analytic function

In the present paper, we introduce and investigate a new result connected to subclasses of normalized and univalent functions in the open unit disk. Some majorization results and geometric properties such as radii of starlikeness, convexity, pre-Schwarzian norm and coefficient estimates are obtained.

The aim of this paper is to solve a class of  auto-convolution Volterra integral equations by the well-known differential transform method. The analytic property of solution and convergence of the method under some assumptions are discussed and some illustrative examples are given to clarify the theoretical results, accuracy and performance of the proposed method.

We define a class of analytic functions, $A(H, n, m, \lambda)$, satisfying the following condition\begin{equation*}\frac{D_{\lambda}^{n+m} f(z)}{D_{\lambda}^{n} f(z)} \prec H(z, t),\end{equation*}where $\lambda \geq 0,n,m\in \mathbb{N}^{\ast }=\mathbb{N}\cup \{0\},t\in\left( \frac{1}{2},1\right] $ and for all $z\in \Omega $. In this study, firstly give estimates for coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ of functions belong to this class. Furthermore, the Fekete- Szeg\"{o} inequality was examined for the functions belonging to this class.

In the present paper, we discuss some differential subordinations and superordinations results for a subclass of analytic univalent functions in the open unit disk U using El-Deeb –Lupa's operator $\mathcal{H}^{n}_{\lambda,\tau}$. Also, we study some sandwich theorems.

In this paper, we consider some differential subordinations and superordinations results for univalent functions by using the operator $(H_{\sigma,\rho,\tau,\mu,y,n})$ Also, we introduce some sandwich theorems.

In the present paper, we introduce a new integral operator involving with Mittag-Leffler function and the Salagean operator. Further, we obtain some sufficient conditions for this integral operator belonging to certain classes of starlike functions.

This paper introduces a new subclass in the open unit disc of analytic functions. It is mainly defined by the modified q-Opoola derivative operator. A coefficient inequality is obtained, and other properties like distortion and closure theorems are derived. Moreover, extreme points of the differential operator are also given. Additionally, Hadamard products (or convolution) of functions respective to the class are also included.

In the present paper, we introduce two new subclasses of the function class P of bi-univalent functions defined in the open unit disc U. Furthermore, we find estimates on the coefficients |a2| and |a3| for functions in these new subclasses.

The purpose of the present paper is to introduce two new subclasses of the function class ∑m  of bi-univalent functions which both f  and f-1  are m-fold symmetric analytic functions. Furthermore, we obtain estimates on the initial coefficients for functions in each of these new subclasses. Also we explain the relation between our results with earlier known results.

In this paper, the authors investigate the bi-univalency of the generalized distribution series associated with quasi-subordination and remodelled $s$-sigmoid function. The early few coefficients are obtained to achieve our goal. The results obtained are new to the history of bi-univalency.

By making use of the Tremblay operator, we introduce new subclasses of analytic functions, for which we obtain some sufficient coefficients estimates, and the consequences are some subordination properties and partial sums inequalities.

In this investigation, we studied a class of Bazilevic type close-to-star functions which is defined by a generalized differential operator. The new class generalizes many known and new subclasses of close-to-star functions. Some of the investigated properties are the coefficient bounds and the Fekete-Szego functional. Our results extend some known and new ones.

In this paper, by employing  sine hyperbolic inverse functions,  we  introduced the generalized  subfamily $\mathcal{RK}_{\sinh}(\beta)$ of analytic functions defined on the open unit disk $\Delta:=\{\xi: \xi \in \mathbb{C} \text{ and } |\xi|<1 \}$ associated with the petal-shaped domain. The bounds of the first three Taylor-Maclaurin's coefficients, Fekete-Szeg\"{o} functional and the second Hankel determinants are investigated for $f\in\mathcal{RK}_{\sinh}(\beta)$. We considered Borel distribution as an application to our main results. Consequently, a number of corollaries have been made based on our results, generalizing previous studies in this direction.

In this paper, we introduce a newly defined  subclass $\mathcal{S}_{\Sigma}(\vartheta,\gamma,\eta;\varphi) $ of bi-univalent functions by using the Tremblay differential operator satisfying subordinate conditions in the unit disk. Moreover, we use the Faber polynomial expansion to derive bounds for the Fekete-Szego problem and first two \emph{Taylor-Maclaurin coefficients} $|a_2|$ and $|a_3|$ for functions of this class.

In this paper, some conditions have been improved so that the function $g(z)$ is defined as $g(z)=1+\sum_{k\ge 2}^{\infty}a_{n+k}z^{n+k}$, which is analytic in unit disk $U$, can be in more specific subclasses of the $S$ class, which is the most fundamental type of univalent function. It is analyzed some characteristics of starlike and convex functions of order $2^{-r}$.

In the present investigation, we introduce the two subclasses SαΣ(γ,ρ,λ,μ,ξ,δ) and SΣ(γ,ρ,λ,μ,ξ,δ;β) of normalized analytic bi-univalent functions defined in the open unit disk and associated with the Ruscheweyh's operator. Further, we obtain bounds for the second and third Taylor-Maclaurin coefficients of the functions belong to these subclasses. We also provide relevant connections with earlier investigations of other researchers.

Considering a function f(z) which is the extremal function for p−valently starlike of order α in the open unit disk, two new classes S∗p(m,α) and Kp(m,α) are introduced. The object of the present paper is to discuss some interesting problems of functions f(z) concerned with S∗p(m,α) and Kp(m,α).

An introduction of a new subclass of bi-univalent functions involving Sakaguchi type functions defined by (p,q)(p,q)-Derivative operators using Gegenbauer polynomials have been obtained. Further, the bounds for initial coefficients |a2||a2|, |a3||a3| and Fekete Szeg\"{o} inequality  have been estimated.

The aim of this study is to introduce the new subclasses of bi-univalent functions coupled with subordination in the mirror of (p,q)-analogue of the modified sigmoid function in the unit disc U={z∈C:|z|<1}. The first two immediate Taylor-Maclaurin coefficients for the function belonging to these newly introduced classes are obtained. The results are new and a number of corollaries are developed by varying the parameters involved.

The main purpose of this article is to introduce and investigate the subcategory $mathcal{H}_{Sigma}(n,beta;phi)$ of bi-univalent functions in the open unit disk $mathbb{U}$ related to subordination. Moreover,  estimates on coefficient $|a_n|$ for functions belong to this subcategory are given applying different technique. In addition,  smaller upper bound and more accurate estimation than the previous outcomes are obtained.