Open Journal of Mathematical Analysis (OMA)

This paper shows how a family of function spaces, coined as Assiamoua spaces, plays a fundamental role in the Fourier analysis of vector-valued functions on compact groups. These spaces make it possible to transcribe the classical results of Fourier analysis in the framework of analysis of vector-valued functions and vector measures. The construction of Sobolev spaces of vector-valued functions on compact groups rests heavily on the members of the aforementioned family.

This work introduces a unique family of bi-univalent functions utilising \(q\)-Gegenbauer polynomials. The estimates of the initial coefficients \(\left\vert a_{2}\right\vert\) and \(\left\vert a_{2}\right\vert\) for functions in this new class, together with the Fekete-Szegö functional, have been obtained. Subsequent to the specialisation of the parameters utilised in our principal findings, many novel outcomes are presented.

In this paper, we establish some new characterizations of a weight \(w\) such that discrete Hardy operator \(\mathcal{H}f(n):=\frac{1}{n}\sum\limits_{s=1}^{n}f(s)\) for quasi-nonincreasing sequence \(f(n)\) is bounded in the Banach space \(\ell _{w}^{p}(\mathbb{Z}_{+})\) when \(0<p<\infty .\) In particular, we will prove that \(\mathcal{H}f\) is bounded in \(\ell _{w}^{p}(\mathbb{Z}_{+})\) if and only if \(w\) belongs to the \(\beta\)-discrete Arino and Muckenhoupt class \(\mathcal{B}_{p,\beta }\). We prove that the self-improving property for the class \(\mathcal{B}_{p,\beta }\) holds, that is we prove that if \(w\in \mathcal{B}_{p,\beta }\) then there exists an \(\varepsilon >0\) such that \(w\in \mathcal{B}_{p-\varepsilon ,\beta }.\)

In this article, we establish new integral inequalities involving sub-multiplicative functions. We first derive several inequalities of primitive type, followed by new inequalities of the convolution product type. We also obtain integral bounds for functions evaluated on the product of two variables. Finally, we study double integral inequalities and their variations. Simple examples are used to illustrate the theory. The understanding of integral inequalities under submultiplicative assumptions is thus deepened, and some new ideas for further research in mathematical analysis are provided.

The current study focuses on the investigation and develop of a new approach called Hussein–Jassim method (HJM), suggested lately by Hassan et al.; specifically, we investigate its applicability to fractional ordinary delay differential equations in the Caputo fractional sense. Several examples are offered to demonstrate the method’s reliability. The results of this study demonstrate that the proposed method is highly effective and convenient for solving fractional delay differential equations.

We define and study the Stockwell transform \(\mathscr{S}_g\) associated with the Whittaker operator
\[\Delta_{\alpha}:=-\frac{1}{4}\left[x^2\frac{\mbox{d}^2}{\mbox{d}x^2}+(x^{-1}+(3-4\alpha)x)\frac{\mbox{d}}{\mbox{d}x}\right],\]
and prove a Plancherel theorem. Moreover, we define the localization operators \(L_{g,\xi}\) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we give a Shapiro-type uncertainty inequality for the modified Whittaker-Stockwell transform \(\mathscr{S}_g\).

Consider a unit disk \(\Omega=\{z:|z|<1\}\). A large subset of the set of analytic-univalent functions defined in \(\Omega\) is examined in this exploration. This new set contains various subsets of the Yamaguchi and starlike functions, both of which have profound properties in the well-known set of Bazilevič functions. The Ma-Minda function and a few mathematical concepts, including subordination, set theory, infinite series formation and product combination of certain geometric expressions, are used in the definition of the new set. The estimates for the coefficient bounds, the Fekete-Szegö functional with real and complex parameters, and the Hankel determinants with a real parameter are some of the accomplishments. In general, when some parameters are changed within their interval of declarations, the set reduces to a number of recognized sets.

Let \( u’ + Au = h(u,t) + f(x,t) \) with the initial condition \( u(x,0) = u_0(x) \), where \( u \in H \), \( u’ := u_t := \frac{du}{dt} \), and \( H \) is a Hilbert space. The nonlinear term satisfies the estimate \( \|h(u,t)\| \le a\|u\|^p (1+t)^{-b} \), and the operator \( A \) satisfies the coercivity condition \( (Au,u) \ge \gamma(t)(u,u) \), where \( \gamma(t) = q_0(1+t)^{-q} \). Here, \( a, p, b, q_0, \) and \( q \) are positive constants. Sufficient conditions are established under which the solution exists and is either bounded or tends to zero as \( t \to \infty \).

In this paper, we derive summation formulae for the generalized Legendre-Gould Hopper polynomials (gLeGHP) \({}_SH^{(m)}_n(x,y,z,w)\) and \(\frac{{}_RH^{(m)}_n(x,y,z,w)}{n!}\) by using different analytical means on their respective generating functions. Further, we derive the summation formulae for polynomials related to \({}_SH^{(m)}_n(x,y,z,w)\) and \(\frac{{}_RH^{(m)}_n(x,y,z,w)}{n!}\) as applications of main results. Some concluding remarks are also given.

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 \(f\).