Strong Differential Sandwich Results for Bazilevic-Sakaguchi Type Functions Associated with Admissible Functions

In the present article, we define a new family for holomorphic functions (so-called Bazilevic-Sakaguchi type functions) and determinate strong differential subordination and superordination results for these new functions by investigating certain suitable classes of admissible functions. These results are applied to obtain strong differential sandwich results.

Solving one extreme problem in a class of locally single-leaf functions

Центральное место в теории конформных отображений занимает решение экстремальных задач на классах однолистных отображений. В известных классах нормированных голоморфных функций $S$ и $C$ решение «проблемы коэффициентов» связано с получением точных оценок модулей тейлоровских коэффициентов элементов классов. Аналогичные задачи ставятся для классов локально однолистных отображений. В.Г.Шеретов ввел в рассмотрение классы локально конформных отображений, генерируемых с помощью интегральных структурных формул из элементов классов $S$ и $C$. В статье решена задача о точной оценке модуля тейлоровского коэффициента в этом классе. The central place in the theory of conformal maps is occupied by the solution of extreme problems on classes of single-leaf maps. In the known classes of normalized holomorphic functions S and C, the solution of the "coefficient problem" is associated with obtaining accurate estimates of the modules of the Taylor coefficients of class elements. Similar problems are posed for classes of locally single-leaf mappings. V.G.Sheretov introduced classes of locally conformal mappings generated using integral structural formulas from elements of classes S and C. The article solves the problem of an accurate estimation of the modulus of the Taylor coefficient in this class.

The Study of the New Classes of m-Fold Symmetric bi-Univalent Functions

In this paper, we introduce three new subclasses of m-fold symmetric holomorphic functions in the open unit disk U, where the functions f and f−1 are m-fold symmetric holomorphic functions in the open unit disk. We denote these classes of functions by FSΣ,mp,q,s(d), FSΣ,mp,q,s(e) and FSΣ,mp,q,s,h,r. As the Fekete-Szegö problem for different classes of functions is a topic of great interest, we study the Fekete-Szegö functional and we obtain estimates on coefficients for the new function classes.

Application of strip domain and parabolic region on univalent holomorphic functions

In this paper, by using univalent functions connected with the strip domain,parabolic starlike and parabolic uniformly convex functions are introduced.Some relations between these classes are proved.

Ribaucour surfaces of harmonic type

We introduce the class of Ribaucour surfaces of harmonic type (in short HR-surfaces) that generalizes the Ribaucour surfaces related to a problem posed by Élie Cartan. We obtain a Weierstrass-type representation for these surfaces which depends on three holomorphic functions. As application, we classify the HR-surfaces of rotation, present examples of complete HR-surfaces of rotation with at most two isolated singularities and an example of a complete HR-surface of rotation with one catenoid type end and one planar end. Also, we present a 5-parameter family of cyclic HR-surfaces foliated by circles in non-parallel planes. Moreover, we classify the isothermic HR-surfaces with planar lines of curvature.

Holographic M5 branes in AdS7 × S4

We study classical M5 brane solutions in the probe limit in the AdS7× S4 background geometry that preserve the minimal amount of supersymmetry. These solutions describe the holography of codimension-2 defects in the 6d boundary dual $$ \mathcal{N} $$ N = (0, 2) supersymmetric gauge theories. The general solution is described in terms of holomorphic functions that satisfy a scaling condition. We show the behavior of the world-volume of a special class of BPS solutions near the AdS boundary region can be characterized by general equations, which describe it as intersections of the zeros of holomorphic functions in three complex variables with a 5-sphere.

A normal criterion concerning zero numbers

Let $$n \ge 4$$ n ≥ 4 be a positive integer, $$\mathcal {F}$$ F be a family of meromorphic functions in D and let $$a(z)(\not \equiv 0), b(z)$$ a ( z ) ( ≢ 0 ) , b ( z ) be two holomorphic functions in D. If, for any function $$f \in \mathcal { F}$$ f ∈ F , (1)$$f(z) \ne \infty $$ f ( z ) ≠ ∞ when $$a(z)=0$$ a ( z ) = 0 , (2) $$f'(z)-a(z)f^{n}(z)-b(z)$$ f ′ ( z ) - a ( z ) f n ( z ) - b ( z ) has at most one zero in D, then $$\mathcal {F}$$ F is normal in D.