In this paper, the differential subordination \( \frac{b}{\phi(z)}+ c~ \phi(z) + d~ \frac{z \phi'(z)}{\phi^{k}(z)} \prec s(z), k\geq 1, z\in\mathbb{E}\) is studied by using Lowner Chain. The corresponding result for differential superordination is also obtained to get sandwich type result. Consequently, we obtain sufficient conditions for Starlikeness and Convexity of analytic function \(f\).

The purpose of this paper is to apply the concept of \(log -\)series in order to determine the sum of certain power series, where the n-th terms involves the factorial mapping, the generalized harmonic numbers and the reciprocals of factorial sums.

This paper investigates the geometry and norm-attainability of operators within various operator ideals, with a particular focus on the role of singular values and compactness. We explore the behavior of norm-attainable operators in the context of classical operator ideals, such as trace-class and Hilbert-Schmidt operators, and examine how their geometric and algebraic properties are influenced by membership in these ideals. A key result of this study is the connection between the singular values of trace-class operators and their operator norm, establishing a foundational relationship for understanding norm-attainment. Additionally, we explore the conditions under which weakly compact and compact operators can attain their operator norm, providing further insights into the structural properties that govern norm-attainability in operator theory. The findings contribute to a deeper understanding of the interplay between operator ideals and norm-attainability, with potential applications in functional analysis and related fields.

The primary objective of this paper is to introduce a novel integral transform, referred to as the Hartley-Bessel-Stockwell transform, and to establish several fundamental results associated with it. Specifically, we derive generalized versions of Parseval’s identity, Plancherel’s theorem, the inversion formula, and Calderon’s reproducing formula for this transform. Furthermore, we investigate the concentration properties of the Hartley-Bessel-Stockwell transform on sets of finite measure and present an uncertainty principle for orthonormal sequences. Finally, leveraging the theory of reproducing kernels and best approximation methods, we examine the extremal functions associated with this transform. We provide their integral representations and derive optimal estimates for these functions within weighted Sobolev spaces.

In this research, we utilize the Opoola differential operator to define new subclasses of starlike and convex functions within the unit disk \(U\): \(S^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\), \(K^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\), \(T^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\), and \(C^{m,t}_{\beta,\mu}(\alpha,\eta,\gamma)\), characterized by parameters \(\alpha\), \(\eta\), and \(\gamma\), which denote their order and type. We investigate various geometric properties of these functions, including characterization properties, growth and distortion theorems, arithmetic mean, and radius of convexity. The results obtained generalize many existing findings, forming a foundation for further research in the theory of geometric functions. Additionally, we present several corollaries and remarks to illustrate extensions of our results.

This paper proposes a new creative modification to the well-known standard Adomian decomposition method (ADM) in order to investigate various types of initial-value problems (IVPs) involving distinct kinds of fourth order ordinary differential equations (ODEs). We demonstrate that the singular point at \(x=0,\) therefore the form factor, could show up in several equations terms. Some non-linear numerical applications that have been studied and explained this method have confirmed its effectiveness and ability to find appropriate solutions for such equations. The outcomes we arrive at with this operator are reliable and converge faster than the exact solution.