Subordination of functions in subclass of Bazilevič Functions B 1(α, β)

Let f be analytic in the unit disc D = {z : |z| < 1} with f ( z ) = z + ∑ n = 2 ∞ a n z n , and for α ≥ 0 and 0 < β ≤ 1, let B 1(α, ß), denote for the class of Bazilevič functions satisfying the expression | arg z 1 − α f ′ ( z ) f ( z ) 1 − α | < β π 2 . We give sharp estimates for various coefficient problems for functions in B 1(α, β), which unify and extend well-known results for starlike functions, strongly starlike functions and functions whose derivative has positive real part in domain D.

On the number of semismooth integers

Let [Formula: see text] be a fixed positive number. Define [Formula: see text] as the number of positive integers [Formula: see text] having no prime factors [Formula: see text], and define [Formula: see text] as the number of positive integers [Formula: see text] having [Formula: see text] prime factors [Formula: see text], with all the other prime factors [Formula: see text]. In this paper, we give an asymptotic estimate for the ratio [Formula: see text], provided that [Formula: see text], [Formula: see text], and [Formula: see text] as [Formula: see text]. Also, combining this estimate with conventional ones for [Formula: see text], we provide sharp estimates for [Formula: see text].

Sharp estimates of approximation of classes of differentiable functions by entire functions

In the paper, we find the sharp estimate of the best approximation, by entire functions of exponential type not greater than $\sigma$, for functions $f(x)$ from the class $W^r H^{\omega}$ such that $\lim\limits_{x \rightarrow -\infty} f(x) = \lim\limits_{x \rightarrow \infty} f(x) = 0$,$$A_{\sigma}(W^r H^{\omega}_0)_C = \frac{1}{\sigma^{r+1}} \int\limits_0^{\pi} \Phi_{\pi, r}(t)\omega'(t/\sigma)dt$$for $\sigma > 0$, $r = 1, 2, 3, \ldots$ and concave modulus of continuity.Also, we calculate the supremum$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

Hermitian Toeplitz determinants for the class S of univalent functions

Introducing a new method, we give sharp estimates of the Hermitian Toeplitz determinants of third order for the class S of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class S.

Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

We consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$ X = { - 1 , + 1 } Λ where $$\varLambda \subset {\mathbb {Z}}^2$$ Λ ⊂ Z 2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$$ { - 1 ̲ , c ̲ o , c ̲ e } to the stable state $$\underline{+1}$$ + 1 ̲ . Here $${\underline{c}}^o,{\underline{c}}^e$$ c ̲ o , c ̲ e denote the odd and the even chessboard respectively. To do this, we identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability.

Sharp asymptotics in a fractional Sturm-Liouville problem

The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.