On classes of analytic functions related to conic domains

We introduce classes of analytic functions related to conic domains, using a new linear multiplier fractional differential operator (n∈N₀={0,1,…}, 0?α<1, λ?0), which is defined as

D⁰f(z)=f(z),     

Characteristic functions as distributional boundary values of analytic functions in tube domains

We propose necessary and sufficient conditions for a complex-valued function f on \( {{\mathbb{R}}^n} \) to be a characteristic function of a probability measure. Certain analytic extensions of f to tubular domains in \( {{\mathbb{C}}^n} \) are studied. In order to extend the class of functions under study, we also consider the case where f is a generalized function (distribution). The main result is given in terms of completely monotonic functions on convex cones in \( {{\mathbb{R}}^n} \) .     

The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered.     

Some results of analytic functions related to concave domains

The main purpose of the present paper is to derive some results for analytic functions whose takes values on concave domains. As special cases of these results, several new sufficient conditions for univalency and starlikeness of analytic functions are obtained.     

Some class of analytic functions related to conic domains

For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R _q ^λ f is defined. Applying R _q ^λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.     

On the summability of spectral expansions of piecewise smooth functions

For piecewise smooth functions of n variables, we prove the uniform Riesz summability of order s > (n ? 3)/2 of their spectral expansions associated with an arbitrary elliptic operator with constant coefficients. For s = (n ? 3)/2, the corresponding Riesz means are bounded.     

On summability of eigenfunction expansions of piecewise smooth functions

We consider two forms of eigenfunction expansions associated with an arbitrary elliptic differential operator with constant coefficients and order m, that is the multiple Fourier series and integrals. For the multiple Fourier integrals, we prove the convergence of the Riesz means of order s?>?(N???3)/2 of piecewise smooth functions of N?≥?2 variables. The same result is proved in the case of the N?≥?3 dimensional multiple Fourier series.