Some sharp bounds of the third-order Hankel determinant for the inverses of the Ozaki type close-to-convex functions
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Cited by 4 publications
References 27 publications
Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function
The aim of this paper is to introduce two new subclasses $\mathcal{R}_{\sin }^{m}(\Im )$ R sin m ( ℑ ) and $\mathcal{R}_{\sin }(\Im )$ R sin ( ℑ ) of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function $\Im (v)=\frac{2}{1+e^{-v}}$ ℑ ( v ) = 2 1 + e − v , $v\geq 0$ v ≥ 0 in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.
Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function
The aim of this paper is to introduce two new subclasses $\mathcal{R}_{\sin }^{m}(\Im )$ R sin m ( ℑ ) and $\mathcal{R}_{\sin }(\Im )$ R sin ( ℑ ) of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function $\Im (v)=\frac{2}{1+e^{-v}}$ ℑ ( v ) = 2 1 + e − v , $v\geq 0$ v ≥ 0 in the open unit disc E. Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.
Upper bound for the second and third Hankel determinants of analytic functions associated with the error function and q-convolution combination
Recently, El-Deeb and Cotîrlă (Mathematics 11:11234834, 2023) used the error function together with a q-convolution to introduce a new operator. By means of this operator the following class $\mathcal{R}_{\alpha ,\Upsilon}^{\lambda ,q}(\delta ,\eta )$ R α , ϒ λ , q ( δ , η ) of analytic functions was studied: $$\begin{aligned} &\mathcal{R}_{\alpha ,\Upsilon }^{\lambda ,q}(\delta ,\eta ) \\ &\quad := \biggl\{ \mathcal{ F}: {\Re} \biggl( (1-\delta +2\eta ) \frac{\mathcal{H}_{\Upsilon }^{\lambda ,q}\mathcal{F}(\zeta )}{\zeta}+(\delta -2\eta ) \bigl(\mathcal{H} _{\Upsilon}^{\lambda ,q}\mathcal{F}(\zeta ) \bigr) ^{{ \prime}}+\eta \zeta \bigl( \mathcal{H}_{\Upsilon}^{\lambda ,q} \mathcal{F}( \zeta ) \bigr) ^{{{\prime \prime}}} \biggr) \biggr\} \\ &\quad >\alpha \quad (0\leqq \alpha < 1). \end{aligned}$$ R α , ϒ λ , q ( δ , η ) : = { F : ℜ ( ( 1 − δ + 2 η ) H ϒ λ , q F ( ζ ) ζ + ( δ − 2 η ) ( H ϒ λ , q F ( ζ ) ) ′ + η ζ ( H ϒ λ , q F ( ζ ) ) ″ ) } > α ( 0 ≦ α < 1 ) . For these general analytic functions $\mathcal{F}\in \mathcal{R}_{\beta ,\Upsilon}^{\lambda ,q}(\delta , \eta )$ F ∈ R β , ϒ λ , q ( δ , η ) , we give upper bounds for the Fekete–Szegö functional and for the second and third Hankel determinants.
On Ozaki Close-to-Convex Functions with Bounded Boundary Rotation
In the present investigation, we introduce a new subclass of univalent functions F(u,λ) and a subclass of bi-univalent function Fo,Σ(u,λ) with bounded boundary and bounded radius rotation. Some examples of the functions belonging to the classes F(u,λ) are also derived. For these new classes, the authors derive many interesting relations between these classes and the existing familiar subclasses in the literature. Furthermore, the authors establish new coefficient estimates for these classes. Apart from the above, the first two initial coefficient bounds for the class Fo,Σ(u,λ) are established.
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