Radius Estimates for Three Leaf Function and Convex Combination of Starlike Functions

  • Shweta GandhiEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 306)


We study radii problems for the class \(\mathcal {S}^*_{3 \mathcal {L}}\) consisting of normalized analytic functions f in the unit disk with \(zf'(z)/f(z)\) subordinate to \(1+ 4z/5+ z^4/5\) and the class associated with convex combination of linear and exponential functions.


Starlike functions Coefficient bounds Growth Radius problems 

2010 Mathematics Subject Classification

30C45 30C50 30C80 


  1. 1.
    R.M. Ali, V. Ravichandran, Uniformly convex and uniformly starlike functions. Math. Newslett. 21(1), 16–30 (2011)Google Scholar
  2. 2.
    R.M. Ali, V. Ravichandran, N. Seenivasagan, Coefficient bounds for p-valent functions. Appl. Math. Comput. 187(1), 35–46 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    R.M. Ali, N.E. Cho, N.K. Jain, V. Ravichandran, Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination. Filomat 26(3), 553–561 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    R.M. Ali, N.K. Jain, V. Ravichandran, Radii of starlikeness associated with the lemniscate of Bernoulli and the left-half plane. Appl. Math. Comput. 218(11), 6557–6565 (2012)MathSciNetzbMATHGoogle Scholar
  5. 5.
    R.M. Ali, N.K. Jain, V. Ravichandran, On the radius constants for classes of analytic functions. Bull. Malays. Math. Sci. Soc. (2) 36(1), 23–38 (2013)Google Scholar
  6. 6.
    D.M. Campbell, A survey of properties of the convex combination of univalent functions. Rocky Mt. J. Math. 5(4), 475–492 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    N.E. Cho, Some new criteria for p-valent meromorphic starlike functions. Nihonkai Math. J. 4(2), 125–132 (1993)Google Scholar
  8. 8.
    N.E. Cho et al., Radius problems for starlike functions associated with the sine function. Bull. Iran. Math. Soc. 45(1), 213–232 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    P.L. Duren, Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259 (Springer, New York, 1983)Google Scholar
  10. 10.
    S. Gandhi, V. Ravichandran, Starlike functions associated with a lune. Asian-Eur. J. Math. 10(4), 1750064 (2017), 12 ppMathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    A.W. Goodman, Univalent Functions, vol. 1–2 (Mariner, Tampa, 1983)Google Scholar
  12. 12.
    W. Janowski, Extremal problems for a family of functions with positive real part and for some related families. Ann. Pol. Math. 23, 159–177 (1970/1971)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    W. Janowski, Some extremal problems for certain families of analytic functions. I. Ann. Pol. Math. 28, 297–326 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    K. Khatter, V. Ravichandran, S. Sivaprasad Kumar, Starlike functions associated with exponential function and the lemniscate of Bernoulli. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(1), 233–253 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    W.C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in Proceedings of the Conference on Complex Analysis (Tianjin, 1992), Conference Proceedings and Lecture Notes in Analysis. I (International Press, Cambridge, 1992), pp. 157–169Google Scholar
  16. 16.
    T.H. MacGregor, Functions whose derivative has a positive real part. Trans. Am. Math. Soc. 104, 532–537 (1962)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    T.H. MacGregor, The radius of univalence of certain analytic functions. Proc. Am. Math. Soc. 14, 514–520 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    T.H. MacGregor, The radius of univalence of certain analytic functions. II. Proc. Am. Math. Soc. 14, 521–524 (1963)Google Scholar
  19. 19.
    R. Mendiratta, S. Nagpal, V. Ravichandran, A subclass of starlike functions associated with left-half of the lemniscate of Bernoulli. Int. J. Math. 25(9), 1450090 (2014), 17 ppMathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    R. Mendiratta, S. Nagpal, V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function. Bull. Malays. Math. Sci. Soc. 38(1), 365–386 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    S. Owa, H.M. Srivastava, Some generalized convolution properties associated with certain subclasses of analytic functions. JIPAM J. Inequal. Pure Appl. Math. 3(3), Article 42 (2002), 13 pp. (electronic)Google Scholar
  22. 22.
    E. Paprocki, J. Sokół, The extremal problems in some subclass of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. 20, 89–94 (1996)MathSciNetzbMATHGoogle Scholar
  23. 23.
    B.N. Rahmanov, On the theory of univalent functions. Dokl. Akad. Nauk SSSR (N.S.) 82, 341–344 (1952)Google Scholar
  24. 24.
    R.K. Raina, J. Sokół, Some properties related to a certain class of starlike functions. C. R. Math. Acad. Sci. Paris 353(11), 973–978 (2015). MR3419845MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    V. Ravichandran, S. Verma, Bound for the fifth coefficient of certain starlike functions. C. R. Math. Acad. Sci. Paris 353(6), 505–510 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    V. Ravichandran, F. Rønning, T.N. Shanmugam, Radius of convexity and radius of starlikeness for some classes of analytic functions. Complex Var. Theory Appl. 33(1–4), 265–280 (1997)MathSciNetzbMATHGoogle Scholar
  27. 27.
    M.S. Robertson, Certain classes of starlike functions. Mich. Math. J. 32(2), 135–140 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    G.M. Shah, On the univalence of some analytic functions. Pac. J. Math. 43, 239–250 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    K. Sharma, N.K. Jain, V. Ravichandran, Starlike functions associated with a cardioid. Afr. Mat. 27(5–6), 923–939 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    J. Sokół, On some subclass of strongly starlike functions. Demonstr. Math. 31(1), 81–86 (1998)MathSciNetzbMATHGoogle Scholar
  31. 31.
    J. Sokół, On sufficient condition to be in a certain subclass of starlike functions defined by subordination. Appl. Math. Comput. 190(1), 237–241 (2007)MathSciNetzbMATHGoogle Scholar
  32. 32.
    J. Sokół, On an application of certain sufficient condition for starlikeness. J. Math. Appl. 30, 131–135 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    J. Sokół, Coefficient estimates in a class of strongly starlike functions. Kyungpook Math. J. 49(2), 349–353 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    J. Sokół, Radius problems in the class \({\cal{SL}}^*\). Appl. Math. Comput. 214(2), 569–573 (2009)Google Scholar
  35. 35.
    J. Sokół, J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions. Zesz. Nauk. Politech. Rzeszowskiej Mat. 19, 101–105 (1996)MathSciNetzbMATHGoogle Scholar
  36. 36.
    B.A. Uralegaddi, M.D. Ganigi, S.M. Sarangi, Univalent functions with positive coefficients. Tamkang J. Math. 25(3), 225–230 (1994)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Mathematics, Miranda HouseUniversity of DelhiDelhiIndia

Personalised recommendations