# Some properties of analytical functions related to Borel distribution series

Volume 26, Issue 4, pp 395--404
Publication Date: January 14, 2022 Submission Date: November 10, 2021 Revision Date: November 23, 2021 Accteptance Date: December 10, 2021
• 441 Views

### Authors

H. Niranjan - Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore--632014, Tamilnadu, India. A. N. Murthy - Department of Mathematics, A. V. V. Junior College, Warangal--506 002, Telangana, India. P. T. Reddy - Department of Mathematics, School of Engineering, NNRESGI, Medichal--500088, Telangana, India.

### Abstract

In this paper, we introduce and study a new subclass of analytic functions which are defined by means of a linear operator. Some results connected to coefficient estimates, growth and distortion theorems, radii of starlikeness, convexity close-to-convexity and integral means inequalities related to the subclass is obtained.

### Share and Cite

##### ISRP Style

H. Niranjan, A. N. Murthy, P. T. Reddy, Some properties of analytical functions related to Borel distribution series, Journal of Mathematics and Computer Science, 26 (2022), no. 4, 395--404

##### AMA Style

Niranjan H., Murthy A. N., Reddy P. T., Some properties of analytical functions related to Borel distribution series. J Math Comput SCI-JM. (2022); 26(4):395--404

##### Chicago/Turabian Style

Niranjan, H., Murthy, A. N., Reddy, P. T.. "Some properties of analytical functions related to Borel distribution series." Journal of Mathematics and Computer Science, 26, no. 4 (2022): 395--404

### Keywords

• Analytic
• starlike
• coefficient bounds
• convexity

•  30C45
•  30C50

### References

• [1] N. Alessa, B. Venkateswarlu, P. Thirupathi Reddy, K. Loganathan, K. Tamilvanan, A New subclass of analytic functions related to Mittag-Leffler type Poisson distribution series, J. Funct. Spaces, 2021 (2021), 7 pages

• [2] A. A. Attiya, Some applications of Mittag-Leffler function in the unit disk, Filomat, 30 (2016), 2075--2081

• [3] D. Bansal, J. K. Prajapat, Certain geometric properties of the Mittag-Leffler functions, Complex Var. Elliptic Equ., 61 (2016), 338--350

• [4] S. M. El-Deeb, G. Murugusundaramoorthy, A. Alburaikan, Bi-Bazilevic functions based on the Mittag-Leffler-type Borel distribution associated with Legendre polynomials, J. Math. Computer Sci., 24 (2022), 235--245

• [5] B. A. Frasin, An application of an operator associated with generalized Mittag-Leffler function, Konuralp J. Math., 7 (2019), 199--202

• [6] B. A. Frasin, T. Al-Hawary, F. Yousef, Some properties of a linear operator involving generalized Mittag-Leffler function, Stud. Univ. Babes-Bolyai Math., 65 (2020), 67--75

• [7] M. Grag, P. Manohar, S. L. Kalla, A Mittag-Leffler-type function of two variables, Integral Transforms Spec. Funct., 24 (2013), 934--944

• [8] J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. (2), 23 (1925), 481--519

• [9] G. Mittag-Leffler, Sur la Nouvelle Fonction $E_{\alpha}(x)$, Comptes Rendus de l'Academie des Sciences Paris, 137 (1903), 554--558

• [10] G. Murugusundaramoorthy, S. M. El-Deeb, Second Hankel determinant for a class of analytic functions of the Mittag-Leffler-type Borel distribution related with Legendre polynomials, Turkish World Math. Soc. J. Appl. Eng. Math., accepted for publications (2021)

• [11] S. M. Popade, R. N. Ingle, P. T. Reddy, On a certain subclass of analytic functions defined by a differential operator, Malaya J. Mat., 8 (2020), 576--580

• [12] K. A. Selvakumaran, H. A. Al-Kharsani, D. Baleanu, S. D. Purohi, K. S. Nisar, Inclusion relationships for some subclasses of analytic functions associated with generalized Bessel functions, J. Comput. Anal. Appl., 24 (2018), 81--90

• [13] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), 109--116

• [14] H. Silverman, A survey with open problems on univalent functions whose coefficient are negative, Rocky Mountain J. Math., 21 (1991), 1099--1125

• [15] H. Silvermani, Integral means for univalent functions with negative coefficient, Houston J. Math., 23 (1997), 169--174

• [16] H. M. Srivastava, G. Murugusundaramoorthy, S. M. El-Deeb, Faber polynomial coefficient estimates of bi-close-to-convex functions connected with the Borel distribution of the Mittag-leffler type, J. Nonlinear Var. Anal., 5 (2021), 103--118

• [17] A. K. Wanas, J. A. Khuttar, Applications of Borel distribution series on analytic functions, Earthline J. Math. Sci., 4 (2020), 71--82

• [18] A. Wiman, Über die Nullstellen der Funktionen $E_a(x)$, Acta Math., 29 (1905), 217--234