SUFFICIENT CONDITIONS FOR STARLIKENESS

Title & Authors
SUFFICIENT CONDITIONS FOR STARLIKENESS
RAVICHANDRAN, V.; SHARMA, KANIKA;

Abstract
We obtain the conditions on $\small{{\beta}}$ so that $\small{1+{\beta}zp^{\prime}(z){\prec}1+4z/3+2z^2/3}$ implies p(z) $\small{{\prec}}$ (2+z)/(2-z), $\small{1+(1-{\alpha})z}$, $\small{(1+(1-2{\alpha})z)/(1-z)}$, ($\small{0{\leq}{\alpha}}$<1), exp(z) or $\small{{\sqrt{1+z}}}$. Similar results are obtained by considering the expressions $\small{1+{\beta}zp^{\prime}(z)/p(z)}$, $\small{1+{\beta}zp^{\prime}(z)/p^2(z)}$ and $\small{p(z)+{\beta}zp^{\prime}(z)/p(z)}$. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition $\small{{\mid}log(zf^{\prime}(z)/f(z)){\mid}}$ < 1 or $\small{{\mid}(zf^{\prime}(z)/f(z))^2-1{\mid}}$ < 1 or zf`(z)/f(z) lying in the region bounded by the cardioid $(9x^2+9y^2-18x+5)^2-16(9x^2+9y^2-6x+1) Keywords convex and starlike functions;lemniscate of Bernoulli;subordination;cardioid; Language English Cited by 1. Subordinations for Functions with Positive Real Part, Complex Analysis and Operator Theory, 2017 References 1. R. M. Ali, N. E. Cho, N. Jain, and V. Ravichandran, Radii of starlikeness and convexity for functions with fixed second coefficient defined by subordination, Filomat 26 (2012), no. 3, 553-561. 2. R. M. Ali, N. E. Cho, V. Ravichandran, and S. Sivaprasad Kumar, Differential subordination for functions associated with the lemniscate of Bernoulli, Taiwanese J. Math. 16 (2012), no. 3, 1017-1026. 3. 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), no. 11, 6557-6565. 4. W. Janowski, Extremal problems for a family of functions with positive real part and for some related families, Ann. Polon. Math. 23 (1970/1971), 159-177. 5. W. Janowski, Some extremal problems for certain families of analytic functions. I, Ann. Polon. Math. 28 (1973), 297-326. 6. 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), 157-169, Conf. Proc. Lecture Notes Anal., I Int. Press, Cambridge, MA, 1992. 7. R. Mendiratta, S. Nagpal, and V. Ravichandran, On a subclass of strongly starlike functions associated with exponential function, Bull. Malays. Math. Sci. Soc. (2) 38 (2015), no. 1, 365-386. 8. S. S. Miller and P. T. Mocanu, Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, 225, Dekker, New York, 2000. 9. E. Paprocki and J. Sokol, The extremal problems in some subclass of strongly starlike functions, Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 20 (1996), 89-94. 10. Y. Polatoglu and M. Bolcal, Some radius problem for certain families of analytic functions, Turkish J. Math. 24 (2000), no. 4, 401-412. 11. V. Ravichandran, F. Ronning, and T. N. Shanmugam, Radius of convexity and radius of starlikeness for some classes of analytic functions, Complex Var. Theory Appl. 33 (1997), no. 1-4, 265-280. 12. M. S. Robertson, Certain classes of starlike functions, Michigan Math. J. 32 (1985), no. 2, 135-140. 13. T. N. Shanmugam and V. Ravichandran, Certain properties of uniformly convex functions, in Computational Methods and Function Theory 1994 (Penang), 319-324, Ser. Approx. Decompos., 5, World Sci. Publ., River Edge, NJ, 1994. 14. K. Sharma, N. K. Jain, and V. Ravichandran, Starlike functions associated with a cardioid, submitted. 15. S. Sivaprasad Kumar, V. Kumar, V. Ravichandran, and N. E. Cho, Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli, J. Inequal. Appl. 2013 (2013) Art. 176, 13pp. 16. J. Sokol, Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J. 49 (2009), no. 2, 349-353. 17. J. Sokol, Radius problems in the class$S_L^*\$, Appl. Math. Comput. 214 (2009), no. 2, 569-573.

18.
J. Sokol and J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike functions, Fol. Sci. Univ. Tech. Res. 147 (1996), 101-105.