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

@inproceedings{cCakmak2021NewSO, title={New subclass of the class of close-to-convex harmonic mappings defined by a third-order differential inequality}, author={Serkan cCakmak and Elif Yacsar and Sibel Yalccin}, year={2021} }

In this paper, we introduce a new subclass of harmonic functions f = s+ t in the open unit disk U = {z ∈ C : |z| < 1} satisfying Re [ γs′(z) + δzs′′(z) + ( δ−γ 2 ) z2s′′′ (z)− λ ] > ∣∣γt′(z) + δzt′′(z) + ( δ−γ 2 ) z2t′′′ (z) ∣∣ , where 0 ≤ λ < γ ≤ δ, z ∈ 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… Expand

#### References

SHOWING 1-10 OF 24 REFERENCES

New subclasses of the class of close-to-convex functions

- Physics
- 1977

In this paper we introduce new subclasses of the class of closeto-convex functions. We call a regular function Az) an alpha-close-to-convex function if (f(z)f'(z)/z) # 0 for z in E and if for some… Expand

Close-to-convexity of a class of harmonic mappings defined by a third-orderdifferential inequality

- Mathematics
- 2021

Abstract: In this paper, we consider a class of normalized harmonic functions in the unit disk satisfying a third-order differential inequality and we investigate several properties of this class… Expand

Stable geometric properties of analytic and harmonic functions

- Physics
- Mathematical Proceedings of the Cambridge Philosophical Society
- 2013

Abstract Given any sense preserving harmonic mapping f=h+ḡ in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λḡ are univalent (resp. close-to-convex, starlike, or convex) if and only… Expand

Some properties for a class of analytic functions defined by a higher-order differential inequality

- Mathematics
- TURKISH JOURNAL OF MATHEMATICS
- 2019

for some λ (λ < p!{α+(p−j)β+(p−j)(p−j−1)(β−α)/2}/(p−j)!) and j = 0, 1, ..., p , where p+1−j+2α/(β−α) > 0 or α = β = 1 . The extreme points of Bp(α, β, λ; j) are determined and various sharp… Expand

Convolution properties of a class of starlike functions

- Mathematics
- 1989

Let R denote the class of functions f(z) = z + a2z2 +* that are analytic in the unit disc E = {z: Izi K 1 } and satisfy the condition Re(f'(z) + zf"(z)) > 0, z E E. It is known that R is a subclass… Expand

A subclass of close-to-convex harmonic mappings

- Mathematics
- 2012

A subclass of complex-valued close-to-convex harmonic functions that are univalent and sense-preserving in the open unit disc is investigated. The coefficient estimates, growth results, area theorem,… Expand

Hadamard Products of Convex Harmonic Mappings

- Mathematics
- 2002

Functions f in the class $ K_H $ are convex, univalent, harmonic, and sense preserving in the unit disk. Such functions can be expressed as $ f = h + \overline {g} $ where h and g are analytic… Expand

Convolutions of planar harmonic convex mappings

- Mathematics
- 2001

Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex… Expand

Inclusion properties for a class of analytic functions defined by a second-order differential inequality

- Mathematics
- 2018

For $$\beta <1$$β<1, and $$\alpha \ge \gamma \ge 0,$$α≥γ≥0, let $$\mathcal {W}_{\beta }(\alpha ,\gamma )$$Wβ(α,γ) consist of normalized analytic functions f in the unit disk satisfying… Expand

Construction of subclasses of univalent harmonic mappings

- Mathematics, Medicine
- 2012

The notion of harmonic Alexander integral operator is introduced and the radius of convexity for certain families of harmonic functions is determined. Expand