We obtain sharp bounds for the second Hankel determinant of logarithmic inverse coefficients for starlike and convex functions.
We obtain sharp bounds for the second Hankel determinant of logarithmic inverse coefficients for starlike and convex functions.
Let 
$[a_1(x),a_2(x),a_3(x),ldots ]$ be the continued fraction expansion of an irrational number 
$xin [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all 
$xin [0,1)$, 
$$ begin{align*} liminf_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n} = 0quad text{and}quad limsup_{n to infty} frac{log (a_n(x)a_{n+1}(x))}{log n}=1. end{align*} $$
We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.
Liu [‘Supercongruences for truncated Appell series’, Colloq. Math. 158(2) (2019), 255–263] and Lin and Liu [‘Congruences for the truncated Appell series 
$F_3$ and 
$F_4$’, Integral Transforms Spec. Funct. 31(1) (2020), 10–17] confirmed four supercongruences for truncated Appell series. Motivated by their work, we give a new supercongruence for the truncated Appell series 
$F_{1}$, together with two generalisations of this supercongruence, by establishing its q-analogues.
Let 
${mathbb {Z}}_{K}$ denote the ring of algebraic integers of an algebraic number field 
$K = {mathbb Q}(theta )$, where 
$theta $ is a root of a monic irreducible polynomial 
$f(x) = x^n + a(bx+c)^m in {mathbb {Z}}[x]$, 
$1leq m<n$. We say 
$f(x)$ is monogenic if 
${1, theta , ldots , theta ^{n-1}}$ is a basis for 
${mathbb {Z}}_K$. We give necessary and sufficient conditions involving only 
$a, b, c, m, n$ for 
让 ${mathbb {Z}}_{K}$ 表示代数数域 $K = {mathbb Q}(theta )$ 的代数整数环，其中 $theta $ 是在 {mathbb {Z}}[x]$, $1leq m<n$ 中的一元不可约多项式 $f(x) = x^n + a(bx+c)^m 的根。如果 ${1, theta , ldots , theta ^{n-1}}$ 是 ${mathbb {Z}}_K$ 的基，我们就说 $f(x)$ 是单源的。我们给出了只涉及 $a，b，c，m，n$ 的 $f(x)$ 单调性的必要条件和充分条件。此外，我们还描述了 ${mathbb {Z}}[theta ]$ 在 ${mathbb {Z}}_K$ 中划分子群 ${mathbb {Z}}[theta ]$ 索引的所有素数的特征。作为应用，我们还提供了一类具有非无平方判别式和伽罗瓦群 $S_n$（n 个字母上的对称群）的单元多项式。
We study the problem of determining the holomorphic self maps of the unit disc that induce a bounded composition operator on Dirichlet-type spaces. We find a class of symbols 
$varphi $ that induce a bounded composition operator on the Dirichlet-type spaces, by applying results of the multidimensional theory of composition operators for the weighted Bergman spaces of the bi-disc.
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