A sharp estimate of the discrepancy of a concatenation sequence of inversive pseudorandom numbers with consecutive primes

We consider an equidistributed concatenation sequence of pseudorandom rational numbers generated from the primes by an inversive congruential method. In particular, we determine the sharp convergence rate for the star discrepancy of said sequence. Our arguments are based on well-known discrepancy estimates for inversive congruential pseudorandom numbers together with asymptotic formulae involving prime numbers.

Geometric convergence results for closed minimal surfaces via bubbling analysis

We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus $$\gamma $$ γ is sequentially compact for any $$\gamma \ge 1$$ γ ≥ 1 . Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity $$m\ge 1$$ m ≥ 1 , away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process.

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$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

On the Boundary Dieudonné–Pick Lemma

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

On Subharmonic and Entire Functions of Small Order: After Kjellberg

We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.

Extreme problems in the space of meromorphic functions of finite order in the half plane. II

The extremal problems in the space of meromorphic functions of order $\rho>0$ in upper half-plane are studed.The method for studying is based on the theory of Fourier coefficients of meromorphic functions. The concept of just meromorphic function of order $\rho>0$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, sharp estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained.