Research Article | Open Access

Volume 2021 |Article ID 9924434 | https://doi.org/10.1155/2021/9924434

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

Accepted03 Jun 2021
Published28 Jun 2021

Abstract

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.

1. Introduction

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 [1] 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.

The classes and were studied by Ma and Minda [2, 3] and Ronning [4, 5]. They provided the analytic characterization of class and as follows:and

In 1999, Kanas et al. [6] 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 [613]. The class [14, 15] denotes the class of strongly starlike functions of order , which is defined as follows:

In 2015, Sokol et al. [16] introduced the class as follows:

For , , in a recent paper, Darus et al. [17] defined a new class as follows:

The class has some interesting relationship with other classes of univalent functions. Motivated from [1721], 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 [22].

Lemma 2. Let and . Then, , where

Proof. Let . Then, . Since .
Therefore, .
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 [23]). 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 [24]). Let and be complex numbers. Then, for , .

Lemma 5. Let be analytic in of the form
, with in . If there exists a point , such that and for some , then we have , where , where . This result is the generalization of Nunokawa’s lemma [25, 26].

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 .

Proof. Let be a complex number such that , for , then .
Since is univalent in so thatSince , thenUsing (21) in (20), , where is defined in (10).

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).

4. Conclusions

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.

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

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.

Acknowledgments

The sixth author was supported by the Universiti Kebangsaan Malaysia grant GUP-2019-032.

References

1. A. W. Goodman, “On uniformly convex functions,” Annales Polonici Mathematici, vol. 56, no. 1, pp. 87–92, 1991. View at: Publisher Site | Google Scholar
2. W. Ma and D. Minda, “Uniformly convex functions,” Annales Polonici Mathematici, vol. 2, no. 57, pp. 165–175, 1992. View at: Publisher Site | Google Scholar
3. W. Ma and D. Minda, “Uniformly convex functions II,” Annales Polonici Mathematici, vol. 3, no. 58, pp. 275–285, 1993. View at: Publisher Site | Google Scholar
4. F. Ronning, “On starlike functions associated with parabolic regions,” Annales Universitatis Mariae Curie-Sklodowska, Sectio A, vol. 45, no. 14, pp. 117–122, 1991. View at: Google Scholar
5. 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. View at: Publisher Site | Google Scholar
6. 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. View at: Publisher Site | Google Scholar
7. S. Kanas, “Subordinations for domains bounded by conic sections,” Bulletin of the Belgian Mathematical Society-Simon Stevin, vol. 15, pp. 589–598, 2008. View at: Publisher Site | Google Scholar
8. 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. View at: Publisher Site | Google Scholar
9. S. Kanas and A. Wiśniowska, “Conic regions and k-uniform convexity II,” Folia Science University Technology Resolve, vol. 22, pp. 65–78, 1998. View at: Publisher Site | Google Scholar
10. 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. View at: Publisher Site | Google Scholar
11. 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. View at: Publisher Site | Google Scholar
12. 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. View at: Google Scholar
13. A. W. Wajnryb, “Some extremal bounds for subclasses of univalent functions,” Applied Mathematics and Computation, vol. 215, pp. 2634–2641, 2009. View at: Google Scholar
14. 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. View at: Publisher Site | Google Scholar
15. 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. View at: Google Scholar
16. 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. View at: Google Scholar
17. M. Darus, S. Hussain, M. Raza, and J. Sokol, “On a subclass of starlike functions,” Results in Mathematics, vol. 73, pp. 1–12, 2018. View at: Publisher Site | Google Scholar
18. 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. View at: Publisher Site | Google Scholar
19. 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. View at: Publisher Site | Google Scholar
20. 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. View at: Publisher Site | Google Scholar
21. 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. View at: Google Scholar
22. S. S. Miller, “Differential inequalities and Carathéodory functions,” Bulletin of the American Mathematical Society, vol. 81, no. 1, pp. 79–82, 1975. View at: Publisher Site | Google Scholar
23. 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. View at: Publisher Site | Google Scholar
24. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, UK, 4th edition, 1958.
25. 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. View at: Publisher Site | Google Scholar
26. 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. View at: Publisher Site | Google Scholar

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