Making use of generalized Dziok-Srivastava operator we introduced a new class of complexvalued harmonic functions which are orientation preserving, univalent and starlike in the unit disc. We investigate the coefficient bounds, distortion inequalities, extreme points and inclusion results for the generalized class of functions.

We obtain the conditions for convergence of operator series.

Let (η n (ω)) be a sequence of independent random variables such that η n (ω) takes the values −1 and 1 with the probabilities p n and 1−p n , respectively. Put q n =min{p n ,1−p n } . Then, for each complex sequence (a n ) such that lim ¯ n→∞ |a n | − − − √ n =1 , the circle {z∈C:|z|=1} is the natural boundary for the function f ω (z)=∑ ∞ n=0 a n η n (ω)z n almost surely if and only if the condition ∑ ∞ k=0 q n k =+∞ holds for every increasing sequence (n k ) of nonnegative integers such that lim − − − k→∞ n k k <+∞

It is proved that the differential equation z n w (n) +(a (n−1) 1 z+a (n−1) 2 )z n−1 w (n−1) +∑ k=0 n−2 (a (k) n−1−k z 2 +a (k) n−k z+a (k) n+1−k )z k w (k) =0 has an entire solution f with a two-member recurrent formula for its Taylor's coefficients. The growth of such function f is studied. The conditions for coefficients a (j) k are obtained, under which the solution f is convex or close-to-convex in D={z:|z|<1} .

Let L be the class of positive continuous functions on (−∞,+∞) and let L 2 + be the class of positive continuous increasing with respect to each variable functions γ in R 2 such that γ(r 1 ,r 2 )→+∞ as r 1 +r 2 →+∞. We prove the following} statement: for all entire functions of the form f(z 1 ,z 2 )=∑ +∞ n+m=0 a nm z n 1 z m 2 such that |a nm |≤exp{−(n+m)ψ(n,m)} for n+m≥k 0 (f) and functions f(z 1 ,1),f(1,z 2 ) are transcendent, ψ∈L 2 + , the inequality M f (r 1 ,r 2 )=O(M f (r 1 ,r 2 )h(lnM f (r 1 ,r 2 ))), h∈L, r ∨ =min{r 1 ,r 2 }→+∞, holds where M f (r 1 ,r 2 )=max{|f(z 1 ,z 2 )|:|z 1 |=r 1 ,|z 2 |=r 2 }, M f (r 1 ,r 2 )=∑ +∞ n+m=0 |a nm |× ×r n 1 r m 2 , if and only if (∀γ∈L 2 + ): r 1 r 2 − − − − √ =O(h(γ(r 1 ,r 2 )ψ(r 1 ,r 2 ))), r ∨ →+∞.

A subset A of a group G is called a kaleidoscopical configuration if there exists a surjective coloring χ:X→κ such that the restriction χ|gA is a bijection for each g∈G . We give two topological constructions of kaleidoscopical configurations and show that each infinite subset of an Abelian group contains an infinite kaleidoscopical configuration.

We present a refinement of the recent Borodin's example of a finite set without a Steiner point. Namely, we show that under a suitable renorming such an example exists in every nonreflexive Banach space.

In the paper we study a combined differential-difference method for solving nonlinear equations with non-differentiable operator. The semilocal convergence of the method is investigated and the order of convergence is established.

The author studies an R G -module A such that R is an integral domain, G is a locally soluble group of infinite section p -rank (or infinite 0-rank), C G (A)=1 , A/C A (G) is not a noetherian R -module, and for every proper subgroup H of infinite section p -rank (or infinite 0-rank respectively), the quotient module A/C A (H) is a noetherian R -module. It is proved that under the above conditions, G is a soluble group. Some properties of soluble groups of this type are obtained.

We introduce a concept of an analytic curve of bounded l-index and investigate possible growth of such curves. Moreover, l-index boundedness of analytic curves satisfying linear differential equations is investigated.

In the paper the role of initial object is played by the positively definite operator L0 acting in a Hilbert space H. The main object of the investigation is operator L~ B: It is interpreted as a perturbation of some proper extension of L0. Using methods of the extension theory the criteria of maximal accretivity and maximal nonnegativity for L~ B are established. In the case when L~ B is a positively definite operator, its energetic space is constructed and the solvability of corresponding variational problem is proved. Moreover, the situation when L0 is minimal differential operator generated in the space of infinite-dimensional vector-functions by the Sturm-Liouville differential expression is considered.