The aim of the present paper is to find the fourth Hankel and Toeplitz determinants for bounded turning functions subordinate to $1+tanhz$. We also consider this problem for Fekete-Szego inequality and Zalcman Conjectures for bounded turning functions subordinate to $1+tanhz$.

In the past several subclasses of starlike functions are defined involving real part and modulus of certain expressions of functions under study, combined by way of an inequality. In the similar fashion, we introduce a new subclass $\mathcal{S}^*(\phi)$ by considering a specific function in place of $\phi$, which is the solution of a differential subordination obtained by reformulating a differential inequality. We also study the class defined by means of a differential inequality and establish the relation between this class and $\mathcal{S}^*(\phi)$. We obtain certain inclusion and radius results for both the classes. Further, we estimate logarithmic coefficients, inverse coefficients and Fekete-Szeg\"o functional bounds for functions in $ \mathcal{S}^*(\phi)$.

The aim of this note is to introduce a topic about some ways to interpolate values of a function, on a finite set of points in the disk. Each interpolating function is built by means of a certain distance between points. We study the behaviour of such interpolations.

In this paper, we study the geometry of the proper curve and proper helix of order 2 lying on the hyperbolic plane $H_{0}^2(-r)$, globally from Minkowski space $E_{1}^3$. We develop the Frenet frame (orthogonal frame) along the proper curve of order 2 using connection $\tilde{\nabla}$ on $E_ {1}^3$ and connection $\nabla$ on $H_ {0} ^ 2(-r)$. The Frenet frame for the proper curve and proper helix of order 2 depends on the curvature of the proper curve and proper helix of order 2 in the hyperbolic plane $ H_ {0} ^ 2(-r)$. Finally, we find the condition for a proper curve of order 2 with non constant curvature to become a $V_{k} -$slant helix in $E_{1}^3$.

In this paper, we obtain confluence formulas, series representations, integral representations and differentiation formulas for Appell-type extension of the $_pR_q(\alpha,\beta;z)$ function.

In this paper we define asymptotically lacunary statistical equivalence set sequences of order $(\alpha,\beta)$ and studied some properties of this concept. We also make an effort to define strongly asymptotically lacunary $p$-equivalence and strongly Cesaro asymptotically $p$-equivalence of order $(\alpha,\beta)$ and examine some topological properties.

The objective of this paper is to present the existence of positive periodic solutions of nonlinear neutral dynamic equations on periodic time scales by applying the Krasnoselskii fixed point theorem. Our findings combine the well-known discrete and continuous cases

In this paper, we consider the efficient numerical approach for the time-fractional convection-diffusion equation with Caputo-Hadamard derivative. This method uses the nonuniform L2-1$_\sigma$ formula for the time-fractional derivative and the local discontinuous Galerkin (LDG) method for the space approximation. In order to analyze the stability and convergence of the algorithm, a new discrete Gronwall inequality related to the discretized model with Caputo-Hadamard derivative is established. The result shows that the method has $\alpha$-robust, i.e., it remains valid when $\alpha\rightarrow 1^-$. Finally, the theoretical results are further verified by a numerical example.

In this paper, we consider a class of first-order differential inclusion, the right-hand side of the problem is governed by the so-called nonconvex state-dependent sweeping process. The perturbation, known as the external forces applied on the system, is general and takes the form of a sum of a single-valued mapping depends on velocity and a set-valued mapping depends on state. We study the existence of solutions and the compactness of the attainable set, where the set-valued mapping is an upper semi-continuous with convex values. Then, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity. The results obtained are applied to the study of a control problem.