CLASSES OF CLOSE-TO-STARLIKE FUNCTIONS BASED ON THE REFERENCE FUNCTION OF A GENERAL FORM

The purpose of the article is to introduce and explore a wide class of doubly close-to-starlike functions, while demonstrating a unified approach to solving a certain range of extreme problems. The article defines a reference function of a general form – a starlike function, on the basis of which classes of close-to-starlike and doubly close-to-starlike functions can be constructed. On the basis of a general support function containing three parameters and new estimates of analytical functions, a generalization of various classes of close-to-starlike and doubly close-to-starlike functions is introduced, considered in a number of articles published in recent years, including the introduced class contains a generalized class of typically real functions. The properties of the introduced class of functions are studied, for example, the growth theorem, estimates of the modulus of the logarithmic derivative of the function and the radius of starlikeness are obtained, in particular cases leading to previously known results and representing new results. All the results of the article are accurate.

1. Avkhadiev F.G., Aksent'ev L.A. The main results on sufficient conditions for an analytic function to be schlicht. Russian Mathematical Surveys, 1975, vol. 30, no. 4, pp. 1–63. https://www.mathnet.ru/links/f5a3866220d851a8713bf8b1673e35c7/rm4232_eng.pdf.

2. Reade M.O. (1955) On close-to-close univalent functions. Michigan Math. J., vol. 3, pp. 59–62.

3. MacGregor T.H. (1963) The radius of univalence of certain analytic functions. Proc. Amer. Math. Soc., vol. 14, pp. 514–520.

4. MacGregor T.H. (1963) The radius of univalence of certain analytic functions, II. Proc. Am. Math. Soc., vol. 14, pp. 521–524. http://dx.doi.org/10.1090/s0002-9939-1963-0148892-5.

5. Chichra P. (1972) On the radii of starlikeness and convexity of certain classes of regular functions. J. of the Australian Math. Soc., vol. 13, no. 2, pp. 208–218 https://doi.org/10.1017/S1446788700011290.

6. Ali R.M., Jain N.K. and Ravichandran V. (2012) On the radius constants for classes of analytic functions. arXiv:1207.4529v1. https://doi.org/10.48550/arXiv.1207.4529.

7. Khatter K., Lee S.K. and Ravichandran V. (2020) Radius of starlikeness for classes of analytic functions. arXiv preprint arXiv:2006.11744. https://doi.org/10.48550/arXiv.2006.11744.

8. El-Faqeer A.S.A., Mohd M.H., Ravichandran V. and Supramaniam S. (2020) Starlikeness of certain analytic functions. arXiv preprint arXiv:2006.11734. https://doi.org/10.48550/arXiv.2006.11734.

9. Sebastianc A. and Ravichandran V. (2021) Radius of starlikeness of certain analytic functions. Math. Slovaca, vol. 71, no. 1, pp. 83–104. https://doi.org/10.1515/ms-2017-0454.

10. Kanaga R. and Ravichandran V. (2021). Starlikeness for certain close-to-star functions. Hacet. J. Math. Stat., vol. 50, no. 2, pp. 414–432. https://doi.org/10.15672/hujms.702703.

11. Sharma M., Jain N.K. and Kumar S. (2023) Constrained radius estimates of certain analytic functions. arXiv:2305.16210v1.

12. Rogosinski W. (1932) Über positive harmonische entwicklungen und typisch-reelle potenzreihen. Math. Zeitschr., vol. 35, no. 1, pp. 93–121. https://doi.org/10.1007/BF01186552.

13. Maiyer F.F., Tastanov M.G., Utemissova A.A. and Ysmagul R.S. (2024) Exact estimates of regular functions and radii of convexity and starlikeness of some classes of starlike and close-to-starlike functions. Herald of the Kazakh-British technical university, vol. 21, no. 2, pp. 127–138 [In Russian]. https://doi.org/10.55452/1998-6688-2024-21-2-127-138.

14. Janowski J. (1973) Some extreme problems for certain families of analytic functions. Ann. Polon. Math, vol. 28, pp. 297–326. https://doi.org/10.4064/ap-28-3-297-326.

15. Anh V.V. and Tuan P.D. (1986) Extremal problems for a class of functions of positive real part and applications. Austral. Math. Soc. (Series A), vol. 41, pp. 152–164. https://doi.org/10.1017/S1446788700033577.

16. Shah G.M. (1972) On the univalence of some analytic functions. Pacific J. Math., vol. 43, no. 1, pp. 239–250. https://doi.org/10.2140/pjm.1972.43.239.

17. Ratti J.S. (1968) The radius of univalence of certain analytic functions. Math. Z., vol. 107, pp. 241–248.

18. Anh V.V. and Tuan P.D. (1977) On starlikeness and convexity of certain pacific. J. of Math., vol. 69, no. 1, pp. 1–9. https://doi.org/10.2140/PJM.1977.69.1.

19. Robertson M.S. (1985) Certain classes of starlike functions. Michigan Math. J., vol. 32, no. 2, pp. 135–140.

20. Maiyer F.F., Tastanov M.G., Utemissova A.A., Kenzhebekova D.S. and Temirbekov N.M. (2024) Estimates and radii of convexity in some classes of regular functions. WSEAS Transactions on Mathematics, E-ISSN: 2224-2880, vol. 23, Art. #18, pp. 446–457. https://doi.org/10.37394/23206.2024.23.47.

21. Gelfer S.A. (1964) Tipichno veshhestvennye funkcii [Typically real functions]. Matem. sb., vol. 64(106), no. 2, pp. 171–184 [In Russian]. https://www.mathnet.ru/rus/sm4441.

22. Libera R.J. (1964) Some radius of convexity problems. Duke Math. J., vol. 31, no. 1, pp. 143–158. https://doi.org/10.1215/S0012-7094-64-03114-X.