Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls γ[Bn(H)], γ>0, and their boundary behavior. These functions have the form

     

A spectral radius formula for the Fourier transform on compact groups and applications to random walks

Let G be a compact group, not necessarily abelian, let ? be its unitary dual, and for f∈L¹(G), let fn?f∗?∗f denote n-fold convolution of f with itself and f? the Fourier transform of f. In this paper, we derive the following spectral radius formula

     

Symmetry and specializability in the continued fraction expansions of some infinite products

Let f(x)∈Z[x]. Set f₀(x)=x and, for n?1, define fn(x)=f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product

     

On deviations and strong asymptotic functions of meromorphic functions of finite lower order

Let f be a transcendental meromorphic function of finite lower order with N(r,f)=S(r,f), and let qν be distinct rational functions, 1?ν?k. For 0<γ<∞ put

     

Global branch from the second eigenvalue for a semilinear Neumann problem in a ball

Let (n?3) be a ball, and let f∈C³. We are concerned with the Neumann problem

     

Regularity results for a class of obstacle problems under nonstandard growth conditions

We prove regularity results for minimizers of functionals in the class , where is a fixed function and f is quasiconvex and fulfills a growth condition of the type

L⁻¹|z|^p(x)?f(x,ξ,z)?L(1+|z|^p(x)),     

On the stability of limit cycles for planar differential systems

We consider a planar differential system , , where P and Q are C¹ functions in some open set U⊆R², and . Let γ be a periodic orbit of the system in U. Let f(x,y):U⊆R²→R be a C¹ function such that

     

A singular Sturm-Liouville equation under homogeneous boundary conditions

Given α>0 and f∈L²(0,1), we are interested in the equation

     

Doubly close-to-convex functions

We introduce the class L(β,γ) of holomorphic, locally univalent functions in the unit disk , which we call the class of doubly close-to-convex functions. This notion unifies the earlier known extensions. The class L(β,γ) appears to be linear invariant. First of all we determine the region of variability for fixed z, |z|=r<1, which give us the exact rotation theorem. The rotation theorem and linear invariance allows us to find the sharp value for the radius of close-to-convexity and bound for the radius of univalence. Moreover, it was helpful as well in finding the sharp region for , for which the integral , f∈L(β,γ), is univalent. Because L(β,γ) reduces to β-close-to-convex functions (γ=0) and to convex functions (β=0 and γ=0), the obtained results generalize several corresponding ones for these classes. We improve as well the value of the radius of univalence for the class considered by Hengartner and Schober (Proc. Amer. Math. Soc. 28 (1971) 519-524) from 0.345 to 0.577.