Abstract
In this present paper, we introduce and explore certain new classes of uniformly convex and starlike functions related to the Liu–Owa integral operator. We explore various properties and characteristics, such as coefficient estimates, rate of growth, distortion result, radii of close-to-convexity, starlikeness, convexity, and Hadamard product. It is important to mention that our results are a generalization of the number of existing results in the literature.
1. Introduction
Let denote the complex plane and assume that Ap denotes the class of -valent function of the formwhich are analytic in the open unit disc and . Specially, for , we denote A=A1.
By , , and , the subclasses of Ap consist of all univalent, convex, and starlike functions . We also denote , the class of starlike function of order , . In 1991, Goodman [1, 2] introduced the classes and of uniformly starlike and uniformly convex functions, respectively. A function is uniformly starlike (uniformly convex) in if is in and has the property that, for every circular arc contained in , with center also in , the arc is starlike (convex) with respect to . A more useful representation of and was given in [3]; see [4, 5], for details:and
In 1999, for , Kanas and Wisniowska [6] introduced the classes and of -uniformly convex and uniformly starlike functions, respectively, see also [7–10].
Let denote the subclass of Ap consisting of functions of form (1) and satisfy the following inequality:
Also, let denote the subclass of Ap consisting of functions of form (1) and satisfy the following inequality:
It follows from (4) and (5) that
Notice that, and , for . The convolution (Hadamard product) for two functions , Ap, is defined bywhere is given in (1) and .
Taking from the above cited work and using Liu–Owa integral operator, we introduce the following class of -valent analytic function. In 2004, Liu and Owa [11] (see also [12–14]) introduced the integral operator : as follows:and
For , given by (1), and using properties of gamma function, we have
Definition 1. For , , , and , a function is in class if and only ifWe also denote , where the class of functions of form (1) for which . For more details, see [15–20].
1.1. Special Cases
Specializing parameters, , and , we obtain the following subclasses studied by various authors:
(1) [21]
(2) [21]
(3) [22]
(4) [23]
(5) [24]
(6) [25]
2. Main Results for the Class
2.1. Coefficient Estimates
In this section, we obtain a necessary and sufficient condition for functions in the classes .
Theorem 1. A function given by (1) is in the class ifwhereand
Proof. It suffices to show that inequality (11) holds true. As we know,Then, inequality (11) may be written aswhich can be written as , whereandThen, we haveNow,Also,Using (23) and (24), then we can obtain the following inequality:The last expression is bounded below by 0 ifwhich complete the proof.
Theorem 2. Let be given by (1) and in ; then, if and only ifwhere , , , and are given by (13)–(16), respectively.
Proof. In view of Theorem 2, we need only to show that satisfies coefficient inequality (27). If , then, by definition, we haveSince is a function of form (1) with the argument properties given in the class and setting in the above inequality, we haveLetting (29) leads to the desired inequality:The function,is an external function for (27).
Corollary 1. Let the function defined by (1) be in the class ; then,with equality in (32), is attained for the function given by (31).
Theorem 3. Let the function with argument property as in class . Define andwhere and . Then, is in the class if and only if it can be expressed aswhere and .
Proof. Assume thatThen, by Theorem 2, . It follows thatConversely, assume that the function defined by (1) belongs to the class ; then,Setting , and , then and this completes the proof.
2.2. Growth and Distortion Result
In this section, we find a growth and distortion bound for functions in the classes .
Theorem 4. Let the function be defined by (1) in the class ; then, for ,andwhere equalities (38) and (39) hold for the function given by (27), for .
Proof. From Theorem 2, we haveThe last inequality follows from Theorem 2 Thus,Similarly,Now, by differentiating (1), we obtainandUsing Theorem 2 in (44), we haveor, equivalentlyUsing (46) into (43) and (44) yields inequality (39).
2.3. Radii of Close-to-Convexity, Starlikeness, and Convexity
In this section, we obtain the radii of close-to-convexity, starlikeness, and convexity for functions in the classes .
Theorem 5. Let ; then,
(i) is starlike of order in the disc , where(ii) is convex of order in the disc , where
These results are sharp for the extremal function given by (31).
Proof. (i) Given and is starlike of order , we haveFor the left-hand side of (49), we haveThe last expression is less than ifUse the fact that if and only ifWe can say (49) is true ifor, equivalently,which is required.
(ii) Using the fact that is convex if and only if is starlike, we can prove (ii) on similar lines to the proof of (i).
2.4. Modified Hadamard Product
Let the function be defined by
Then, we define the modified Hadamard product of and by
Now, we prove the following.
Theorem 6. Let given by (55) be in the class ; then, for
Proof. We need to prove the largest , such thatFrom Theorem 2, we haveBy Cauchy–Schwarz inequality, we haveThus, it is sufficient to show thatThat is,Note thatConsequently, from (62) and (63), we obtainor, equivalently,Since is an increasing function for , letting in (65), we obtainThe proof of our theorem is now completed.
Theorem 7. Let given by (55) be in the class . If the sequence is nondecreasing, then functionbelongs to the class , where
Proof. We need to prove the largest .
From Theorem 2, we haveandIt follows from (69) and (70) thatTherefore, we need to find the largest , such thatThat is,Since is an increasing function for , letting in (73), we readily haveThe proof of our theorem is now completed.
3. Conclusion
In our current investigation, we have presented and studied thoroughly some new subclasses of valent functions related with uniformly convex and starlike functions, in connection with the Liu–Owa integral operator given by (8). We have obtained sufficient and necessary conditions in relation to these classes, including growth, distortion theorem, and radius problem. Some special cases have been discussed as applications of our main results. The techniques and ideas of this paper may stimulate for further research in this area of knowledge.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
SH came with the main thoughts and helped to draft the manuscript, SGAS and IA proved the main theorems, and SN and MD revised the paper. All authors read and approved the final manuscript.
Acknowledgments
The fourth author was supported by UKM (Grant GUP-2019-032).