Received July 29, 2020. Published online September 23, 2021.
Abstract: The aim of this paper is to find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in the classes $\mathcal{SP}_p(\alpha,\beta)$ and $\mathcal{UCV}_p(\alpha,\beta)$ of uniformly spirallike functions. Further, we consider an integral operator related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.
Keywords: analytic function; Hadamard product; uniformly spirallike function; Pascal distribution series
References: [1] T. Al-Hawary, B. A. Frasin: Uniformly analytic functions with varying arguments. An. Univ. Oradea, Fasc. Mat. 23 (2016), 37-44. MR 3496011 | Zbl 1349.30023
[2] N. E. Cho, S. Y. Woo, S. Owa: Uniform convexity properties for hypergeometric functions. Fract. Calc. Appl. Anal. 5 (2002), 303-313. MR 1922272 | Zbl 1028.30013
[3] K. K. Dixit, S. K. Pal: On a class of univalent functions related to complex order. Indian J. Pure Appl. Math. 26 (1995), 889-896. MR 1347537 | Zbl 0835.30006
[4] R. M. El-Ashwah, W. Y. Kota: Some condition on a Poisson distribution series to be in subclasses of univalent functions. Acta Univ. Apulensis, Math. Inform. 51 (2017), 89-103. DOI 10.17114/j.aua.2017.51.08 | MR 3711116 | Zbl 1413.30053
[5] S. M. El-Deeb, T. Bulboacă: Differential sandwich-type results for symmetric functions connected with a $q$-analog integral operator. Mathematics 7 (2019), 17 pages, Article ID 1185. DOI 10.3390/math7121185
[6] S. M. El-Deeb, T. Bulboacă, J. Dziok: Pascal distribution series connected with certain subclasses of univalent functions. Kyungpook Math. J. 59 (2019), 301-314. DOI 10.5666/KMJ.2019.59.2.301 | MR 3987710 | Zbl 1435.30045
[7] B. A. Frasin: Comprehensive family of uniformly analytic functions. Tamkang J. Math. 36 (2005), 243-254. DOI 10.5556/j.tkjm.36.2005.117 | MR 2182243 | Zbl 1086.30012
[8] B. A. Frasin: Quasi-Hadamard product of certain classes of uniformly analytic functions. Gen. Math. 16 (2008), 29-35. MR 2439231 | Zbl 1240.30047
[9] B. A. Frasin: On certain subclasses of analytic functions associated with Poisson distribution series. Acta Univ. Sapientiae, Math. 11 (2019), 78-86. DOI 10.2478/ausm-2019-0007 | MR 3995739 | Zbl 1432.30010
[10] B. A. Frasin: Two subclasses of analytic functions associated with Poisson distribution. Turkish J. Ineq. 4 (2020), 25-30.
[11] B. A. Frasin, I. Aldawish: On subclasses of uniformly spiral-like functions associated with generalized Bessel functions. J. Funct. Spaces 2019 (2019), Article ID 1329462, 6 pages. DOI 10.1155/2019/1329462 | MR 3998948 | Zbl 1431.30010
[12] B. A. Frasin, T. Al-Hawary, F. Yousef: Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions. Afr. Mat. 30 (2019), 223-230. DOI 10.1007/s13370-018-0638-5 | MR 3927324 | Zbl 1438.30042
[13] B. A. Frasin, S. M. Aydoğan: Pascal distribution series for subclasses of analytic functions associated with conic domains. Afr. Mat. 32 (2021), 105-113. DOI 10.1007/s13370-020-00813-1 | MR 4222458 | Zbl 07363570
[14] B. A. Frasin, M. M. Gharaibeh: Subclass of analytic functions associated with Poisson distribution series. Afr. Mat. 31 (2020), 1167-1173. DOI 10.1007/s13370-020-00788-z | MR 4154545 | Zbl 1463.30053
[15] A. W. Goodman: On uniformly convex functions. Ann. Pol. Math. 56 (1991), 87-92. DOI 10.4064/ap-56-1-87-92 | MR 1145573 | Zbl 0744.30010
[16] A. W. Goodman: On uniformly starlike functions. J. Math. Anal. Appl. 155 (1991), 364-370. DOI 10.1016/0022-247X(91)90006-L | MR 1097287 | Zbl 0726.30013
[17] S. Kanas, A. Wiśniowska: Conic regions and $k$-uniform convexity. J. Comput. Appl. Math. 105 (1999), 327-336. DOI 10.1016/S0377-0427(99)00018-7 | MR 1690599 | Zbl 0944.30008
[18] S. Kanas, A. Wiśniowska: Conic domains and starlike functions. Rev. Roum. Math. Pures Appl. 45 (2000), 647-657. MR 1836295 | Zbl 0990.30010
[19] E. P. Merkes, W. T. Scott: Starlike hypergeometric functions. Proc. Am. Math. Soc. 12 (1961), 885-888. DOI 10.1090/S0002-9939-1961-0143950-1 | MR 143950 | Zbl 0102.06502
[20] G. Murugusundaramoorthy: Subclasses of starlike and convex functions involving Poisson distribution series. Afr. Mat. 28 (2017), 1357-1366. DOI 10.1007/s13370-017-0520-x | MR 3714601 | Zbl 1381.30015
[21] G. Murugusundaramoorthy, K. Vijaya, S. Porwal: Some inclusion results of certain subclass of analytic functions associated with Poisson distribution series. Hacet. J. Math. Stat. 45 (2016), 1101-1107. DOI 10.15672/HJMS.20164513110 | MR 3585439 | Zbl 1359.30022
[22] S. Porwal: Mapping properties of generalized Bessel functions on some subclasses of univalent functions. An. Univ. Oradea, Fasc. Mat. 20 (2013), 51-60. MR 3154762 | Zbl 1313.30070
[23] S. Porwal: An application of a Poisson distribution series on certain analytic functions. J. Complex Anal. 2014 (2014), Article ID 984135, 3 pages. DOI 10.1155/2014/984135 | MR 3173344 | Zbl 1310.30017
[24] S. Porwal, M. Kumar: A unified study on starlike and convex functions associated with Poisson distribution series. Afr. Mat. 27 (2016), 1021-1027. DOI 10.1007/s13370-016-0398-z | MR 3536083 | Zbl 1352.30014
[25] V. Ravichandran, C. Selvaraj, R. Rajagopal: On uniformly convex spiral functions and uniformly spirallike functions. Soochow J. Math. 29 (2003), 392-405. MR 2021539 | Zbl 1047.30007
[26] F. Rønning: On starlike functions associated with parabolic regions. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 45 (1991), 117-122. MR 1322145 | Zbl 0769.30011
[27] F. Rønning: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc. 118 (1993), 189-196. DOI 10.1090/S0002-9939-1993-1128729-7 | MR 1128729 | Zbl 0805.30012
[28] C. Selvaraj, R. Geetha: On subclasses of uniformly convex spirallike functions and corresponding class of spirallike functions. Int. J. Contemp. Math. Sci. 5 (2010), 1845-1854. MR 2733653 | Zbl 1218.30049
[29] H. Silverman: Starlike and convexity properties for hypergeometric functions. J. Math. Anal. Appl. 172 (1993), 574-581. DOI 10.1006/jmaa.1993.1044 | MR 1201006 | Zbl 0774.30015
[30] H. M. Srivastava, G. Murugusundaramoorthy, S. Sivasubramanian: Hypergeometric functions in the parabolic starlike and uniformly convex domains. Integral Transforms Spec. Funct. 18 (2007), 511-520. DOI 10.1080/10652460701391324 | MR 2348596 | Zbl 1198.30013
Affiliations: Gangadharan Murugusundaramoorthy, School of Advanced Sciences, Vellore Institute of Technology (Deemed to be University), Vellore 632014, Tamil Nadu, India, e-mail: gmsmoorthy@yahoo.com; Basem Aref Frasin, Faculty of Science, Department of Mathematics, Al al-Bayt University, Mafraq 25113, Jordan, e-mail: bafrasin@yahoo.com; Tariq Al-Hawary, Department of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan, e-mail: tariq_amh@bau.edu.jo
