Radius of starlikeness through subordination

Authors

DOI:

https://doi.org/10.24193/subbmath.2023.1.12

Keywords:

univalent functions, convex functions, starlike functions, subordination, radius of starlikeness

Abstract

A normalized function \(f\) on the open unit disc is starlike (or convex) univalent if the associated function \(zf'(z)/f(z)\)  (or \(1+zf''(z)/f'(z)\)) is a function with positive real part. The radius of starlikeness or convexity is usually obtained by using the estimates for functions with positive real part. Using subordination, we examine  the radius of various starlikeness, in particular, radii of Janowski starlikeness and  starlikeness of order beta, for the function f when the function f is either convex or  \((zf'(z)+\alpha z^2f''(z))/f(z)\) lies in the right-half plane. Radii of starlikeness associated with lemniscate of Bernoulli and exponential functions are also considered.

Author Biography

  • Vaithiyanathan Ravichandran, NATIONAL INSTITUTE OF TECHNOLOGY TIRUCHIRAPPALLI

    Dr. V. Ravichandran joined as Professor of Mathematics in April 2018 at NIT Trichy. He has studied at National College, Trichy for his B.Sc., M.Sc. and M.Phil. degrees and completed his Ph.D. degree from Anna University, Chennai. He started his career as Lecturer in 1996 and spent the period between 2004-2007 at Universiti Sains Malaysia where he was a Visiting Professor later in 2011-12. He joined as Reader at University of Delhi in 2007 and appointed there as Professor in 2014. He was HoD, Department of Mathematics, DU from 21.12.2015 - 12.04.2018. He has published more than 150 research papers; more than 50 of these papers are with Prof. Dato' Indera Rosihan M. Ali of Universiti Sains Malaysia. He has supervised 15 Ph.D. and 9 M.Phil. scholars. He serves as an editor of Bulletin of the Malaysian Mathematical Sciences Society since 2004.

    He was a member for op-option of the Programme Advisory Committee (PAC): Mathematical Sciences of DST-SERB. He was earlier the Chairman, Governing Body, Shyama Prasad Mukherji College for Women, University of Delhi from June 2017 to April 2018. Being HoD Mathematics, DU, he been part of University Court, Academic Council, Board of Studies in Mathematical Sciences at DU. He was also member of Business Advisory Committee of the Academic Council, DU and Technical Advisory Committee, University Computer Center. He was also a member of Joint Consultative Group for establishing a Centre for Social Applications of Mathematics, Ambedkar University, Delhi.

    He held Adjunct Professorship at Shri Mata Vaishno Devi University between August 2016-August 2018. He has been a Visiting Professor at Division of Mathematical Sciences, Pukyong National University, Busan, South Korea and at School of Mathematical Sciences, Universiti Sains Malaysia.

    He was also reviewer for Mathematical Review published American Mathematical Society and for Zentralblatt Math published by European Mathematical Society. He has reviewed around 100 papers for the later. He regularly acts as a referee for more than 50 reputed journals.

    He has got several research grants as PI or CoPI. Several of his papers won merit rewards from USM. For three consecutive years from 2008 - 2010, his papers won the Best Research Paper Award in the Category of Science, Technology and Medicine awarded by the Malaysian Council of Scholarly Publications (Majlis Penerbitan Ilmiah Malaysia MAPIM).

    As he is not keen on travelling, his participation in conferences were limited. However, he has organized few conferences/workshops/refresher courses. Notable among them are the two conferences he has organized in honor of the two retiring (now retired) faculty members at DU.

    He has been a popular speaker at various mathematics festivals/events at University of Delhi. Being a user of LaTeX from 1993, he is always happy to train anyone this wonderful typesetting software.

References

R. M. Ali, N. E. Cho, N. K. Jain, and V. Ravichandran, Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination, Filomat, 26 (2012), pp. 553–561.

R. M. Ali, N. K. Jain, and V. Ravichandran, Radii of starlikeness associated with the lemniscate of bernoulli and the left-half plane, Appl. Math. Comput., 218 (2012), pp. 6557–6565.

M. K. Aouf, J. Dziok, and J. Sokó l, On a subclass of strongly starlike functions, Appl. Math. Lett., 24 (2011), pp. 27–32.

L. de Branges, A proof of the Bieberbach conjecture, Acta Math., 154 (1985), pp. 137–152.

A. Gangadharan, V. Ravichandran, and T. N. Shanmugam, Radii of convexity and strong starlikeness for some classes of analytic functions, J. Math. Anal. Appl., 211 (1997), pp. 301–313.

K. Khatter, V. Ravichandran, and S. Sivaprasad Kumar, Starlike functions associated with exponential function and the lemniscate of Bernoulli, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM, 113 (2019), pp. 233–253.

Z. a. Lewandowski, S. Miller, and E. Z l otkiewicz, Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc., 56 (1976), pp. 111–117.

J.-L. Li and S. Owa, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math., 33 (2002), pp. 313–318.

E. Lindelöf, M´ emoire sur certaines in´ egalit´ es dans la th´ eorie des fonctions monog` enes et sur quelques

propri´ et´ es nouvelles de ces fonctions dans le voisinage d’un point singulier essentiel, par Ernst Lindelöf..., la Soci´ et´ e de litt´ erature finnoise, 1908.

M.-S. Liu, Y.-C. Zhu, and H. M. Srivastava, Properties and characteristics of certain subclasses of starlike functions of order β, Math. Comput. Modelling, 48 (2008), pp. 402–419.

W. C. Ma and D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conf. Proc. Lecture Notes Anal., I, Int. Press, Cambridge, MA, 1994, pp. 157–169.

R. Mendiratta, S. Nagpal, and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc., 38 (2015), pp. 365–386.

S. S. Miller and P. T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J., 28 (1981), pp. 157–172.

A. Naz, S. Nagpal, and V. Ravichandran, Star-likeness associated with the exponential function, Turkish J. Math., 43 (2019), pp. 1353–1371.

M. Nunokawa, On a sufficient condition for multivalently starlikeness, Tsukuba J. Math., 18 (1994), pp. 131–134.

M. Obradovi´ c and S. B. Joshi, On certain classes of strongly starlike functions, Taiwanese J. Math., 2 (1998), pp. 297–302.

K. S. Padmanabhan, On sufficient conditions for starlikeness, Indian J. Pure Appl. Math., 32 (2001), pp. 543–550.

E. Paprocki and J. Sokó l, The extremal problems in some subclass of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat., (1996), pp. 89–94.

C. Ramesha, S. Kumar, and K. S. Padmanabhan, A sufficient condition for starlikeness, Chinese J. Math., 23 (1995), pp. 167–171.

V. Ravichandran, Certain applications of first order differential subordination, Far East J. Math. Sci. (FJMS), 12 (2004), pp. 41–51.

V. Ravichandran, M. Hussain Khan, H. Silverman, and K. G. Subramanian, Radius problems for a class of analytic functions, Demonstratio Math., 39 (2006), pp. 67–74.

V. Ravichandran, F. Rø nning, and T. N. Shanmugam, Radius of convexity and radius of starlikeness for some classes of analytic functions, Complex Variables Theory Appl., 33 (1997), pp. 265–280.

V. Ravichandran, C. Selvaraj, and R. Rajalaksmi, Sufficient conditions for starlike functions of order α, JIPAM. J. Inequal. Pure Appl. Math., 3 (2002), pp. Article 81, 6.

M. I. S. Robertson, On the theory of univalent functions, Ann. of Math. (2), 37 (1936), pp. 374–408.

S. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv., 48 (1973), pp. 119–135.

T. N. Shanmugam, Convolution and differential subordination, Internat. J. Math. Math. Sci., 12 (1989), pp. 333–340.

S. Singh and S. Gupta, A differential subordination and starlikeness of analytic functions, Appl. Math. Lett., 19 (2006), pp. 618–627.

J. Sokó land J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat., (1996), pp. 101–105.

Downloads

Published

2023-03-17

Issue

Vol. 68 No. 1 (2023)

Section

Articles
Crossref
Scopus
Google Scholar
Europe PMC