Research Article | Open Access
Syed Ghoos Ali Shah, Saima Noor, Saqib Hussain, Asifa Tasleem, Akhter Rasheed, Maslina Darus, "Analytic Functions Related with Starlikeness", Mathematical Problems in Engineering, vol. 2021, Article ID 9924434, 5 pages, 2021. https://doi.org/10.1155/2021/9924434
Analytic Functions Related with Starlikeness
The aim of present investigation is to study a new class of analytic function related with the Sokol-Nunokawa class. We derived relationships of this class with strongly starlike functions and obtained many interesting results.
Let be the class of functions having the series representation,which are analytic in the open unit disk and normalized by and .
A function is in class of univalent function if and only if implies , for all , . An analytic function subordinate to an analytic function denoted as if and only if there exists an analytic function in with and , , such that , . In particular, if is univalent in , then if and only if and .
Let denote the class of analytic functions of the form such that in . A function is in class of normalize convex function if and only if . Also, a function is in class if and only if .
In 1991, Goodman  introduced the class of uniformly convex function . A function is in class if, for every circular arc with center in , the arc is convex. Also, a function is in class if, for every circular arc contained in with center in , the arc is starlike.
In 1999, Kanas et al.  introduced the class , , of -uniformly convex functions as follows:
Furthermore, the class of -uniformly starlike functions is defined as follows:
Many authors had investigated the interesting properties of the abovementioned classes. For details, see [6–13]. The class [14, 15] denotes the class of strongly starlike functions of order , which is defined as follows:
In 2015, Sokol et al.  introduced the class as follows:
For , , in a recent paper, Darus et al.  defined a new class as follows:
The class has some interesting relationship with other classes of univalent functions. Motivated from [17–21], we introduced a new class of analytic functions and found its relation with some other existing classes of analytic functions.
Definition 1. Let , . Then, is in class if and only ifFor special values of parameter , we obtained many known classes of analytic functions. For example, if we take , then , the class of convex functions.
2. Set of Lemmas
To prove our main results, we need the following lemmas.
Lemma 1. Let and and let be a complex-valued function satisfying the conditions: (i) is continuous in a domain .(ii) and .(iii) whenever and . If is a function that is analytic in such that and for , then . This result is due to Miller .
Lemma 2. Let and . Then, , where
Proof. Let . Then, . Since .
We formulate a functional by taking , , thus . Clearly, (i) is continuous in domain (ii)Consider . Then, , where we have used the fact that . This implies that , where .
Clearly, , if and only if and . From , .
From , we have . Now, using Lemma 1, we have the required result.
Lemma 3 (see ). If and the complex number satisfies , then the differential equation , has a univalent solution in given by
If is analytic in and satisfies , then , and is the best dominant.
Lemma 4 (see ). Let and be complex numbers. Then, for , .
3. Main Results
In this section, we derived our main results.
Theorem 1. The function is in the class if and only if .
Proof. Let , then some simple calculation gives usBy using (9) and (12), we obtainUsing (12) and (13), it is clear that if and only ifIt is suffices to study , , and in (14).Then, equation (15) leads to a simple computation which shows that
orThe right-hand side of the above inequality (16), is seen to have minimum for and minimum value is . Hence, a necessary and sufficient conditions for (16) is or . This completes the proof.
Corollary 1. The function belongs to a class if and only if .
Remark 1. It is important to note that if , then .
Theorem 2. Let . Then, , where is the class of starlike functions of order , where is defined in (10) and and .
Proof. Let , where is analytic in unit disk and . Now, using (9), we obtain . This implies that , where . The above relation can be written in the following Briot Bouquet differential subordination . Now, using Lemma 3, we have this complete the proof.
Special case: for and , we have , that is, implies .
Theorem 3. Let of the form ; then, is strongly starlike of order , where .
Proof. Let then, is of the form . Now using the definition of class , we have .
If there exists a point , we have . Then, from Lemma 5, we have , where andFor the case , we obtainAlso, we have .
Then, from for , we haveTherefore, .
By using (18) and (19), we have , which is a contradiction, therefore, for . Similarly, we can prove the case , by using the same method as the above and will get a contradiction. This proves that is strongly starlike of order .
Theorem 4. If , then , where , with and .
Proof. Let , where is analytic in and . .
This implies that , where , . The above relation can be written in the following Briot–Bouquet differential subordination, . Using Lemma 4, for and , we have , which is required.
Remark 2. If , then and , where is defined in (10).
Theorem 5. Let and ; then, contains an open disk of radius .
Theorem 6. Let such that . Then,where is defined in (10).
Proof. As , and . Now, , and . Then, , this implies , where , and , in which is defined in (10).
In this article, we studied a new class of analytic functions. We derived certain necessary and sufficient condition, order of strongly starlike. We also pointed out some existing results in literature.
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
SGAS and SH came with the main thoughts and helped to draft the manuscript, AT wrote, reviewed, and edited the manuscript, AR proved the main theorems, and MD and SN are revised the paper. All authors read and approved the final manuscript.
The sixth author was supported by the Universiti Kebangsaan Malaysia grant GUP-2019-032.
- A. W. Goodman, “On uniformly convex functions,” Annales Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991.
- W. Ma and D. Minda, “Uniformly convex functions,” Annales Polonici Mathematici, vol. 2, no. 57, pp. 165–175, 1992.
- W. Ma and D. Minda, “Uniformly convex functions II,” Annales Polonici Mathematici, vol. 3, no. 58, pp. 275–285, 1993.
- F. Ronning, “On starlike functions associated with parabolic regions,” Annales Universitatis Mariae Curie-Sklodowska, Sectio A, vol. 45, no. 14, pp. 117–122, 1991.
- F. Ronning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993.
- S. Kanas and A. Wisniowska, “Conic regions and k-uniform convexity,” Journal of Computational and Applied Mathematics, vol. 105, no. 1-2, pp. 327–336, 1999.
- S. Kanas, “Subordinations for domains bounded by conic sections,” Bulletin of the Belgian Mathematical Society-Simon Stevin, vol. 15, pp. 589–598, 2008.
- S. Kanas and H. M. Srivastava, “Linear operators associated with k-uniformly convex functions,” Integral Transforms and Special Functions, vol. 9, no. 2, pp. 121–132, 2000.
- S. Kanas and A. Wiśniowska, “Conic regions and k-uniform convexity II,” Folia Science University Technology Resolve, vol. 22, pp. 65–78, 1998.
- A. Lecko and A. Wiśniowska, “Geometric properties of subclasses of starlike functions,” Journal of Computational and Applied Mathematics, vol. 155, no. 2, pp. 383–387, 2003.
- A. Rasheed, S. Hussain, S. G. A. Shah, M. Darus, and S. Lodhi, “Majorization problem for two subclasses of meromorphic functions associated with a convolution operator,” AIMS Mathematics, vol. 5, no. 5, pp. 5157–5170, 2020.
- J. Sokol and M. Nunokawa, “On some class of convex functions,” Comptes Rendus de l’Académie des Sciences-Series I, vol. 353, pp. 427–431, 2015.
- A. W. Wajnryb, “Some extremal bounds for subclasses of univalent functions,” Applied Mathematics and Computation, vol. 215, pp. 2634–2641, 2009.
- D. A. Brannan and W. E. Kirwan, “On some classes of bounded univalent functions,” Journal of the London Mathematical Society, vol. 1, no. 2, pp. 431–443, 1969.
- J. Stankiewicz, “Quelques problemes extremaux dans les classes des fonctions -angulairement etoilees,” Annales Universitatis Mariae Curie-Sklodowska, Sectio A, vol. 20, pp. 59–75, 1966.
- J. Sokoł and A. Wiśniowska, “On some classes of starlike functions related with parabola,” Folia Science University Technology Resolve, vol. 28, pp. 35–42, 1993.
- M. Darus, S. Hussain, M. Raza, and J. Sokol, “On a subclass of starlike functions,” Results in Mathematics, vol. 73, pp. 1–12, 2018.
- Z.-G. Wang, M. Naeem, S. Hussain, T. Mahmood, and A. Rasheed, “A class of analytic functions related to convexity and functions with bounded turning,” AIMS Mathematics, vol. 5, no. 3, pp. 1926–1935, 2020.
- S. G. A. Shah, S. Hussain, A. Rasheed, Z. Shareef, and M. Darus, “Application of quasisubordination to certain classes of meromorphic functions,” Journal of Function Spaces, vol. 2020, 8 pages, 2020.
- S. Hussain, S. Khan, M. Asad Zaighum, and M. Darus, “Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator,” AIMS Mathematics, vol. 2, no. 4, pp. 622–634, 2017.
- S. G. A. Shah, S. Noor, M. Darus, W. Ul Haq, and S. Hussain, “On meromorphic functions defined by a new class of liu-srivastava integral operator,” International Journal of Analysis and Applications, vol. 18, no. 6, 2020.
- S. S. Miller, “Differential inequalities and Carathéodory functions,” Bulletin of the American Mathematical Society, vol. 81, no. 1, pp. 79–82, 1975.
- S. S. Miller and P. T. Mocanu, “Univalent solutions of Briot-Bouquet differential equations,” Journal of Differential Equations, vol. 56, no. 3, pp. 297–309, 1985.
- E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, UK, 4th edition, 1958.
- M. Nunokawa, “On properties of non-Carathéodory functions,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol. 68, no. 6, pp. 152-153, 1992.
- M. Nunokawa, “On the order of strongly starlikeness of strongly convex functions,” Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol. 69, no. 7, pp. 234–237, 1993.
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