Related papers: The Real Schwarz Lemma: The Sequel

For analytic functions f(z) in the open unit disk U with f(0)=f'(0)-1=0, S. S. Miller and P. T. Mocanu (Integral Transform. Spec. Funct. 19(2008)) have considered some sufficient problems for starlikeness. The object of the present paper is…

In this paper a finite dimensional unital associative algebra is presented, and its group of algebra automorphisms is detailed. The studied algebra can physically be understood as the creation operator algebra in a formal quantum field…

Let function $f$ be analytic in the unit disk ${\mathbb D}$ and be normalized so that $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper we give sharp bounds of the modulus of its second, third and fourth coefficient, if $f$ satisfies \[…

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is…

Suppose that $k$ and $N$ are positive integers. Let $f$ be a newform on $\Gamma_0(N)$ of weight $k$ with $L$-function $L_f(s)$. Previous works have studied the zeros of the period polynomial $r_f(z)$, which is a generating function for the…

One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $\alpha$-valuations of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide a…

For $0\neq x>-1$ let $$\Delta(x)={{\ln \Gamma(x+1)} \over x}.$$ Recently Adell and Alzer proved the complete monotonicity of $\Delta'$ on $(-1,\infty)$ by giving an integral representation of $(-1)^n \Delta^{(n+1)}(x)$ in terms of the…

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…

The main purpose of this paper is to establish a Schwarz lemma for the solutions to the Dirichlet problems for the invariant Laplacians. The obtained result of this paper is a generalization of the corresponding known results [11, Theorem…

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on $\Gamma$, which is part of the boundary of $\Omega$. Sakai in 1991 gave a complete characterization of the boundary of a…

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…

We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstneri\v c and Kalaj (F.~Forstneri{\v{c}} and D.~Kalaj. \newblock…

Let (X,d) be a metric space and $ \alpha > 0 $. In this paper, we study extensions of some complex-valued Lipschitz functions, from some special subset $ X_0 $ to X. These extensions are with no-increasing Lipschitz number or the smallest…

In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li…

This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a…

We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the existence of a discontinuous homomorphism of the additive group of real numbers induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$.…

In this paper we consider certain 1-parametric family of Dirichlet series. For a particular value of the parameter the series turns into the Dirichlet series for the Riemann zeta function. We prove that almost every series of the family has…

This self-contained paper is part of a series \cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1) Any homomorphism of (almost any) mapping class group or…

A type of fractional derivative, referred to as \alpha-derivative, is studied. The \alpha-derivative of fractional type obeys Leibnitz rule. Based on the definition of \alpha-derivative the operations of analysis and differential geometry…

We prove a density theorem for the auxiliar function $\mathop{\mathcal R}(s)$ found by Siegel in Riemann papers. Let $\alpha$ be a real number with $\frac12< \alpha\le 1$, and let $N(\alpha,T)$ be the number of zeros $\rho=\beta+i\gamma$ of…