A set of closed unit disks in the Euclidean plane is said to be double-saturated packing if no two disks have inner points
in common and any closed unit disk intersects at least two disks of the set. We prove that the density of a double saturated
packing of unit disks is ≥
, what is the length of the shortest path connecting them that stays within the doubly covered region? This is a problem of G. Fejes-Tóth and he conjectured that if the distance between A and B is d, then the length of this path is at most
Получен ряд теорем об аналитичности функц ии, непрерывной в единич ном круге и имеющей нулев ые интегралы по некот орым окружностям. Ранее по добные результаты были изве стны лишь на всей комп лексной плоскости. Решена задача Фаркаша: доказ ано, что непрерывная в единичном круге функция, имеюща я нулевые интегралы по всем окружностям, кас ающимся границы этого круга, является аналитичес кой.
Analytic functions with bounded Mocanu variation generalize the concept of α-convexity in the unit disc. We introduce and
study certain classes of such functions. Inclusion results are obtained and a sharp radius problem is solved.
In this study, a normalized form of regular Coulomb wave function is considered. By using the differential subordinations method due to Miller and Mocanu, we determine some conditions on the parameters such that the normalized regular Coulomb wave function is lemniscate starlike and exponential starlike in the open unit disk, respectively. In additon, by using the relationship between the regular Coulomb wave function and the Bessel function of the first kind we give some conditions for which the classical Bessel function of the first kind is lemniscate and exponential starlike in the unit disk 𝔻.
In this paper, we introduce a new concept of q-bounded radius rotation and define the class R*m(q), m ≥ 2, q ∈ (0, 1). The class R*2(q) coincides with S*q which consists of q-starlike functions defined in the open unit disc. Distortion theorems, coefficient result and radius problem are studied. Relevant connections to various known results are pointed out.
In this study, we investigate approximation properties and obtain Voronovskaja type results for complex modified Szász-Mirakjan operators. Also, we estimate the exact orders of approximation in compact disks and prove that the complex modified Szász-Mirakjan operators attached to an analytic function preserve the univalence, starlikeness, convexity and spirallikeness in the unit disk.
Let ℌ be a Hilbert space and F(ℌ) be the full Fock space generated by ℌ. For v ∈ ℌ, the (left) creation operator l(v) : F(ℌ)
→ F(ℌ),f↦ v ⊗ f, and its adjoint, the (left) annihilation operator l(v)*, are defined. If v, w ∈ℌ are orthonormal, it is proved that the spectrum of the operator l(v) + l(w)* is purely continuous and conincides with the closed unit disk.