardi-Libera-Livingston integral operator (see );(iv) , where is the fractional integral of of order when and fractional derivative of of order when . The extended fractional differintegral operator was introduced and studied by Patel and Mishra . The fractional differential operator with was investigated by Srivastava and Aouf . The operator was introduced by Owa and Srivastava  (see also [13–15]).(v) = , where the linear operator was studied by Saitoh  which yields the operator introduced by Carlson and Shaffer  for ;(vi) = , where is the Choi-Saigo-Srivastava operator  which is closely related to the Carlson-Shaffer  operator ;(vii) = , where the operator was considered by Liu and Noor ;(viii) = , where is the Cho-Kwon-Srivastava operator .
In recent years, many interesting subclasses of analytic functions, associated with the Dziok-Srivastava operator and its many special cases, were investigated by, for example, Dziok and Srivastava [5, 20], Gangadharan et al. , Liu and Noor , Liu , Liu and Srivastava , and others see also [19, 24–26]). In the present paper, we shall use the method based upon the differential subordination to derive inclusion relationships and other interesting properties and characteristics of the Dziok-Srivastava operator .
2. Main Results
Unless otherwise mentioned, we assume throughout the sequel that ; ; ; and .
Let denote the class of functions of the form that are analytic in , we write . In our present investigation, we shall require the following lemmas.
Lemma 1 (see ). Let be analytic and convex (univalent) in with and . If then, for and , and is the best dominant.
Lemma 2 (see ). Let be a set in the complex plane and be a complex number satisfying . Suppose that the function satisfies the condition for all real and for all . If the functions and , then in .
Lemma 3 (see ). Let be analytic in with and for all . If there exist two points such that for some and and for all , then where
Theorem 4. Let , . Let , then implies The bound is the best possible.
Proof. It easily follows from (13) that From (20) and (22), we have That is, Let then (24) may be written as By using a well-known result (see ) to (26) we obtain that or, equivalently, where is analytic in , and for . Since for and , (28) yields To see that the bound cannot be increased, we consider the function Since we easily have that satisfies (20) and as . This completes the proof of Theorem 4.
Theorem 5. Let , . If satisfies the following inequality then where is the positive root of the equation
then is analytic in and . Differentiating (36) and using (22), we obtain that
Using (33) and (38), we have
Now for all real , we have
where is the positive root of (35).
Note that for , , and we have and . This shows . Hence for each , . By Lemma 2, we get , and this proves (34).
Theorem 6. Suppose that ; and , . If given by satisfies then where and are the solution of the equations: where is given by (19).
Proof. Using (42) and the identity (22), it follows that
for . Putting
On differentiating (47) followed by a simple calculation, we get
Let be the function which maps onto the angular domain with . By using (43) in (48), we get
Further, an application of Lemma 1 yields in and hence for .
Suppose there exist two points such that the condition (28) is satisfied. Then by Lemma 3, we obtain (18) under the constraint (19). Therefore, we have which contradicts the assumption (43). This proves the assertion (44) of the Theorem 6.
For , Theorem 6 reduces to the following corollary.
Corollary 7. Suppose that and . If defined by (42) satisfies then where is the solution of the equation:
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