Integral Transforms of Functions to Be in a Class of Analytic Functions Using Duality Techniques
Let , denote the class of all normalized analytic functions in the unit disc such that for some with , and . Let , , denote the Pascu class of -convex functions given by the analytic condition which unifies the classes of starlike and convex functions. The aim of this paper is to find conditions on so that the integral transform of the form carry functions from into . As for the applications, for specific values of , it is found that several known integral operators carry functions from into . The results for a more generalized operator related to are also given.
Let denote the class of all functions analytic in the open unit disc with the normalization , and let be the class of functions that are univalent in . A function is said to be starlike or convex , if maps conformally onto the domains, respectively, starlike with respect to origin and convex. Note that in , if follows from the well-known Alexander theorem (see  for details). A useful generalization of the class is the class that has the analytic characterization and . Various generalizations of classes and are abundant in the literature. One such generalization is the following.
A function is said to be in the Pascu class of -convex functions of order if  or in other words This class is denoted by . Even though this class is known as Pascu class of -convex functions of order , since we use the parameter for another important class, we denote this class by , , and we remark that, in the sequel, we only consider the class . Clearly, and , which implies that this class is a smooth passage between the classes of starlike and convex functions.
The main objective of this work is to find conditions on the nonnegative real valued integrable function satisfying , such that the operator is in the class . Note that this operator was introduced in . To investigate this admissibility property, the class to which the function belongs is important. Let , where , , and , denote the class of all normalized analytic functions in the open unit disc such that for some . This class and its particular cases were considered by many authors so that the corresponding operator given by (3) is univalent and in for some particular values of , , , and . This work was motivated in  by studying the conditions under which and was generalized in  by studying the case . Similar situation for the convex case, namely, , was initiated in . After several generalizations by many authors, recently, the conditions under which were obtained in  and the corresponding results for the convex case so that were obtained in . Applications involving several well-known integral transforms were studied in [6, 7] (see also ). For all of the literatures involving the complete study in this direction so far, we refer to [6–9] and the references therein.
In this work, we find conditions on so that using duality techniques which are presented in Section 2. As applications, in Section 3, we consider particular values for in (3) so that results for some of the well-known integral operators can be deduced. A more generalized operator introduced in  is considered in Section 4 for similar type of results.
When , then is chosen to be , in which case . When , (5) yields , or , and leads to two cases as follows.(i)For , then, choosing gives .(ii)For , then, and .
Next, we introduce two known auxiliary functions . Let
Here, denotes the convolution inverse of such that . By , we mean the following: if and are in with the power series expansions and , respectively, then the convolution or Hadamard product of and is given by .
Since , , when , making the change of variables , in (7) results in writing as
Now, let be the solution of the initial value-problem satisfying . The series solution is given by Let be the solution of the differential equation satisfying . The series solution of is given by Note that also satisfies .
Our main result is the generalization of the following results given in [6, 7]. The necessary and sufficient conditions under which the operator carries the function from to the classes and , respectively, are given in the next two results.
2. Main Results
We start with a result that gives both necessary and sufficient conditions for an integral transform that satisfies the admissibility property of the class , which contains nonunivalent functions also, to the Pascu class .
Since ,  we may assume that , . Further, assuming that , from (5) we get
using (6) and (7). This gives
Since implies that , by the well-known result [11, page 94] of convolution theory we get
This condition holds if and only if [11, page 23]
Using the results (14) and (19) from Theorems 1 and 2, respectively, we obtain
which is the required result.
To verify sharpness, let be the solution of the differential equation where satisfies (22). Further simplification using (10) and (12) gives where .
Clearly, . Using and the series expansion of , we get
This means that Using (37), a simple computation gives This means that Hence, for some , so is not even locally univalent in . This shows that the result is sharp for .
Remark 6. This result generalizes various results known in this direction. For example, gives Theorem 1 [6, Theorem 3.1], and gives Theorem 2 [7, Theorem ]. For other particular cases with or , we refer to [6, 7] and the references therein.
Proof. Consider the following: Integration by parts gives by a simple computation. It is easy to see that, from Theorems 3 and 4, only when is decreasing on which is nothing but (40), and the proof is complete by applying Theorem 5.
Remark 8. Even though we did not consider the case , even at , Theorem 7 does not reduce to a similar result given in . This is due to the fact that our condition (40) has the term in the denominator, whereas the corresponding result in  has the term and hence has different condition.
It is difficult to check the condition given in Section 2, for . In order to find applications, simplified conditions are required. For this purpose, from (40), it is enough to show that is increasing on which is equivalent to having the fact that is decreasing on , where and are defined in (15) and (16). It is enough to have .
Let , where . So to satisfy the above condition, we need to have Since and , we get . This implies that it suffices to have that is increasing on , which means that The above equation also holds if which is equivalent to the condition This inequality can further be reduced to if the inequality is true. So, in order to obtain further results, we check conditions (48) and (49). Note that, whenever , from (47) we see that it is sufficient to check (48) as there is no necessity for the condition given by (49).
As the first application, we consider , so that the Bernardi operator of the function in is in .
Theorem 9. Let , , and let satisfy (22). If , then the function, given by the Bernardi operator, belongs to the class if
Proof. With , using (51), we get So, inequality (49) is satisfied. If inequality (48) holds, then , which means that On further simplification, this inequality reduces to Clearly, . Hence, using and substituting in (54), we get which is true by the hypothesis.
Remark 11. We recall that, throughout this paper, we use , as case is considered in . But we note that, for the Bernardi operator, the result given in  uses the fact , which is not true. Hence, in order to make completion of the work for in this direction, we give the result related to Bernardi operator for . Since the condition for given in  is different from the one given by (49) and (48), we explicitly prove this result.
Theorem 12. Let , , and let satisfy (22). If , then the function belongs to the class if and for .
Proof. Note that, for the Bernardi operator, the case is considered in [6, Theorem 5.1] and is considered in [7, Theorem 5.1]. We consider only the case .
For the case , the integral operator is in if and only if which can be easily obtained as in Theorem 7.
Let , where for the integral operator to be in , .
Substituting the value of and in the above equation, we have Consider the case when , where . So inequality (59) on simplification reduces to If and , then, for , (60) is satisfied.
For , , which clearly implies that if is decreasing on , that is, , then also inequality (60) holds. Further simplifying (59), for Bernardi operator, we need to show that On substituting the values of , , and in (61), we get To satisfy (62), should hold: Since , we get satisfying the hypothesis of the theorem which gives that given by (56) is in .
Theorem 16. Let , , , , and , and let satisfy (18). If , then the function belongs to the class if
Proof. For to be in , should satisfy (47), where
Since , and hold. Hence it is enough to check (48). If we substitute the values of , , and in (48), a simple computation shows that (48) is equivalent to