Hankel Determinant Problem for q-strongly Close-to-Convex Functions

In this paper, we introduce a new class $K_{q}(\alpha), \quad 0<\alpha \leq1, \quad 0<q<1, $ of normalized analytic functions $f $ such that $\big|\arg\frac{D_qf(z)}{D_qg(z)}\big| \leq \alpha \frac{\pi}{2},$ where $g$ is convex univalent in $E= \{z: |z|<1\} $ and $D_qf $ is the $q$-derivative of $f $ defined as: $$D_qf(z)= \frac{f(z)-f(qz)}{(1-q)z}, \quad z\neq0\quad D_qf(0)= f^{\prime}(0). $$ The problem of growth of the Hankel determinant $H_n(k) $ for the class $K_q(\alpha) $ is investigated. Some known interesting results are pointed out as applications of the main results.

On a Subclass of Analytic Functions That Are Starlike with Respect to a Boundary Point Involving Exponential Function

In the present exploration, the authors define and inspect a new class of functions that are regular in the unit disc D ≔ ς ∈ ℂ : ς < 1 , by using an adapted version of the interesting analytic formula offered by Robertson (unexploited) for starlike functions with respect to a boundary point by subordinating to an exponential function. Examples of some new subclasses are presented. Initial coefficient estimates are specified, and the familiar Fekete-Szegö inequality is obtained. Differential subordinations concerning these newly demarcated subclasses are also established.

Some effective solutions for Prandtl's type integro-differential equation

Prandtl’s type integro-differential equations with a different coefficient are investigated. Using the methods of the theory of analytic functions and integral transformations, the singular integro-differential equation is reduced to boundary value problems of the theory of analytic functions. Effective solutions of this equation and the corresponding asymptotic estimates are obtained

Third-Order Differential Subordination Results for Analytic Functions Associated with a Certain Differential Operator

In this research, we study suitable classes of admissible functions and establish the properties of third-order differential subordination by making use a certain differential operator of analytic functions in U and have the normalized Taylor–Maclaurin series of the form: f(z)=z+∑n=2∞anzn, (z∈U). Some new results on differential subordination with some corollaries are obtained. These properties and results are symmetry to the properties of the differential superordination to form the sandwich theorems.

Certain Properties of a Class of Functions Defined by Means of a Generalized Differential Operator

In this article, we construct a new subclass of analytic functions involving a generalized differential operator and investigate certain properties including the radius of starlikeness, closure properties and integral means result for the class of analytic functions with negative coefficients. Further, the relationship between the results and some known results in literature are also established.