New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality

Abstract

In this paper, we introduce a new subclass of harmonic functions $f=s+\overline{t}$ in the open unit disk $U=\{z\in C:\left|z\right|<1\}$ satisfying

${\text{Re}}\left[ \gamma \mathfrak{s}^{\prime }(z)+\delta z\mathfrak{s}^{\prime \prime }(z)+\left( \frac{\delta -\gamma }{2}\right) z^{2}\mathfrak{s}^{\prime \prime \prime }\left( z\right) -\lambda \right]>\left \vert \gamma \mathfrak{t}^{\prime }(z)+\delta z\mathfrak{t}^{\prime\prime }(z)+\left( \frac{\delta -\gamma }{2}\right) z^{2}\mathfrak{t}^{\prime \prime \prime }\left( z\right) \right \vert,$

where $0\le \lambda <\gamma \le \delta ,z\in U.$ We determine several properties of this class such as close-to-convexity, coefficient bounds, and growth estimates. We also prove that this class is closed under convex combination and convolution of its members. Furthermore, we investigate the properties of fully starlikeness and fully convexity of the class.

Keywords

• harmonic
• univalent
• close-to-convex
• coefficient estimates
• convolution

References

1. [1] R.M. Ali, M.M. Nargesi and V. Ravichandran, Convexity of integral transforms and duality, Complex Var. Elliptic Equ. 58 (11), 1569–1590, 2013.
2. [2] R.M. Ali, D. Satwanti and A. Swaminathan, Inclusion properties for a class of analytic functions defined by a second-order differential inequality, RACSAM, 112, 117–133, 2018.
3. [3] P.N. Chichra, New subclasses of the class of close-to-convex functions, Proc. Am. Math. Soc. 62 (1), 37-43, 1976.
4. [4] M. Chuaqui, P. Duren and B. Osgood, Curvature properties of planar harmonic map- pings, Comput. Methods Funct. Theory, 4 (1), 127-142, 2004.
5. [5] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I 9, 3-25, 1984.
6. [6] M. Dorff, Convolutions of planar harmonic convex mappings, Complex Var. Theory Appl., 45 (3), 263–271, 2001.
7. [7] P. Duren, Univalent Functions, in: Grundlehren Der Mathematischen Wis- senschaften, vol. 259, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.
8. [8] P. Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, 156, Cambridge Univ. Press, Cambridge, 2004.

9. [9] L. Fejér, Über die Positivitat von Summen, die nach trigonometrischen oder Legen- dreschen Funktionen fortschreiten, Acta Litt. Ac Sei. Szeged, 75-86, 1925.
10. [10] N. Ghosh and A. Vasudevarao, On a subclass of harmonic close-to-convex mappings, Monatsh. Math., 188, 247-267, 2019.
11. [11] N. Ghosh and A. Vasudevarao, The radii of fully starlikeness and fully convexity of a harmonic operator, Monatsh Math., 188, 653-666, 2019.
12. [12] M. Goodloe, Hadamard products of convex harmonic mappings, Complex Var. Theory Appl., 47 (2), 81–92, 2002.
13. [13] R. Herandez and M.J. Martin, Stable geometric properties of analytic and harmonic functions, Math. Proc. Cambridge Philos. Soc. 155, 343–359, 2013.
14. [14] S.S. Miller and P.T. Mocanu, Differential subordinations and univalent functions, Michigan Math. J. 28 (2), 157–171, 1981.
15. [15] S.S. Miller and P.T Mocanu, Differential Subordinations, Theory and Applications, Marcel Dekker, New York, Basel, 1999.
16. [16] S. Nagpal and V. Ravichandran, Fully starlike and fully convex harmonic mappings of order , Ann. Polon. Math. 108 (1), 85-107, 2013.
17. [17] S. Nagpal and V. Ravichandran, Construction of subclasses of univalent harmonic mappings, J. Korean Math. Soc., 53, 567–592, 2014.
18. [18] Rajbala, J.K. Prajapat, On a subclass of close-to-convex harmonic mappings, Asian- European Jour Math., 14 (06), 2150102, 2021.
19. [19] O. Al-Refai, Some properties for a class of analytic functions defined by a higher-order differential inequality, Turkish J. Math., 43, 2473-2493, 2019.
20. [20] R.M. Ali, S.K. Lee, K.G. Subramanian and A. Swaminathan, A third order differential equation and starlikeness of a double integral operator, Abst. Appl. Anal., Article ID 901235, 2011.
21. [21] H, Silverman, Harmonic univalent functions with negative coefficients, Jour. Math. Anal. Appl., 220, 283-289, 1998.
22. [22] R. Singh and S. Singh, Convolution properties of a class of starlike functions, Proc. Amer. Math. Soc., 106, 145-152, 1989.
23. [23] E. Yaşar and S. Yalçın, Close-to-convexity of a class of harmonic mappings defined by a third-order differential inequality, Turkish J. Math., 45 (2), 678-694, 2021.