Research article

Some results for the family of univalent functions related with Limaçon domain

  • Received: 07 October 2020 Accepted: 30 November 2020 Published: 19 January 2021
  • MSC : 30C45, 30C50, 30C55

  • The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

    Citation: Afis Saliu, Khalida Inayat Noor, Saqib Hussain, Maslina Darus. Some results for the family of univalent functions related with Limaçon domain[J]. AIMS Mathematics, 2021, 6(4): 3410-3431. doi: 10.3934/math.2021204

    Related Papers:

  • The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.


    加载中


    [1] R. M. Ali, V. Ravichandran, N. Seenivasagan, Coefficient bounds for $p$-valent functions, Appl. Math. Comput., 187 (2007), 35–46.
    [2] K. O. Babalola, On $H_{3}(1)$ Hankel determinant for some classes of univalent functions, Inequality Theory and Applications, 6 (2010), 1–7.
    [3] N. E. Cho, V. Kumar, S. S. Kumar, V. Ravichandran, Radius problems for starlike functions associated with the sine function, B. Iran. Math. Soc., 45 (2019), 213–232. doi: 10.1007/s41980-018-0127-5
    [4] M. Darus, I. Faisal, Some subclasses of analytic functions of complex order defined by new differential operator, Tamkang J. Math., 43 (2012), 223–242. doi: 10.5556/j.tkjm.43.2012.740
    [5] M. Darus, S. Hussain, M. Raza, J. Sokół, On a subclass of starlike functions, Results Math., 73 (2018), 22. doi: 10.1007/s00025-018-0771-3
    [6] R. M. El-Ashwash, M. K. Aouf, M. Darus, Differential subordination results for analytic functions, Asian-Eur. J. Math., 6 (2013), 1350044. doi: 10.1142/S1793557113500447
    [7] A. W. Goodman, Univalent functions, Vols. I and II, Polygonal Publishing House, Washinton, New Jersey, 1983.
    [8] A. Goodman, On uniformly convex functions, Ann. Polon. Math., 56 (1991), 87–92. doi: 10.4064/ap-56-1-87-92
    [9] S. Hussain, A. Rasheed, M. A. Zaighum, M. Darus, A Subclass of Analytic Functions Related to $k$-Uniformly Convex and Starlike Functions, J. Funct. Spaces, 2017 (2017), 9010964.
    [10] S. Hussain, S. Khan, M. A. Zaighum, M. Darus, Certain subclass of analytic functions related with conic domains and associated with Salagean q-differential operator, AIMS Math., 2 (2017), 622–634. doi: 10.3934/Math.2017.4.622
    [11] I. S. Jack, Functions starlike and convex of order $\alpha$, J. Lond. Math. Soc., 2 (1971), 469–474.
    [12] W. Janowski, Some extremal problems for certain families of analytic functions I, Ann. Polon. Math., 28 (1973), 297–326. doi: 10.4064/ap-28-3-297-326
    [13] S. Kanas, A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105 (1999), 327–336. doi: 10.1016/S0377-0427(99)00018-7
    [14] S. Kanas, A. Wisniowska, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl., 45 (2000), 647–658.
    [15] S. Kanas, V. S. Masih, A. Ebadian, Relations of a planar domains bounded by hyperbolas with families of holomorphic functions, J. Inequal. Appl., 2019 (2019), 246. doi: 10.1186/s13660-019-2190-8
    [16] S. Kanas, V. S. Masih, A. Ebadian, Coefficients problems for families of holomorphic functions related to hyperbola, $\overset\frown{\text{M}}$ath. Slovaca, 70 (2020), 605–616.
    [17] W. Ma, D. Minda, A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 157–169, Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA.
    [18] V. S. Masih, S. Kanas, Subclasses of Starlike and Convex Functions Associated with the Limaçon Domain, Symmetry, 12 (2020), 942. doi: 10.3390/sym12060942
    [19] S. S. Miller, P. T. Mocanu, Differential Subordinations, Theory and Applications, Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 225, Marcel Dekker Inc., New York/Basel, 2000.
    [20] G. Murugusundaramoorthy, T. Bulboaca, Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region, Mathematics, 8 (2020), 1041. doi: 10.3390/math8061041
    [21] Z. Nehari, Conformal mapping, McGraw-Hill, New York, 1952.
    [22] J. W. Noonan, D. K. Thomas, On the Hankel determinants of areally mean p-valent functions, Proc. Lond. Math. Soc., 3 (1972), 503–524.
    [23] K. I. Noor, S. N. Malik, On coefficient inequalities of functions associated with conic domains, Comput. Math. Appl., 62 (2011), 2209–2217. doi: 10.1016/j.camwa.2011.07.006
    [24] R. K. Raina, J. Sokól, On coefficient estimates for a certain class of starlike functions, Hacet. J. Math. Stat., 44 (2015), 1427–1433.
    [25] R. K. Raina, J. Sokól, On a class of analytic functions governed by subordination, Acta Univ. Sapientiae Math., 11 (2019), 144–155.
    [26] A. Rasheed, S. Hussain, M. A. Zaighum, M. Darus, Class of analytic function related with uniformly convex and Janowski's functions, J. Funct. Spaces, 2018 (2018), 4679857.
    [27] V. Ravichandran, F. Rønning, T. N. Shanmugam, Radius of convexity and radius of starlikeness for some classes of analytic functions, Complex Var. Elliptic Equ., 33 (1997), 265–280.
    [28] M. Raza, K. I. Noor, Subclass of Bazilevic functions of complex order, AIMS Math., 5 (2020), 2448–2460. doi: 10.3934/math.2020162
    [29] W. Rogosinski, On the coefficients of subordinate functions, Proc. Lond. Math. Soc., 2 (1945), 48–82.
    [30] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196. doi: 10.1090/S0002-9939-1993-1128729-7
    [31] A. Saliu, On Generalized k-Uniformly Close-to-Convex Functions of Janowski Type, International Journal of Analysis and Applications, 17 (2019), 958–973.
    [32] A. Saliu, K. I. Noor, On Janowski Close-to-Convex Functions Associated with Conic Regions, International Journal of Analysis and Applications, 18 (2020), 614–623.
    [33] A. Saliu, K. I. Noor, S. Hussain, M. Darus, On Quantum Differential Subordination Related with Certain Family of Analytic Functions, J. Math., 2020 (2020), 6675732.
    [34] A. Saliu, K. I. Noor, On subclasses of functions with boundary and radius rotations associated with crescent domains, Bull. Korean Math. Soc., 57 (2020), 1529–1539.
    [35] P. Sharma, R. K. Raina, J. Sokól, Certain Ma-Minda type classes of analytic functions associated with the crescent-shaped region, Anal. Math. Phys., 9 (2019), 1887–1903. doi: 10.1007/s13324-019-00285-y
    [36] S. Siregar, M. Darus, Certain conditions for starlikeness of analytic functions of Koebe type, Int. J. Math. Math. Sci., 2011 (2011), 679704.
    [37] H. M. Srivastava, K. Bilal, K. Nazar, Q. Z. Ahmad, Coefficient inequalities for $ q $-starlike functions associated with the Janowski functions, Hokkaido Math. J., 48 (2019), 407–425. doi: 10.14492/hokmj/1562810517
    [38] N. Tuneski, M. Darus, E. Gelova, Simple sufficient conditions for bounded turning, Rend. Semin. Mat. Univ. Padova, 132 (2014), 231–238. doi: 10.4171/RSMUP/132-11
    [39] Y. Yunus, S. A. Halim, A. B. Akbarally, Subclass of starlike functions associated with a limaçon, AIP Conference Proceedings, 1974 (2018), 030023. doi: 10.1063/1.5041667
  • Reader Comments
  • © 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3382) PDF downloads(299) Cited by(10)

Article outline

Figures and Tables

Figures(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog