We investigate integral means over spherical shell of holomorphic functions in the unit ball $\mathbb {B}_n$ of $\mathbb {C}^n$ with respect to the weighted volume measures and their relation with the weighted Hadamard product. The main result of this paper has many consequences which improve some well-known estimates related to the Hadamard product in Hardy spaces and weighted Bergman spaces.

In this study, a spherical indenter mounted on an atomic force microscope (AFM) was used to compress a Nannochloropsis oculata (N. oculata) cell on a poly-l-lysine coated slide. A mathematical model of the cell, which was derived by considering a fluid-filled spherical shell with axisymmetric compression between a sphere and an infinite flat plate, is proposed. In the construction of this mathematical model, the spherical shell was assumed to be a homogenous, isotropic, and elastic material. Thin-film theory was applicable to the spherical shell because the thickness of the shell was nearly negligible compared with its diameter. The governing equations of the contact and noncontact regions were converted from a boundary condition problem to an initial value problem. Then, the fourth-order Runge–Kutta method was applied to solve the transformed governing equations. The force curve obtained from the compression experiment was compared with the theoretical results derived from the proposed model. Furthermore, the numerical solution of the proposed model was verified to be consistent with the experimental data. The mechanical properties of cell walls were confirmed by applying the least square error method. Subsequently, the contact radius, inner pressure and tension distribution of the cell wall could be determined using the proposed model. The models proposed in other studies are suitable for analyzing the compression characteristics of cells whose size is of the order of tens of micrometers and millimeters. By contrast, the model proposed in this study can analyze the compression characteristics of N. oculata, which is only a few micrometers in diameter. Furthermore, a force curve that accurately describes the deformation behavior of N. oculata under strain levels of 25% was established.

In the present paper, the problem on normal low-velocity impact of a solid upon an isotropic spherical shell is studied without considering the changes in the geometrical dimensions of the contact domain. At the moment of impact, shock waves (surfaces of strong discontinuity) are generated in the target, which then propagate along the shell during the process of impact. Behind the wave fronts up to the boundary of the contact domain, the solution is constructed with the help of the theory of discontinuities and one-term ray expansions. The ray method is used outside the contact spot, but the Laplace transform method is applied within the contact region. As a result, the exact solution of the contact force is determined as a function of time. This model is intended to be used in simulating crash scenarios in frontal impacts, and to provide an effective tool to estimate the severity of effect on the human head and to estimate brain injury risks.