Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function

Muhammad Azam; Naseer Ullah

doi:doi:10.11648/j.ijamtp.20210703.13

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Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function

Muhammad Azam, Naseer Ullah

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 3)

Received: 2 June 2021 Accepted: 19 July 2021 Published: 11 September 2021

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Abstract

Abdullah Alotaibi defined a starlike function connected with a cosine hyperbolic function in the year 2020. We establish some appropriate conditions for several features of multivalent analytic starlike function subordinated with cosine hyperbolic function in this article. We determine conditions on α are subordinated by Janowski function. We acquire some suitable conditions by selecting specific values for functions we get some adequate conditions for multivalent starlik function related with cosine hyperbolic. Over the last decade, starlike functions have grown in popularity in both literature and application. Our goal in this work is look at some practical challenges with q-starlike functions. Moreover, we will show that the class described in this research, as well as the results gained, generalizes numerous previously published papers. We need to add some fundamental Geometric function theory literature here to comprehend the notions employed in our work in a straightforward way. To do so, we'll start with the notation, which signifies the class of holomorphic or analytic functions in the holomorphic or analytic functions. Then the relationships must be stable. In addition, all univalent functions will belong to the subfamily. Furthermore, the possibility of subjections between analytic functions and, as shown by, as; the functions, are related by the connection of subjection, if there exists an analytic function with restrictions and such that in addition, if the function is in, we get The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domains and to study some of its useful properties of multivalent analytic functions subordinated cosine hyperbolic function.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 3)
DOI	10.11648/j.ijamtp.20210703.13
Page(s)	80-87
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2021. Published by Science Publishing Group

Keywords

Multivalent Functions, Janowski Function, Trigonometric Functions

References

[1]	Ali, R. M., Cho, N. E., Ravichandran, V., & Kumar, S. S. (2012). Differential subordination for functions for functions associated with lemniscate of Bernoulli’s. Taiwanese journal of mathematics, 16 (3): 1017-1026.
[2]	Arif, M., Ahmad, K., Liu, J. L., Sokol, J. (2019): A new class of analytic functions associated with Salagean operator. Journal of Function Spaces, Volume 2019, Article ID 5157394, 8 pages.
[3]	Cho, N. E., Kumar, V., Kumar, S. S., & Ravichandran. V. (2019): Radius problems for starlike functions associated with the sine function. Bulletin of the Iranian Mathematical Society, 45 (1) (2019): 213-232.
[4]	Cho, N. E., Kumar, S., Kumar, V., Ravichandran, V., Srivastava, H. M. Starlike functions related to the Bell numbers. Symmetry, 11 (2) (2019): 219; 17 pages. doi: 10.3390/sym11020219.
[5]	Dziok, J., Raina, R. K., Sok, J. (2013): On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers. Mathematical and Computer Modelling, 57 (56): 12031211.
[6]	Halim, S. A., & Omar, R. (2012): Applications of certain functions associated with lemniscte of Bernoulli. Journal of the Indonesian Mathematical Society, 18 (2): 9399.
[7]	Haq, M., Raza, M., Arif, M., Khan, Q., & Tang, H, (2019): Q-analogue of di⁄erential subordinations. Mathematics, 7 (8): 724: Doi: 10:3390=math7080724.
[8]	Hayman, W., H. (1994): Multivalent functions, Second Edition, Cambridge Uni. Press.
[9]	Jack, I. S. (1971): Functions starlike and convex of order alpha. Journal of the London Mathematical Society, 2 (3): 469474.
[10]	Janowski, W. (1970): Extremal problems for a family of functions with positive real part and for some related families. Annales Polonici Mathematici. 23 (2): 159-177.
[11]	Kanas, S., R…aducanu, D. (2014): Some class of analytic functions related to conic domains. Mathematica Slovaca, 64 (5): 11831196.
[12]	Kargar, R., Ebadian, A. & Sokół, J. On Booth lemniscate and starlike functions. Anal. Math. Phys. 9, 143–154 (2019).
[13]	Kargar, R., Ebadian, A. & Sokół, J. Radius Problems for Some Subclasses of Analytic Functions. Complex Anal. Oper. Theory 11, 1639–1649 (2017).
[14]	Kumar, S., Ravichandran, V., & Cho, N. E. (2013): Su¢ cient conditions for starlike functions associated with the lemniscate of Bernoulli. Journal of Inequalities and Applications. 2013 (1); Article Id. 176; 13 pages.
[15]	Kumar, S., Ravichandran, V. (2016): A subclass of starlike functions associated with a rational function. Southeast Asian Bulletin of Mathematics, 40 (2): 199-212.
[16]	Kumar, S., Ravichandran, V. (2018): Subordinations for functions with positive real part. Complex Analysis and Operator Theory. 12 (5): 11791191.
[17]	Ma, W., Minda, D. (1994) A unied treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis; Li, Z., Ren, F., Yang, L., Zhang, S. (Eds), Int. Press: Cambridge, MA, USA, 157169.
[18]	Mendiratta, R., Nagpal, S., Ravichandran, V. (2015): On a subclass of strongly starlike functions associated with exponential function. Bulletin of the Malaysian Mathematical Sciences Society, 38 (1): 365-386.
[19]	Paprocki, E., & Sok, J. (1996): The extremal problem in some subclass of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 20: 89-94.
[20]	Raina, R. K.; Sok, J. (2015) On coe¢ cient estimates for a certain class of starlike functions. Hacet. J. Math. Stat. 44: 14271433.
[21]	Raza, M., Sok, J., & Mushtaq, S. (2019): Di⁄erential subordination for analytic functions. Iranian Journal of science and technology, Transactions A: Science. 43 (3): 883890.
[22]	Sharma, K., Jain, N. K., Ravichandran, V. (2016): Starlike functions associated with a cardioid. Afrika Matematika, 27 (56): 923-939.
[23]	Sharma, K., Ravichandran, V. (2016): Application of subordination theory to starlike functions, Bull. Iran. Math. Soc. 42 (3): 761-777.
[24]	Shi, L., Ali, I., Arif, M., Cho, N. E., Hussain, S., & Khan, H. (2019): A study of third Hankel determinant problem for certain subfamilies of analytic functions involving cardioid domain. Mathematics, 7 (5): 418: Doi: 10:3390=math7050418.
[25]	Shi, L., Srivastava, H. M., Arif, M., Hussain, S., & Khan H. (2019): An investigation of the third Hankel determinant problem for certain subfamilies of univalent functions involving the exponential function. Symmetry, 11 (5); 598: Doi: 10:3390=sym11050598.
[26]	Sok, J., Stankiewicz, J. (1996) Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 19: 101-105.

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APA Style

Muhammad Azam, Naseer Ullah. (2021). Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function. International Journal of Applied Mathematics and Theoretical Physics, 7(3), 80-87. https://doi.org/10.11648/j.ijamtp.20210703.13

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ACS Style

Muhammad Azam; Naseer Ullah. Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function. Int. J. Appl. Math. Theor. Phys. 2021, 7(3), 80-87. doi: 10.11648/j.ijamtp.20210703.13

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AMA Style

Muhammad Azam, Naseer Ullah. Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function. Int J Appl Math Theor Phys. 2021;7(3):80-87. doi: 10.11648/j.ijamtp.20210703.13

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@article{10.11648/j.ijamtp.20210703.13,
  author = {Muhammad Azam and Naseer Ullah},
  title = {Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {7},
  number = {3},
  pages = {80-87},
  doi = {10.11648/j.ijamtp.20210703.13},
  url = {https://doi.org/10.11648/j.ijamtp.20210703.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210703.13},
  abstract = {Abdullah Alotaibi defined a starlike function connected with a cosine hyperbolic function in the year 2020. We establish some appropriate conditions for several features of multivalent analytic starlike function subordinated with cosine hyperbolic function in this article. We determine conditions on α are subordinated by Janowski function. We acquire some suitable conditions by selecting specific values for functions we get some adequate conditions for multivalent starlik function related with cosine hyperbolic. Over the last decade, starlike functions have grown in popularity in both literature and application. Our goal in this work is look at some practical challenges with q-starlike functions. Moreover, we will show that the class described in this research, as well as the results gained, generalizes numerous previously published papers. We need to add some fundamental Geometric function theory literature here to comprehend the notions employed in our work in a straightforward way. To do so, we'll start with the notation, which signifies the class of holomorphic or analytic functions in the holomorphic or analytic functions. Then the relationships must be stable. In addition, all univalent functions will belong to the subfamily. Furthermore, the possibility of subjections between analytic functions and, as shown by, as; the functions, are related by the connection of subjection, if there exists an analytic function with restrictions and such that in addition, if the function is in, we get The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domains and to study some of its useful properties of multivalent analytic functions subordinated cosine hyperbolic function.},
 year = {2021}
}

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TY  - JOUR
T1  - Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function
AU  - Muhammad Azam
AU  - Naseer Ullah
Y1  - 2021/09/11
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210703.13
DO  - 10.11648/j.ijamtp.20210703.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 80
EP  - 87
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210703.13
AB  - Abdullah Alotaibi defined a starlike function connected with a cosine hyperbolic function in the year 2020. We establish some appropriate conditions for several features of multivalent analytic starlike function subordinated with cosine hyperbolic function in this article. We determine conditions on α are subordinated by Janowski function. We acquire some suitable conditions by selecting specific values for functions we get some adequate conditions for multivalent starlik function related with cosine hyperbolic. Over the last decade, starlike functions have grown in popularity in both literature and application. Our goal in this work is look at some practical challenges with q-starlike functions. Moreover, we will show that the class described in this research, as well as the results gained, generalizes numerous previously published papers. We need to add some fundamental Geometric function theory literature here to comprehend the notions employed in our work in a straightforward way. To do so, we'll start with the notation, which signifies the class of holomorphic or analytic functions in the holomorphic or analytic functions. Then the relationships must be stable. In addition, all univalent functions will belong to the subfamily. Furthermore, the possibility of subjections between analytic functions and, as shown by, as; the functions, are related by the connection of subjection, if there exists an analytic function with restrictions and such that in addition, if the function is in, we get The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domains and to study some of its useful properties of multivalent analytic functions subordinated cosine hyperbolic function.
VL  - 7
IS  - 3
ER  -

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Author Information

Muhammad Azam

Faculty of Applied Science, School of Mathematics, Beijing University of Technology, Beijing, China
Naseer Ullah

Faculty of Applied Science, School of Mathematics, Beijing University of Technology, Beijing, China

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APA Style

Muhammad Azam, Naseer Ullah. (2021). Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function. International Journal of Applied Mathematics and Theoretical Physics, 7(3), 80-87. https://doi.org/10.11648/j.ijamtp.20210703.13

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ACS Style

Muhammad Azam; Naseer Ullah. Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function. Int. J. Appl. Math. Theor. Phys. 2021, 7(3), 80-87. doi: 10.11648/j.ijamtp.20210703.13

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AMA Style

Muhammad Azam, Naseer Ullah. Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function. Int J Appl Math Theor Phys. 2021;7(3):80-87. doi: 10.11648/j.ijamtp.20210703.13

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@article{10.11648/j.ijamtp.20210703.13,
  author = {Muhammad Azam and Naseer Ullah},
  title = {Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {7},
  number = {3},
  pages = {80-87},
  doi = {10.11648/j.ijamtp.20210703.13},
  url = {https://doi.org/10.11648/j.ijamtp.20210703.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210703.13},
  abstract = {Abdullah Alotaibi defined a starlike function connected with a cosine hyperbolic function in the year 2020. We establish some appropriate conditions for several features of multivalent analytic starlike function subordinated with cosine hyperbolic function in this article. We determine conditions on α are subordinated by Janowski function. We acquire some suitable conditions by selecting specific values for functions we get some adequate conditions for multivalent starlik function related with cosine hyperbolic. Over the last decade, starlike functions have grown in popularity in both literature and application. Our goal in this work is look at some practical challenges with q-starlike functions. Moreover, we will show that the class described in this research, as well as the results gained, generalizes numerous previously published papers. We need to add some fundamental Geometric function theory literature here to comprehend the notions employed in our work in a straightforward way. To do so, we'll start with the notation, which signifies the class of holomorphic or analytic functions in the holomorphic or analytic functions. Then the relationships must be stable. In addition, all univalent functions will belong to the subfamily. Furthermore, the possibility of subjections between analytic functions and, as shown by, as; the functions, are related by the connection of subjection, if there exists an analytic function with restrictions and such that in addition, if the function is in, we get The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domains and to study some of its useful properties of multivalent analytic functions subordinated cosine hyperbolic function.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Some Properties of Multivalent Analytic Starlike Function Subordinated with Cosine Hyperbolic Function
AU  - Muhammad Azam
AU  - Naseer Ullah
Y1  - 2021/09/11
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210703.13
DO  - 10.11648/j.ijamtp.20210703.13
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 80
EP  - 87
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210703.13
AB  - Abdullah Alotaibi defined a starlike function connected with a cosine hyperbolic function in the year 2020. We establish some appropriate conditions for several features of multivalent analytic starlike function subordinated with cosine hyperbolic function in this article. We determine conditions on α are subordinated by Janowski function. We acquire some suitable conditions by selecting specific values for functions we get some adequate conditions for multivalent starlik function related with cosine hyperbolic. Over the last decade, starlike functions have grown in popularity in both literature and application. Our goal in this work is look at some practical challenges with q-starlike functions. Moreover, we will show that the class described in this research, as well as the results gained, generalizes numerous previously published papers. We need to add some fundamental Geometric function theory literature here to comprehend the notions employed in our work in a straightforward way. To do so, we'll start with the notation, which signifies the class of holomorphic or analytic functions in the holomorphic or analytic functions. Then the relationships must be stable. In addition, all univalent functions will belong to the subfamily. Furthermore, the possibility of subjections between analytic functions and, as shown by, as; the functions, are related by the connection of subjection, if there exists an analytic function with restrictions and such that in addition, if the function is in, we get The aim of this paper is to define a family of multivalent q-starlike functions associated with circular domains and to study some of its useful properties of multivalent analytic functions subordinated cosine hyperbolic function.
VL  - 7
IS  - 3
ER  -

Copy | Download