Simple criteria for univalence and coefficient bounds for a certain subclass of analytic functions

Abstract

In the first part of this work we present several new geometric properties of analytic functions by applying the differential subordination. In addition, several results in the geometric functions theory pointed out. In the second part we find upper bounds for coefficients of functions in class $\mathcal{B}_\Sigma^{q,\mu}(\beta,\lambda,h)$ which is defined by fractional $q$-calculus operators.

Keywords

• Univalent function,differential subordination,starlike functions,close-to-convex functions,bi-univalent functions,fractional $q$-calculus operators,Faber polynomials

References

1. Airault, H. and Bouali, A. Differential calculus on the Faber polynomials, Bull. Sci. Math., 130 (2006), 179-222.
2. Airault, H. and Ren, J. An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math., 126 (2002), 343-367.
3. Alamoush, A. G. and Darus, M. On coefficient estimates for bi-univalent functions of Fox-Wright functions, Far East J. Math. Sci. (FJMS), 89 (2014), 249-262.
4. Algahtani, O. Estimates of initial coefficients for certain subclasses of bi-univalent functions involving quasi-subordination, J. Nonlinear Sci. Appl., 10 (2017), 1004-1011.
5. Aouf, M. K. and Bulboacă, T. Subordination and superordination properties of multivalent functions defined by certain integral operator, J. Franklin Institute, 347 (3) (2010), 641-653.
6. Aouf, M. K. and Dziok, J., Distortion and convolution theorems for operators of generalized fractional calculus involving Wright function, J. Appl. Anal., 14 (2008), 183-192.
7. Bukhari, S. Z. H., Bulboacă, T., Shabbir, M. S., Subordination and superordination results for analytic functions with respect to symmetrical points, Quaest. Math., 41 (1) (2018) 65-79.
8. Bulboacă, T. On some classes of differential subordinations, Studia Univ. Babeş-Bolyai Math., 31 (1986), 45-50.

9. Bulboacă, T. On some classes of differential subordinations (II), Studia Univ. Babeş-Bolyai Math., 31 (1986), 13-21.
10. Bulut, S. Coefficient estimates for initial Taylor-Maclaurin coefficients for a subclass of analytic and bi-univalent functions defined by Al-Oboudi differential operator, Sci. World J., (2013), Art. ID 171039, 6 pages.
11. Duren, P. L. Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, New York, (1983).
12. El-Ashwah, R. M., Aouf, M. K., Bulboacă, T. Differential subordinations for classes of meromorphic p-valent functions defined by multiplier transformations, Bull. Aust. Math. Soc., 83 (2011), 353-368.
13. Kanas, S., Tudor, A. E. Differential subordinations and harmonic means, Bull. Malays. Math. Sci. Soc. (2), 38 (2015), 1243-1253.
14. Keogh, F. R., Merkes, E. P. A coeffcient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., 20 (1969), 8-12.
15. Kumar, S., Ravichandran, V. Subordinations for functions with positive real part, Complex Anal. Oper. Theory, (2017), 1-15. DOI:10.1007/s11785-017-0690-4
16. Hamidi, S. G., Jahangiri, J. M. Faber polynomial coefficient estimates for analytic bi-close-toconvex functions, C. R. Math. Acad. Sci. Paris, 352 (2014), 17-20.
17. Hamidi, S. G., Jahangiri, J. M. Faber polynomial coefficients of bi-subordinate functions, C. R. Math. Acad. Sci. Paris, 354 (2016), 365-370.
18. Miller, S. S., Mocanu, P. T. Second-order differential inequalities in the complex plane, J. Math. Anal. Appl., 65 (1978), 298-305.
19. Nunokawa, M., Sokół, J. On multivalent starlike functions and Ozaki condition, Complex Var. Elliptic Equ., (2018), 1-15, DOI:10.1080/17476933.2017.1419209
20. Nunokawa, M., Sokół, J. On some differential subordinations, Studia Sci. Math. Hungar., 54 (2017), 436-445.
21. Murugusundaramoorthy, G. Coefficient estimates of bi-Bazilević functions defined by Srivastava-Attiya operator, Matematiche (Catania), 69 (2014), 43-56.
22. Murugusundaramoorthy, G., Janani, T., Purohit, S. D. Coefficient estimate of bi-Bazilević functions associated with fractional q-calculus operators, Fund. Inform., 151 (2017), 49-62.
23. Orhan, H., Magesh, N., Balaji, V. K. Initial coeffcient Bounds for a general class of bi-univalent functions, Filomat, 29 (2015), 1259-1267.
24. Owa, S., Srivastava, H. M. Univalent and starlike generalized hypergeometric functions, Canad. J. Math., 39 (1987), 1057-1077.
25. Purohit, S. D., Raina, R. K. Certain subclasses of analytic functions associated with fractional q-calculus operators, Math. Scand., 10 (2011), 55-70.
26. Srivatava, H. M., Aouf, M. K. Some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients, Comput. Math. Appl., 30 (1) (1995), 53-61.
27. Srivastava, H. M., Gaboury, S., Ghanim, F. Coefficient estimates for a general subclass of analytic and bi-univalent functions of the Ma-Minda type, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., (2017), DOI:10.1007/s13398-017-0416-5
28. Srivastava, H. M., Murugusundaramoorthy, G., Vijaya, K. Coefficient estimates for some families of bi-Bazilević functions of the Ma-Minda type involvig the Hohlov operator, J. Classical Anal., 2 (2013), 167-181.
29. Todorov, P. G., On the Faber polynomials of the univalent functions of class Σ, J. Math. Anal. Appl., 162 (1991), 268-276.
30. Zireh, A., Adegani, E. A., Bulut, S. Faber polynomial coefficient estimates for a comprehensive subclass of analytic bi-univalent functions defined by subordination, Bull. Belg. Math. Soc. Simon Stevin, 23 (2016), 487-504.