New properties of the generalized Dini function

Year 2020, , 1753 - 1760, 06.10.2020

Ebrahim Analouei Adegani , Teodor Bulboaca

https://doi.org/10.15672/hujms.552260

Cited By: 1

Abstract

In this work we study some properties of the normalized form of generalized Dini function like close-to-convexity of some order and close-to-convex with respect to another convex function. Furthermore, we investigate sufficient conditions which these functions are uniformly $k$-starlike functions of complex order $b$ in the open unit disk, and some consequences of the main results are also presented.

****************************************************************************************************************************

Keywords

Analytic functions, univalent function, Bessel functions of first kind of order $\nu$, modified Dini function, starlike, convex and close-to-convex functions of order $\alpha$

References

• [1] D. Bansal, M.K. Soni and A. Soni, Certain geometric properties of the modified Dini function, Anal. Math. Phys. 9, 1383–1392, 2019.
• [2] Á. Baricz, E. Deniz and N. Yagmur, Close-to-convexity of normalized Dini functions, Math. Nachr. 289, 1721–1726, 2016.
• [3] Á. Baricz, P.A. Kupán and R. Szász, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc. 142 (6), 2019–2025, 2014.
• [4] Á. Baricz, S. Ponnusamy and S. Singh, Modified Dini functions: monotonicity pat- terns and functional inequalities, Acta Math. Hungar. 149 (1), 120–142, 2016.
• [5] Á. Baricz, E. Toklu and E. Kadioğlu, Radii of starlikeness and convexity of Wright functions, Math. Commun. 23 (1), 97–117, 2018.
• [6] S.Z.H. Bukhari, J. Sokól and S. Zafar, Unified approach to starlike and convex func- tions involving convolution between analytic functions, Results Math. 73, Article num- ber: 30, 2018.
• [7] M.U. Din, M. Raza, S. Hussain and M. Darus, Certain geometric properties of gen- eralized Dini functions, J. Funct. Spaces, 2018, Art. ID 2684023, 1–9, 2018.
• [8] A.W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56, 87–92, 1991.
• [9] A.W. Goodman, On uniformly starlike functions, Ann. Polon. Math. 155, 364–370, 1991.
• [10] S. Kanas and A. Wiśniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105, 327–336, 1999.
• [11] S. Kanas and A. Wiśniowska, Conic regions and k-starlike functions, Rev. Roumaine Math. Pures Appl. 45 (4), 647–657, 2000.
• [12] J.K. Prajapat, Certain geometric properties of the Wright functions, Integral Trans- forms Spec. Funct. 26 (3), 203–212, 2015.
• [13] D. Răducanu, Geometric properties of Mittag-Leffler functions, in: Models and The- ories in Social Systems, Studies in Systems, Decision and Control 79, 403–415, Springer, Cham, 2018.
• [14] F. Rönning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 45, 117–122, 1991.
• [15] F. Rönning, Uniformly convex functions and a corresponding class of starlike func- tions. Proc. Amer. Math. Soc. 1 (18), 189–196, 1993.
• [16] S. Owa, M. Nunokawa, H. Saitoh and H.M. Srivastava, Close-to-convexity, starlike- ness and convexity of certain analytic functions, Appl. Math. Lett. 15, 63–69, 2002.
• [17] S.K. Sahoo and N.L. Sharma, On a generalization of close-to-convex functions, Ann. Polon. Math. 113 (1), 93–108, 2015.