Kyungpook Mathematical Journal 2021; 61(2): 269-278
Published online June 30, 2021
Copyright © Kyungpook Mathematical Journal.
Coefficient Bounds for a Subclass of Harmonic Mappings Convex in One Direction
Mohammad Mehdi Shabani, Maryam Yazdi, Saeed Hashemi Sababe*
Department of Mathematics, University of Shahrood, Shahrood, Iran
e-mail : Mohammadmehdishabani@yahoo.com
Young Researchers and Elite Club, Malard Branch, Islamic Azad University, Malard, Iran
e-mail : Msh_yazdi@yahoo.com
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, and Young Researchers and Elite Club, Malard Branch, Islamic Azad University, Malard, Iran
e-mail : Hashemi_1365@yahoo.com and S.Hashemi@ualberta.ca
Received: August 6, 2020; Revised: December 14, 2020; Accepted: December 14, 2020
In this paper, we investigate harmonic univalent functions convex in the direction
Keywords: harmonic, univalent, convex, shear construction
A continuous function
If a univalent harmonic mapping
Theorem 2.1.() A sense-preserving harmonic function
Moreover, they proved by the following theorem that Theorem 2.1 would be generalized to a convex domain in the direction θ.
Theorem 2.2.() A harmonic function
Hengartner and Schober  studied analytic functions
Actually, this method deals with existing the sequences
Theorem 2.3.() Suppose
Then we have
ψis univalent on ,
ψis normalized by (2.1).
Using this characterization of functions, Hengartner and Schober then proved the next theorem:
Theorem 2.4.() If
The upper bound is sharp for
To be able to manipulate these consistent results for the specific functions that are convex in the proper direction of
In this case, we have
Similarly, one can shows that
which shows that ψ defined in (2.4) satisfies (2.1). Therefore
Therefore we can apply Theorem 2.4 to
3. Growth and Distortion Theorems
In this section we study the class
Theorem 3.1. Let
Equality is obtained for both upper bounds when
and for the lower bounds when
Using Theorem 2.4 gives inequality (3.1). Similarly,
Applying Theorem 2.4 again yields (3.1).
The sharpness of the functions comes from examining the sharpness of the functions for Theorem 2.4. Let
Figure 1. The shear of
Figure 2. The shear of
Theorem 3.2. Let
Corollary 3.3. In Thorem 3.2, if
It is easy to check that if
Sheil-Small  proved that if
In Theorems 3.1 and 3.2, we described how the geometry of the related analytic function
Theorem 3.4.() If
Equality is obtained in all three inequalities by
Again, suppose that
Therefore, we can apply Theorem 2.4 to
Theorem 3.5. Let
Similarly, we have
The authors thanks to the referee, especially for his comment on Corollary 3.3. A part of this research was carried out while the third author was visiting the university of Alberta. The author is grateful to his colleagues in the department of mathematical and statistical siecnces for their kind hosting.
- P. Ahuja, Use of theory of conformal mappings in harmonic univalent mappings with directional convexity, Bull. Malays. Math. Sci. Soc, 35(2)(2012), 775-784.
- J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 9(1984), 3-25.
- W. Hengartner and G. Schober, On schlicht mappings to domains convex in one direction, Comment. Math. Helv., 45(1970), 303-314.
- R. Hernández and O. Venegas, Distortion Theorems Associated with Schwarzian Derivative for Harmonic Mappings, Complex Anal. Oper. Theory, 13(2019), 1783-1793.
- X. Huang, Harmonic quasiconformal homeomorphisms of the unit disk, Chinese Ann. Math. Ser. A, 29(2008), 519-524.
- D. Kalaj and M. Pavlović, Boundary correspondence under quasiconformal harmonic diffeomorphisms of a half-plane, Ann. Acad. Sci. Fenn. Math., 30(2005), 159-165.
- Z. Nehari, Conformal mappings, Dover Publications, New York, 1975.
- M. Pavlović, Boundary correspondence under harmonic quasiconformal homeomorphisms of the unit disk, Ann. Acad. Sci. Fenn. Math., 27(2002), 365-372.
- M. M. Shabani and S. Hashemi Sababe, On some classes of spiral-like functions deﬁned by the Salagean operator, Korean J. Math., 28(2020), 137-147.
- M. M. Shabani, M. Yazdi and S. Hashemi Sababe, Some distortion theorems for new subclass of harmonic univalent functions, Honam Math. J., 42(4)(2020), 701-717.
- T. Sheil-Small, Constants for planar harmonic mappings, J. London Math. Soc., 42(2)(1990), 237-248.
- X. Zhang, J. Lu and X. Li, Growth and distortion theorems for almost starlike mappings of complex order λ, Acta Math Sci. Ser. B, 28(3)(2018), 769-777.